![the stimulus but on account of fatigue the latent period was prolonged, from the normal 5 seconds to as
many minutes.
SUMMARY.
Plants not only perceive, but also respond to long ether waves employed in signalling through space.
Mechanical response to wireless stimulation is exhibited by the leaf of Mimosa pudica.
All plants give electric response to the stimulus of long ether waves.
Growing plants exhibit response to electric waves by modification of rate of growth. Feeble stimulus
induces an acceleration, while strong stimulus causes a retardation of the rate of growth.
The perceptive range of the plant is far greater than ours; it not only perceives but responds to the
different rays of the vast ethereal spectrum.
[26] Pfeffer—Vol. II, p. 104.
[27] Plant Response—p. 618 (1905).
[28] Comparative Electro-Physiology—p. 149.](https://image.slidesharecdn.com/17570788-250606120645-7312a02c/85/Modelling-Hydrology-Hydraulics-And-Contaminant-Transport-Systems-In-Python-1st-Edition-Kuiry-27-320.jpg)
![XXXIX.—GEOTROPISM
By
Sir J. C. Bose.
No phenomenon of tropic movement appears so inexplicable as that of geotropism. There are two
diametrically opposite effects induced by the same stimulus of gravity, in the root a movement
downwards, and in the shoot a movement upwards. The seeming impossibility of explaining effects so
divergent by the fundamental reaction of stimulus, has led to the assumption that the irritability of stem
and root are of opposite character. I shall, however, be able to show that this assumption is
unnecessary.
The difficulty of relating geotropic curvature to a definite reaction to stimulus is accentuated by the
fact that the direction of the incident stimulus, and the side which responds effectively to it are not
clearly understood; nor is it known, whether the reaction to this stimulus is a contraction, or its very
opposite, an expansion.
Taking the simple case of a horizontally laid shoot, the geotropic up-curvature is evidently due to
differential effect of the stimulus on upper and lower sides of the organ. The up-curvature may be
explained by one or the other of two suppositions: (1) that the stimulus of gravity induces contraction of
the upper side; or (2) that the fundamental reaction is not a contraction but an expansion and this of
the lower side. The second of these two assumptions has found a more general acceptance.
Tropic curvatures in general are brought about by the differential effect of stimulus on two sides of
the organ. Thus light falling on one side of a shoot induces local contraction, the rays being cut off from
acting on the further side by the opacity of the intervening tissue. But there is no opaque screen to cut
off the vertical lines of gravity,[29] which enter the upper side of a horizontally laid shoot and leave it by
the lower side. Though lines of force of gravity are transmitted without hindrance, yet a differential
action is found to take place, for the upper side, where the lines of force enter, becomes concave, while
the lower side where they emerge becomes convex. Why should there be this difference?
For the removal of various obscurities connected with geotropism it is therefore necessary to
elucidate the following:
1. The sign of excitation is, as we found, a contraction and concomitant galvanometric negativity.
Does gravitational stimulus, like stimulus in general, induce this excitatory reaction?
2. What is the effective direction of geotropic stimulus? In the case of light, we are able to trace the
rays of light which is incident on the proximal side and measure the angle of inclination. In the case of
gravity, the invisible lines of force enter by one side of the organ and leave by the other side. Assuming
that the direction of stimulus is coincident with the vertical lines of gravity, is it the upper or the lower
side of the organ that undergoes effective stimulation?
3. What is the law relating to the 'directive angle' and the resulting geotropic curvature? By the
directive angle (sometimes referred to as the angle of inclination) is meant, as previously explained, the
angle which direction of stimulus makes with the responding surface.
4. We have finally to investigate, whether the assumption of opposite irritabilities of the root and the
shoot is at all justifiable. If not, we have to find the true explanation of the opposite curvatures
exhibited by the two types of organs.](https://image.slidesharecdn.com/17570788-250606120645-7312a02c/85/Modelling-Hydrology-Hydraulics-And-Contaminant-Transport-Systems-In-Python-1st-Edition-Kuiry-28-320.jpg)
![side is one of galvanometric positivity, which indicates an increase of turgor and expansion. The tropic
effect of geotropic stimulus is thus similar to that of any other mode of stimulation, i.e., a contraction of
the upper (which in the present case is the proximal) and expansion of the lower or the distal side. The
vertical lines of gravity impinge on the upper side of the organ which undergoes effective stimulation.
In order to show that the concavity of the upper side is not due to the passive yielding to the
expansion of the lower half, I restrained the organ from any movement. I have explained that excitatory
electric response is manifested even in the absence of mechanical expression of excitation; and under
geotropic stimulus, the securely held shoot gave the response of galvanometric negativity of the upper
side. Hence the fundamental reaction under geotropic stimulus is excitatory contraction as under other
modes of stimulation.
Finally, I employed the additional test of induced paralysis by application of intense cold. Excitatory
physiological reaction is, as we know, abolished temporarily by the action of excessive cold.
Experiment 163.—I obtained records of mechanical response to determine the side which undergoes
excitation under geotropic stimulus, the method of discrimination being local paralysis induced by cold. I
took the flower-scapes of Amarayllis and of Uriclis, and holding them vertical applied fragments of ice
on one of the two sides. I then laid the scape horizontal, first with cooled side below, the record showed
that this did not affect the geotropic movement. But on cooling the upper side, the geotropic movement
became arrested, and it was not till the plant had assumed the temperature of the surroundings that
the geotropic movement became renewed. Figure 158 shows the effect of alternate application of cold,
on the upper and lower sides of the organ.[30] In the shoot, therefore, it is the upper side of the organ
that becomes effectively stimulated. Before proceeding further I shall make brief reference to the highly
suggestive statolithic theory of gravi-perception.
Fig. 158.—Effect of alternate application of cold on the upper and lower sides of
the organ. Application of cold on upper side (down-pointing arrow) induces arrest of
geotropic movement. Application below (up-pointing arrow) causes no arrest.
THEORY OF STATOLITHS.
With regard to the perception of geotropic stimulus there can be no doubt that this must be due to
the effect of weight of cell contents, whether of the sap itself, or of the heavy particles contained in the
cells, exerting pressure on the sensitive plasma. The theory of statoliths advocated by Noll, Haberlandt
and Nemec (in spite of certain difficulties which further work may remove) is the only rational
explanation hitherto offered for gravi-perception. The sensitive plasma is the ectoplasm of the entire
cell, and statoliths are relatively heavy bodies, such as crystals and starch grains. Haberlandt has found](https://image.slidesharecdn.com/17570788-250606120645-7312a02c/85/Modelling-Hydrology-Hydraulics-And-Contaminant-Transport-Systems-In-Python-1st-Edition-Kuiry-30-320.jpg)
![statoliths in the apo-geotropic organs like stems.[31] When the cell is laid horizontal, it is the lower
tangential wall which has to support the greater weight, and thus undergo excitation. In the case of
multicellular plants laid horizontally, the excitation on the upper side is, as we have seen, the more
effective than on the lower side. This inequality, it has been suggested, is probably due to this
difference that the statoliths on the upper side press on the inner tangential walls of the cells while
those on the lower side rest on the outer tangential walls.
When the organ is held erect, the action of statoliths would be symmetrical on the two sides. But
when it is laid horizontal a complete rearrangement of the statoliths will take place, and the differential
effects on the upper and lower sides will thus induce geotropic reaction. This period of migration must
necessarily be very short; but the reaction time, or the latent period, is found to be of considerable
duration. Even in rapidly reacting organs there is always an interval of about one to one and a half
hours, before the horizontally placed organ shows a noticeable curvature, and this latent period may in
other cases be extended to several hours.[32] This great difference between the period of migration
and the latent period offers a serious difficulty in the acceptance of the theory of statoliths. But it may
be urged that the latent period has hitherto been obtained by relatively crude methods, and I therefore
undertook a fresh determination of its value by a sensitive and accurate means of record.
DETERMINATION OF THE LATENT PERIOD.
As regards the interpretation of the record of geotropic movement, it should be borne in mind that
after the perception of stimulus a certain time must elapse before the induced growth-variation will
result in curvature. There is again another factor which causes delay in the exhibition of true geotropic
movement; for the up-movement of stems, in response to the stimulus of gravity, has to overcome the
opposite down movement, caused by weight, before it becomes at all perceptible. On account of the
bending due to weight there is a greater tension on the upper side, which as we have seen (p. 193),
enhances the rate of growth, and thus tends to make that side convex. The exhibition of geotropic
response by induced contraction of the excited upper side thus becomes greatly delayed. In these
circumstances I tried to discover specimens in which the geotropic action would be quick, and in which
the retarding effect of weight could be considerably reduced.
Geotropic response of flower stalk of Tuberose: Experiment 164.—For this I took a short length of
flower stalk of tuberose in a state of active growth; the flower head itself was cut off in order to remove
unnecessary weight. After a suitable period of rest for recovery from the shock of operation, the
specimen was placed in a horizontal position, and its record taken. The successive dots in the curve are
at intervals of 20 seconds, and the geotropic up-movement is seen to be initiated (Fig. 159) after the
tenth dot, the latent period being thus 3 minutes and 20 seconds, the greater part of which was spent
in overcoming the down-movement caused by the weight of the organ.
Fig. 159. Fig. 160.](https://image.slidesharecdn.com/17570788-250606120645-7312a02c/85/Modelling-Hydrology-Hydraulics-And-Contaminant-Transport-Systems-In-Python-1st-Edition-Kuiry-31-320.jpg)
![In geotropic curvature we may for all practical purposes regard the direction of stimulus as
coinciding with the vertical lines of gravity. The analogy between the effects of light and of gravity is
very close[33]; in both the induced curvature is such that the organ moves so as to meet the stimulus.
This will be made still more evident in the investigations on torsional geotropic response described in a
subsequent chapter. The tropic curve under geotropic stimulus is similar to that under photic stimulus.
The tropic reaction, both under the stimulus of light and of gravity, increases similarly with the 'directive'
angle. These real analogies are unfortunately obscured by the use of arbitrary terminology used in
description of the geotropic curvature of the shoot. In figure 163 records are given of the effects of
vertical light and of vertical stimulus of gravity, on the responses of the horizontally laid bud of Crinum.
In both, the upper side undergoes contraction and the movement of response carries the organ
upwards so as to place it parallel to the incident stimulus. Though the reactions are similar in the two
cases, yet the effect of light is termed positive phototropism, that of gravity negative geotropism. I
would draw the attention of plant-physiologists to the anomalous character of the existing
nomenclature. Geotropism of the shoot should, for reasons given above, be termed positive instead of
negative, and it is unfortunate that long usage has given currency to terms which are misleading, and
which certainly has the effect of obscuring analogous phenomena. Until the existing terminology is
revised, it would perhaps be advisable to distinguish the geotropism of the shoot as Zenithotropism and
of the root as Nadirotropism.
RELATION BETWEEN THE DIRECTIVE ANGLE AND GEOTROPIC REACTION.
When the main axis of the shoot is held vertical, the angle made by the surface of the organ with
lines of force of gravity is zero, and there is no geotropic effect. The geotropic reaction increases with
the directive angle; theoretically the geotropic effect should vary as the sine of the angle. I shall in the
next chapter describe the very accurate electrical method, which I have been able to devise for
determination of relative intensities of geotropic action at various angles. Under perfect conditions of
symmetry, the intensity of effect is found to vary as the sine of the directive angle. This quantitative
relation fully demonstrates that geotropic stimulus acts in a definite direction which coincides with the
vertical lines of gravity.
The conditions of perfect symmetry for study of geotropic action at various angles will be fully
described in the next chapter. In the ordinary method of experimentation with mechanical response the
organ is rotated in a vertical plane. The geotropic movement is found increased as the directive angle is
increased from zero to 90°.
DIFFERENTIAL GEOTROPIC EXCITABILITY.
It has been shown that geotropic stimulus acts more effectively on the upper side of the organ. The
intensity of geotropic reaction is, moreover, modified by the excitability of the responding tissue. It is
easy to demonstrate this by application of depressing agents on the more effective side of the organ.
The rate of geotropic up-movement will be found reduced, or even abolished by the local application of
cold, anæsthetics like chloroform, and of poisonous potassium cyanide solution.
The different sides of a dorsiventral organ are unequally excitable to different forms of stimuli. I
have already shown (p. 85) that the lower side of the pulvinus of Mimosa, is about 80 times more
excitable to electric stimulus than the upper side. Since the effect of geotropic stimulus is similar to that
of other forms of stimuli, the lower side of the pulvinus should prove to be geotropically more excitable
than the upper side. This I have been able to demonstrate by different methods of investigation which
will be described in the following chapters.
Under ordinary circumstances, the upper half of the pulvinus is, on account of its favourable
position, more effectively stimulated by geotropic stimulus; in consequence of this the leaf assume a
more or less horizontal position of dia-geotropic equilibrium. But when the plant is inverted the more
excitable lower half of the organ now occupies the favourable position for geotropic excitation. The leaf](https://image.slidesharecdn.com/17570788-250606120645-7312a02c/85/Modelling-Hydrology-Hydraulics-And-Contaminant-Transport-Systems-In-Python-1st-Edition-Kuiry-34-320.jpg)
![now erects itself till it becomes almost parallel to the stem. The response of the same pulvinus which
was formerly dia-geotropic now becomes negatively geotropic; but an identical organ cannot be
supposed to possess two different specific sensibilities. The normal horizontal position assumed by the
leaf is, therefore, due to differential geotropic excitabilities of the two sides of a dorsiventral organ.
I have explained (p. 401) that when the pulvinus of Mimosa is subjected to lateral stimulation of any
kind, it undergoes a torsion, in virtue of which the less excitable half of the organ is made to face the
stimulus. Experiments will be described in a subsequent chapter which show that geotropic stimulus
also induces similar torsional response. The results obtained from this method of enquiry give
independent proof: (1) that the lower half of the pulvinus is geotropically the more excitable, and (2)
that the direction of incident geotropic stimulus is the vertical line of gravity which impinges on the
upper surface of the organ.
SUMMARY.
The stimulus of gravity is shown to induce an excitatory reaction which is similar to that induced by
other forms of stimulation. The direct effect of geotropic stimulus is an incipient contraction and
retardation of rate of growth.
The upper side of a horizontally laid shoot is more effectively stimulated than the lower side, the
excited upper side becoming concave. Electrical investigation also shows that it is the upper side that
undergoes direct stimulation.
Tropic reactions are said to be positive, when the directly stimulated side undergoes contraction with
the result that the organ moves to meet the stimulus. According to this test, the geotropic response of
the stem is positive.
The geotropic response is delayed by the bending down of the horizontally laid shoot. Reduction of
weight is found to shorten the latent period; in the case of the petiole of Tropæolum this is shorter than
20 seconds. The latent period of geotropic response is found to be of the same order as the migration
period of the hypothetical statoliths.
The complete geotropic curve shows characteristics which are similar to tropic curves in general.
In a dorsiventral organ the geotropic excitabilities of the upper and lower sides are different. In the
pulvinus of Mimosa the geotropic excitability of the lower half is greater than that of the upper half. The
differential excitabilities of a dorsiventral organ modifies its position of geotropic equilibrium.
[29] I shall in what follows take the direction of vertical lines of gravity as that of movement of
falling bodies, from above towards the centre of the earth.
[30] Plant Response—p. 505.
[31] Haberlandt—Physiological Plant Anatomy—p. 603.
[32] Jost—Ibid, p. 437.
[33] Exception to this will be found in page 336, where explanation is offered for the difference.](https://image.slidesharecdn.com/17570788-250606120645-7312a02c/85/Modelling-Hydrology-Hydraulics-And-Contaminant-Transport-Systems-In-Python-1st-Edition-Kuiry-35-320.jpg)
![XL.—GEO-ELECTRIC RESPONSE OF SHOOT
By
Sir J. C. Bose,
Assisted by
Satyendra Chandra Guha, M.Sc.
The experiments that have been described in the preceding chapter show that the upper side of a
horizontally laid shoot undergoes excitatory contraction, in consequence of which the organ bends
upwards. The fundamental geotropic reaction is, therefore, not expansion, but contraction which results
from all modes of stimulation.
In confirmation of the above, I wished to discover and employ new means of detecting excitatory
reaction under geotropic stimulus. In regard to this, I would refer to the fact which I have fully
established that the state of excitation can be detected by the induced electromotive change of
galvanometric negativity. This electrical indication of excitation may be observed even in plants
physically restrained from exhibiting response by mechanical movement.[34]
ELECTRIC RESPONSE TO STIMULUS.
Before giving account of the results of investigations on the detection of geotropic excitation by
means of electric response, I shall describe a few typical experiments which will fully explain the
method of the electrical investigation, and show the correspondence of mechanical and electric
responses. I have explained how tropic curvatures are brought about by the joint effects, of contraction
of the directly excited proximal side A, and the expansion of the distal side B. In the diagram of
mechanical response to stimulus (Fig. 164a) the excitatory contraction is indicated by - sign, and the
expansion, by + sign. The resulting movement is, therefore, towards the stimulus as shown by the
curved arrow.
I shall now describe the corresponding electric effects in response to unilateral stimulus. We have to
determine the induced electrical variation at the proximal side A, and at the distal side B.](https://image.slidesharecdn.com/17570788-250606120645-7312a02c/85/Modelling-Hydrology-Hydraulics-And-Contaminant-Transport-Systems-In-Python-1st-Edition-Kuiry-36-320.jpg)
![Fig. 165.—Diagrammatic representation of geo-electric response. The middle figure represents vertical
position. In figure to the right rotation through +90° has placed A above with induced electric change of
galvanometric negativity of A. In the figure to the left, rotation is through -90° A being below; the electric
response is by induced galvanometric positivity of A. For simplification of diagram, vertical position of sepal
is not always shown in the figure.
GEO-ELECTRIC RESPONSE OF THE UPPER AND LOWER SIDES OF THE ORGAN.
We have next to discover the electric change induced by geotropic stimulus on the upper and lower
sides of the organ. For this purpose it is necessary to find a neutral point which is not affected by the
inclination of the organ from vertical to horizontal position. For the present experiment, I employed the
flower of the water lily Nymphæa, the peduncle of which is sensitive to geotropic action. One electrical
contact is made with a sepal, which is always kept vertical; the other electric contact is made at the
point A, on one side of the flower stalk (Fig. 165). On making connections with a sensitive galvanometer
a very feeble current was found, which was due to slight physiological difference between the neutral
point, N, and A. This natural current may be allowed to remain, the action current due to geotropism
being superposed on it; or the natural current may be neutralised by means of a potentiometer and the
reflected spot of light brought to zero of the scale.
Induced electric variation on upper side of the organ: Experiment 170.—While the sepal is held
vertical, the stalk is displaced through +90° so that the point A is above. Geotropic stimulation is at
once followed by a responsive current which flows through the galvanometer from N to A, the upper
side of the organ thus exhibiting excitatory reaction of galvanometric negativity (Right-hand figure of
166). When the stalk is brought back to vertical position geotropic stimulation disappears, and with it
the responsive current.
Electric response of the lower side: Experiment 171.—The stalk is now displaced through -90°; the
point A, which under rotation through +90° pointed upwards, is now made to point downwards. The
direction of the current of response is now found to have undergone a reversal; it now flows from A on
the lower side to the neutral point N; thus under geotropic action the lower side of the organ exhibits
galvanometric positivity indicative of increase of turgor and expansion (Left-hand figure 166).[35]
Having thus found that the upper side of the organ under geotropic stimulus becomes
galvanometrically negative, and the lower side, galvanometrically positive, we make electric connections
with two diametrically opposite points of the shoot A and B, and subject the organ to alternate rotation
through +90° and -90°. The electro-motive changes induced at the two sides now became algebraically
summated. I employ two methods for geotropic stimulation: that (1) of Axial Rotation, and (2) of
Vertical Rotation.](https://image.slidesharecdn.com/17570788-250606120645-7312a02c/85/Modelling-Hydrology-Hydraulics-And-Contaminant-Transport-Systems-In-Python-1st-Edition-Kuiry-39-320.jpg)
![The differential stimulation of A and B disappears on rotation of the axis back to zero position, and
the induced electro-motive response also disappears at the same time. If now the axis be rotated
through -90°, A will become the lower, and B the upper and the excited side. The electro-motive change
is now found to have undergone a reversal, B becoming galvanometrically negative. This induced
electro-motive variation under geotropic stimulus is of considerable intensity often exceeding 15
millivolts. The characteristic electric change is shown diagrammatically in figure 167 in which the middle
figure shows the symmetrical or zero position. On rotation through +90° (figure to the right) A occupies
the upper and B the lower position. A is seen to exhibit induced change of galvanometric negativity.
Rotation through -90° reverses the current of response, as B now occupies the upper and A the lower
position.
CHARACTERISTICS OF GEO-ELECTRIC RESPONSE.
There are certain phenomena connected with the electric response under geotropic stimulus which
appear to be highly significant. According to statolithic theory
Geotropic response begins as soon as an organ is deflected from its stable position, so that a few
starch-grains press upon the ectoplasts occupying the walls which are underneath in the new position;
an actual rearrangement of the starch-grains is therefore not an essential condition of stimulation. As a
matter of fact, the starch-grains do very soon migrate on to the physically lower walls, when a positively
or negatively geotropic organ is placed horizontally, with the result that the intensity of stimulation
gradually increases attaining its maximum value when all the falling starch-grains have moved on to the
lower region of the ectoplast. The time required for the complete rearrangement of the statoliths may
be termed the period of migration; its average length varies from five to twenty minutes in different
organs.[36]
Stimulation, according to the statolithic theory, is induced by the displacement of the particles. The
diameter of the geotropically sensitive cells is considerably less than 0·1 mm.; and the stimulus will be
perceived after the very short interval taken by the statoliths to fall through a space shorter than 0·1
mm. This may be somewhat delayed by the viscous nature of the plasma, but in any case the period for
perceptible displacement of the statoliths should be very short, about a second or so, and the latent
period of perception of stimulus should be of this order.
The mechanical indication of response to stimulus is delayed by a period which is somewhat
indefinite; for the initiation of responsive growth variation will necessarily lag behind the perception of
stimulus.](https://image.slidesharecdn.com/17570788-250606120645-7312a02c/85/Modelling-Hydrology-Hydraulics-And-Contaminant-Transport-Systems-In-Python-1st-Edition-Kuiry-41-320.jpg)
![Angle of inclination +45°+90°+45°0°-45°-90°
Galvanometer deflection-19 -35 -18 0 +14 +25
RELATION BETWEEN ANGLE OF VERTICAL ROTATION AND INTENSITY OF GEOTROPIC REACTION.
The relation between the angle of inclination and the resulting geotropic action has already been
determined by the Method of Axial Rotation. The ratio between the geotropic effects at 90° and 45°
was thus found to be 1·49, which is nearly the same as Sin 90°
⁄Sin 45°. I was next desirous of
determining the relative excitations at the two angles by the Method of Vertical Rotation. It is
necessary here to refer to certain differences of condition in the two methods. In the Axial Method,
the hypothetical statoliths are distributed uniformly through the length of the cell, and rotation round
the long axis causes displacement of the statoliths, the resulting pressure thus increasing with the
sine of the angle of inclination. But in the case of vertical rotation through 45° to the right, the
statoliths originally at the base of the cell accumulate to the right hand corner of the cell; a portion
of the basal side of the cell is thus subjected to pressure. When the angle is increased to 90° the
statoliths pass along the whole length including the basal and apical sides of the cell; but the
excitability of the apical half may prove to be greater than that of the basal half. Hence excitatory
geotropic effect is not likely to vary strictly as in sine of angle of inclination.
Whatever the reason may be, I find as a result of experiments with 12 different specimens that
the mean ratio of the effects at 90° and 45°, obtained by the Method of Vertical Rotation, is 1·8:1
which is greater than 1·49:1 obtained by the Method of Axial Rotation, this latter value being
practically the same as Sin 90°
⁄Sin 45°.
SUMMARY.
It is shown that the state of excitation under direct stimulus is exhibited by an electrical change
of galvanometric negativity; the effect of indirect stimulus induces, on the other hand, an electrical
change of galvanometric positivity. The negative electric change corresponds to contraction and
diminution of turgor; the positive electric change indicates, on the other hand, an expansion and
increase of turgor.
The electric response to geotropic stimulus is studied by the two methods of Axial and Vertical
Rotation. The upper side of a horizontally laid shoot is found to undergo an excitatory change of
galvanometric negativity.
In quick reacting organs the latent period of geo-electric response is about 5 seconds, and the
maximum excitation is induced in the course of 2 minutes.
The geo-electric response is due to physiological reaction. The intensity of response declines with
age and is abolished at the death of the plant.
Under symmetrical conditions, the intensity of geotropic reaction is found proportional to the sine
of the angle of inclination.
Electric investigation shows that the lower half of the pulvinus of Mimosa is geotropically more
excitable than the upper half.
[34] Comparative Electro-Physiology, p. 20.
[35] For detailed account cf. Chapter XLIII.
[36] Haberlandt—Ibid—p. 598.](https://image.slidesharecdn.com/17570788-250606120645-7312a02c/85/Modelling-Hydrology-Hydraulics-And-Contaminant-Transport-Systems-In-Python-1st-Edition-Kuiry-46-320.jpg)
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- 6. Modelling Hydrology, Hydraulics and Contaminant Transport Systems in Python
- 8. Modelling Hydrology, Hydraulics and Contaminant Transport Systems in Python Soumendra Nath Kuiry Dhrubajyoti Sen
- 9. First edition published 2022 by CRC Press 6000 Broken Sound Parkway NW, Suite 300, Boca Raton, FL 33487-2742 and by CRC Press 2 Park Square, Milton Park, Abingdon, Oxon, OX14 4RN © 2022 Taylor & Francis Group, LLC First edition published by CRC Press 2022 CRC Press is an imprint of Taylor & Francis Group, LLC Reasonable efforts have been made to publish reliable data and information, but the author and publisher cannot assume responsibility for the validity of all materials or the consequences of their use. The authors and publishers have attempted to trace the copyright holders of all material reproduced in this publication and apologize to copyright holders if permission to publish in this form has not been obtained. If any copyright material has not been acknowledged please write and let us know so we may rectify in any future reprint. Except as permitted under U.S. Copyright Law, no part of this book may be reprinted, reproduced, transmitted, or utilized in any form by any electronic, mechanical, or other means, now known or hereafter invented, including photocopying, microfilming, and recording, or in any information storage or retrieval system, without written permission from the publishers. For permission to photocopy or use material electronically from this work, access www.copyright.com or contact the Copyright Clearance Center, Inc. (CCC), 222 Rosewood Drive, Danvers, MA 01923, 978- 750-8400. For works that are not available on CCC please contact mpkbookspermissions@tandf.co.uk Trademark notice: Product or corporate names may be trademarks or registered trademarks and are used only for identification and explanation without intent to infringe. ISBN: 978-0-367-25578-7 (hbk) ISBN: 978-1-032-12989-1 (pbk) ISBN: 978-0-429-28857-9 (ebk) DOI: 10.1201/9780429288579 Typeset in Times by SPi Technologies India Pvt Ltd (Straive)
- 10. v Contents Preface........................................................................................................................xi About the Authors....................................................................................................xiii Chapter 1 Introduction to Modelling in Hydrology, Hydraulics, and Contaminant Transport..........................................................................1 1.1 Examples of Different Types of Models in Water Systems: Deterministic, Stochastic, Data-Based, and Others...................................................................................1 1.2 Choosing a Numerical Approach for Flow and Transport Modelling...................................................................................2 1.3 Python as the Preferred Programming Platform.........................2 1.4 Pedagogical Emphasis................................................................3 1.5 Types of Models Treated in the Book.........................................3 Chapter 2 Non-Linear and Simultaneous Equations..............................................5 2.1 Examples of Non-Linear Functions............................................5 2.1.1 Normal Depth of Flow in a Trapezoidal Channel.........5 2.1.2 Height and Velocity of a Surge Wave............................6 2.1.3 Depth of Flow in a Constricted and Raised Channel Section.............................................................7 2.2 System of Equations...................................................................8 2.2.1 System of Reactors – Steady-State Analysis.................8 2.2.2 Steady-State Distribution of Flow in Pipe Networks.....................................................................10 2.2.3 Derivation of the Unit Hydrograph.............................13 2.3 Solution Techniques.................................................................14 2.3.1 Non-Linear Equations in One Variable.......................14 2.3.2 Linear Simultaneous Equations...................................15 2.3.3 Non-Linear Simultaneous Equations..........................16 2.4 Python Programs......................................................................17 2.4.1 Non-Linear Equations in One Variable: Finding Uniform Flow Depth in a Channel..............................17 2.4.2 Non-Linear Equations in One Variable: Finding the Height and Velocity of a Surge Wave....................20 2.4.3 Non-Linear Equations in One Variable: Finding the Depth of flow above a Hump in a Contraction......23 2.4.4 Solution of Linear Simultaneous Equations: Concentrations in Interconnected Reactors.................25 2.4.5 Solution of Linear Simultaneous Equations: Derivation of the Unit Hydrograph.............................27
- 11. viContents 2.4.6 Solution of Non-Linear Simultaneous Equations: Flow Distribution in a Three-Pipe Network................30 2.4.7 Solution of Non-Linear Simultaneous Equations: Flow Distribution in a General Pipe Network.............33 References...........................................................................................36 Chapter 3 Ordinary Differential Equations..........................................................37 3.1 Examples of Ordinary Differential Equations in Hydrology, Hydraulics, and Water Resources Engineering.....37 3.1.1 Emptying of a Water Tank...........................................37 3.1.2 Computing Flood Outflow from the Spillway of a Dam by the Level-Pool Routing Method..................38 3.1.3 Water Surface Profile for Steady-State Gradually Varied Flows................................................................39 3.1.4 Steady-State Concentration Profile for Dissolved Oxygen and Biochemical Oxygen Demand in One-Dimensional Flows..............................................41 3.1.5 Oscillations of Water Level in a Surge Tank...............42 3.1.6 Recharge of Rainwater into Ground and Steady- State Groundwater-Table Profile.................................44 3.1.7 Steady-State Concentration Profile for Contaminant Injection in One-Dimensional Channel Flows.............................................................45 3.2 Solution Techniques.................................................................46 3.2.1 First-Order Ordinary Differential Equations...............47 3.2.1.1 Euler’s Method...............................................47 3.2.1.2 Fourth-Order Runge–Kutta Method...............48 3.2.1.3 Accuracy and Stability...................................49 3.2.2 Second-Order Ordinary Differential Equations..........50 3.2.3 Two-Point Boundary Value Problems.........................51 3.3 Python Programs......................................................................51 3.3.1 First-Order ODE: Solving the Tank Filling and Emptying Problem Using Heun’s Method..................52 3.3.2 First-Order ODE: Flood Routing through a Reservoir and Spillway Using Heun’s Method...........54 3.3.3 First-Order ODE: Computation of the Back- Water Gradually Varied Flow Profile Using Fourth-Order Runge–Kutta (RK4) Method.................58 3.3.4 First-Order ODE: Computing the Steady-State BOD and DO Concentration Profiles in a One- Dimensional Stream Using Heun’s Method................61 3.3.5 Second-Order ODE: Surge-Tank Oscillation Problem Solved Using Heun’s Method.......................64 3.3.6 Second-Order ODE: Steady-State Groundwater Table Profile for Recharge and Withdrawal................66
- 12. Contents vii 3.3.7 Second-Order ODE: Computing the Steady-State Concentration Profile for Point Loadings in One- Dimensional Channel Flow.........................................70 References...........................................................................................74 Chapter 4 Partial Differential Equations in Surface Hydrology, Free Surface Flows, and Ideal Fluid Flows.................................................75 4.1 Governing Equations of Free Surface Flow.............................76 4.1.1 Governing Equations of Flow in a Prismatic Channel........................................................................76 4.1.2 Ideal Fluid Flow..........................................................78 4.1.3 Governing Equations of Two-Dimensional Depth-Averaged Flows................................................78 4.2 Numerical Methods for Solving the Flow Equations...............79 4.2.1 Solving the Kinematic Wave Equation for Flow in a Prismatic Channel with Lateral Inflows...............79 4.2.2 Routing a Flood Wave by the Kinematic Wave Approximation in a Triangular Channel......................81 4.2.3 Open-Book Catchment Hydrograph with the Kinematic Wave Approximation.................................82 4.2.4 Simulation of Unsteady Flows in a Channel Using the St. Venant Equations...................................82 4.2.5 Ideal Fluid Flow Equation Solving.............................84 4.2.6 Simulation of Two-Dimensional Depth-Averaged Flows in a Shallow Basin............................................86 4.3 Python Programs......................................................................88 4.3.1 Flow in a Rectangular Channel with Lateral Inflows Solved by the Kinematic Wave Equation.......88 4.3.2 Routing a Flood Hydrograph by the Kinematic Wave Approximation in a Triangular Channel............91 4.3.3 Simulation of a Simplified Open-Book Catchment Hydrograph with the Kinematic Wave Approximation............................................................94 4.3.4 Simulation of a Surge Wave in a Trapezoidal Channel Using the St. Venant Equations.....................96 4.3.5 Simulation of Streamlines in an Ideal Fluid Flow.....102 4.3.6 Two-Dimensional Depth-Averaged Flows in a Shallow Basin............................................................104 References.........................................................................................110 Chapter 5 Partial Differential Equations in Subsurface Flows..........................111 5.1 Governing Equations of Subsurface Flows............................111 5.1.1 Governing Equations of Flow in an Unconfined Aquifer......................................................................112
- 13. viiiContents 5.1.2 Governing Equations of Flow in a Confined Aquifer......................................................................113 5.1.3 Governing Equation of Steady-State Seepage in the Vertical Plane.......................................................114 5.2 Numerical Methods for Solving the Groundwater and Seepage Flow Equations.........................................................114 5.2.1 Solving the Unsteady One-Dimensional Groundwater Flow in an Unconfined Aquifer...........114 5.2.2 Solving the Unsteady Two-Dimensional Groundwater Flow in an Unconfined Aquifer...........116 5.2.3 Steady-State Seepage below Floors and Piles...........118 5.3 Python Programs....................................................................119 5.3.1 Unsteady One-Dimensional Groundwater Flow in an Unconfined Aquifer..........................................119 5.3.2 Unsteady Two-Dimensional Groundwater Flow in an Unconfined Aquifer..........................................122 5.3.3 Steady Seepage below a Weir Floor and Sheet Pile...................................................................127 References.........................................................................................131 Chapter 6 Partial Differential Equations in Contaminant Transport..................133 6.1 Governing Equations..............................................................135 6.1.1 Governing Equations for Reaction-Diffusion, without Advection.....................................................135 6.1.2 Governing Equations for Advection and Diffusion...136 6.1.3 Governing Equations for Advection, Diffusion, and Reaction..............................................................137 6.2 Numerical Methods for Finding the Fate of a Contaminant....................................................................137 6.2.1 Solving the One-Dimensional Unsteady Reaction-Diffusion Problem......................................137 6.2.2 Solving the One-Dimensional Unsteady Advection-Diffusion Problem...................................139 6.2.3 Solving the One-Dimensional Combined Unsteady Advection, Diffusion, and Reaction Equation....................................................................139 6.2.4 Solving the Two-Dimensional Unsteady Advection and Diffusion Equation............................140 6.3 Python Programs....................................................................141 6.3.1 One-Dimensional Unsteady Reaction-Diffusion Problem.....................................................................141 6.3.2 One-Dimensional Unsteady Advection-Diffusion Problem.....................................................................143
- 14. Contents ix 6.3.3 Two-Dimensional Unsteady Advection-Diffusion Problem.....................................................................146 6.3.4 Contaminant Dispersion for Seepage below Sheet Pile and Floor..................................................148 References.........................................................................................152 Chapter 7 Simple Data-Based Models...............................................................153 7.1 Environmental Data and Motivation for Data Analysis..........154 7.1.1 Time-Series Data: Variations in Time.......................154 7.1.1.1 Hourly Record of Temperature and Humidity....................................................154 7.1.1.2 Record of Daily River Stage and Discharge...................................................155 7.1.1.3 Variation of the Rate of Infiltration in Soil with Time............................................157 7.1.2 Data Recorded in One-Dimensional Space...............159 7.1.2.1 Velocity at a Point in a Channel.................159 7.1.2.2 Elevation Versus Reservoir Capacity.........160 7.1.2.3 Variation of Sediment Concentration with Depth in a Channel............................160 7.1.3 Area of a Closed Polygon..........................................161 7.2 Solution Techniques...............................................................162 7.2.1 Interpolation..............................................................162 7.2.2 Regression.................................................................164 7.2.3 Area-Finding and Numerical integration..................166 7.3 Python Programs....................................................................168 7.3.1 Interpolation..............................................................169 7.3.2 Regression.................................................................170 7.3.3 Computation of Area.................................................172 7.3.4 Numerical Integration................................................173 References.........................................................................................176 Index����������������������������������������������������������������������������������������������������������������������177
- 16. xi Preface The need for a book on the use of numerical techniques in environmental and civil engineering was felt during the course of our teaching allied subjects at the senior undergraduate and graduate levels. Although the first- and second-year level under- graduate syllabi on engineering mathematics deal extensively with a variety of mod- elling tools and solution techniques for handling different types of scientific and engineering equations, the specific use of numerical techniques in solving problems of hydraulics, hydrology, and contaminant transport is found lacking in the gradua- tion level courses in most colleges. Nonetheless, there has been an ever-growing interest among the student commu- nity on the use of the open-source programming language Python, especially for data handling and data-based computations, thanks to the availability of a wealth of data from the environmental, especially, hydro-meteorological fields. In this book, we intend to introduce the reader to the rudiments of the application of numerical tech- niques to a variety of problems encountered in the field of water-environment through the use of coding. Preference in using the programming language Python has mostly been for the reason that it has become, at least at the time of writing this book, quite popular among students in addition to it being open source with a wealth of helpful online resources. It is possible that over the coming years, yet another programming language may become popular just as has been witnessed over the past few decades with the popularity of scientific coding shifting from FORTRAN to C/C++, and then to computing platforms like the MATLAB®. Nevertheless, it is hoped that the section of the book demonstrating the use of numerical techniques in finding solu- tions to the physical problems of hydrology, hydraulics, and transport would still remain helpful to the future reader. On the whole, it must be emphasized that this book is meant to be used more as a teaching aid for the senior undergraduate and graduate classes and for them to gain an understanding of the physical processes in the world of the water-environment through computer-based hands-on problem solving. It is hoped that on gaining an interest in the field of modelling, the keen reader would be inspired to venture for more accurate solution techniques or start using better and efficient coding platforms for advanced research projects. The book is divided into seven chapters, of which Chapter 1 gives an overview of the models and their types in the field of hydrology, hydraulics, and contami- nant transport, which have been the focus of this book. Chapter 2 elaborates on the solution of non-linear simultaneous equations, taking up examples from the field of simple hydraulic and water-engineering problems. Chapter 3 provides examples of ordinary differential equations from the field of the water-environ- ment. The next three chapters, Chapters 4, 5, and 6, discuss the application of numerical techniques to surface flows, subsurface flows, and contaminant trans- port, respectively. The final chapter (Chapter 7) serves as an exposure to the rudi- mentary quantitative analysis of different data sets, as encountered in the field of the hydro-environment. This chapter, therefore, differs from the others in not
- 17. xiiPreface dealing with physics-based models but, at the same time, exposes the reader to a variety of quantitative methods commonly used in analysing observational data. In fact, some of the techniques demonstrated in Chapter 7, though simple, may be used to process a wide body of field and laboratory data and then apply these for calibrat- ing and validating one or more of the physical models introduced in the previous chapters. Finally, the authors would like to thank the CRC Press for providing them the opportunity and encouragement to script their ideas that otherwise would probably never have seen the light of the day. Dr. Gagandeep Singh and Mr. Lakshay Gaba of the CRC Press are to be specially acknowledged for constantly monitoring the edito- rial process and bringing the project to fruition in a short time. Soumendra Nath Kuiry Dhrubajyoti Sen MATLAB® is a registered trademark of The Math Works, Inc. For product information, please contact: The Math Works, Inc. 3 Apple Hill Drive Natick, MA 01760-2098 Tel: 508-647-7000 Fax: 508-647-7001 E-mail: info@mathworks.com Web: http://www.mathworks.com
- 18. xiii About the Authors Soumendra Nath Kuiry is a faculty at the Indian Institute of Technology Madras, with expertise in developing computational techniques in the different processes of free surface flows. Specializing in flood modelling, his research interests extend over simulations of tsunami wave propagation and dam break phenomena, modelling of storm surges due to cyclones and simulation of sediment transport in rivers, estuar- ies, and coasts. Dhrubajyoti Sen is a faculty at the Indian Institute of Technology Kharagpur, with research interests in experimental and numerical modelling of surface flows, dam break incidents, storm surges, and contaminant transport. His interests in practical projects have also led him to explore the application of digital methods and elec- tronic sensing technologies in environmental sciences and hydraulic engineering.
- 20. 1 DOI: 10.1201/9780429288579-1 Introduction to Modelling in Hydrology, Hydraulics, and Contaminant Transport 1.1 EXAMPLES OF DIFFERENT TYPES OF MODELS IN WATER SYSTEMS: DETERMINISTIC, STOCHASTIC, DATA-BASED, AND OTHERS Water systems vary widely in applications and the present text is aligned mostly towards the hydraulic and hydrologic phenomena as dealt with in civil, environmen- tal, or agricultural science and engineering disciplines. The phenomena discussed include those from the surface and subsurface phases of the hydrologic cycle, and open channel flows through natural or engineered structures. From its initial touch- down till its eventual dispersal to the ocean, the rainwater collects and transmits pol- lutants, as it flows through the different pathways. The text therefore also discusses the simulation techniques for predicting the fate and transport of the water pollutants in channels, shallow water bodies, soil-seepage, and groundwater flows. The hydrologic surface and subsurface flows are also related to conventional hydraulics and may be approximated through suitable mathematical relations or equations. The different chapters of this book demonstrate the numerical techniques by which these equations may be solved on a personal computer. In the present age of “big-data”, there is an increasing trend of modelling the hydrologic and hydraulic variables through data analysis – such as by using stochastic or data-driven tech- niques. These tools are available through various platforms and help in obtaining the “bigger picture” of the environmental processes occurring around us. However, on certain smaller scales, it is still necessary to understand the physics behind a hydrau- lic phenomenon and predict its behaviour using compatible models. For example, atmospheric models and data analytics may help predict future rainfall over a catch- ment but precise delineation of the flood-inundated areas may only be possible by solving the free-surface flow equations using suitable computational methods. Similarly, computing the extent of surge that may move up a tidal river under pre- dicted future conditions of sea level rise may only be possible using numerical mod- els to solve the flow equations with the help of a computer. For predicting the movement and extent of spread of a contaminant along with the flow of water, 1
- 21. 2 Modelling Flow and Contaminant Transport whether over the earth’s surface or below, would likewise require solving the appro- priate equations numerically on a computer. 1.2 CHOOSING A NUMERICAL APPROACH FOR FLOW AND TRANSPORT MODELLING Many of the physical processes of environmental and engineered flows may broadly be described mathematically by one simple equation, a set of simultaneous equa- tions, or differential equations. In each of these types of equations, the different flow variables are interrelated in terms of the rate of change of one or several variables as a function of time and/or space variables. Under certain simplified conditions, the equation(s) may be solved analytically, and may not require a numerical solution to solve using computers. Some of such examples include the formulae for evapotrans- piration and infiltration, expressed in terms of the independent variables. However, in many practical situations, the unique geometry of the flow domain may make it dif- ficult for applying analytical techniques, as the processes can only be described by non-linear partial differential equations. As a result, only numerical solutions of the equations offer the feasible means for obtaining the desired results. Oftentimes, the variation of flow with time may also demand the application of specific numerical techniques. It is important to note that numerical solutions, though producing approx- imate results, may yield better accuracy either by refining the equations themselves or by adopting more accurate numerical techniques. Quite often, however, simple approximations may also yield reasonably acceptable results, which may help in understanding the occurrence of a particular phenomenon or aid in taking a rapid decision. The contents of this book do not venture into very accurate numerical approximations of the different hydraulic phenomena as encountered in the natural or built environment. Rather, they attempt to demonstrate the general methods that may be selected for tackling a given problem and obtaining an initial solution. Further improvement and refinement of the models are possible by expanding on the building blocks demonstrated here on the application of numerical techniques to different geophysical and engineered flows. 1.3 PYTHON AS THE PREFERRED PROGRAMMING PLATFORM For solving the numerically approximated equations, one needs to write computer programs or codes, which may be run on a suitable computational platform. The traditionally used computer language for scientific code writing since the middle of the last century had been FORTRAN, and sometimes BASIC and others. However, these were taken over by the end of the century by C/C++, and to some extent by Java. However, the last century also saw the rise of computational platforms like MATLAB, or other similar interpretable programming platforms, which became popular among researchers and still remain so. For this book, however, we have cho- sen the language Python, which appears to have been gaining popularity lately, espe- cially among the student and academic community. Since the purpose of this text is more pedagogical than a research exposition, we have also chosen Python as the preferred coding language because of its relatively easy learning curve. Python is
- 22. Models for Flow and Contaminant Transport 3 also a completely open-source computing platform and is rather easy to install and start working. An added advantage of using Python as the coding language is that it provides readily accessible functions for plotting graphs. Thus, it may be used to display and save graphical outputs of the variables which are used as inputs or those which are obtained from running a code. Codes written in FORTRAN or C/C++ first require the variables to be stored in files and then plotted graphically using another software package, such as the Paraview. Although Python may have become popular for its varied applications in non-scientific and scientific but non-numerical applica- tions, Python has also been demonstrated for use in numeric computations, as proved by the appearance of books on the subject over the past decade. Although one may use Python to write codes on the go, such as on interpretable computation platforms, in this book we shall encourage the readers to write scripts (equivalent to computer codes) and then run the codes using the Python command. This is definitely required for longer programs since the errors in the script may be corrected or modified for future reuse. Further, since there are several books and online resources guiding the installation of the Python computing platform, it is not discussed in this book. 1.4 PEDAGOGICAL EMPHASIS This book is primarily intended for the students of hydraulics, hydrology, environ- mental, and water resources engineering who may like to start writing their own codes for the problems at hand. Though nowadays many open-source software pack- ages are available for the accomplishment of different tasks in the fields described, quite often, especially when the problem is simple in terms of geometry and other physical properties, it is enough to write a simple computer program and obtain the solution. The same program may be later improved and adapted to more complicated inputs and geometries. Since the book also discusses the popular numerical schemes used for solving problems in the field of hydrology, hydraulics, and transport of con- taminants by flowing water it may be used for teaching a semester-long course on numerical methods for the students of such specializations. The codes are not written in the most optimum way and may be modified by the student in order to make it more efficient in terms of memory management or time of execution. Similarly, the student may experiment with other advanced forms of graphical outputs, including animations. 1.5 TYPES OF MODELS TREATED IN THE BOOK The models presented in this book on hydrology, hydraulics, and contaminant trans- port are demonstrated for the following physical processes: 1. Surface runoff generation by rainfall 2. Flows in one-dimensional open channels 3. Depth change in reservoirs from inflows and outflows 4. Flows in shallow lakes and water bodies 5. Flows in pipe networks 6. Shallow groundwater flow, considered two-dimensional in the horizontal plane
- 23. 4 Modelling Flow and Contaminant Transport 7. Saturated seepage flow through soils in the vertical two-dimensional plane 8. Contaminant transport in one-dimensional open channel flows, shallow two-dimensional surface flows, and seepage flows in soils. The equations involved are of the following types: 1. Non-linear equation 2. Simultaneous linear and non-linear equations 3. Ordinary differential equations 4. Partial differential equations The numerical techniques discussed are: 1. The Newton–Raphson method of finding the roots of non-linear equations 2. Solution of simultaneous linear equations using the Gaussian elimination method 3. Solution of simultaneous non-linear equations by a combination of the above 4. Solution of ordinary and partial differential equations using the method of finite differences This book is not intended to elaborate on the numerical techniques, as many useful texts are widely available, some of these being listed in the bibliography. Nor is this book meant to be a programming guide for the language Python, for which again several books and web resources are available. The reader is thus encouraged to pro- ceed only after acquiring a basic knowledge of coding in Python in order to find the book useful. Nonetheless, it is emphasized that this book should be found useful to the science and engineering students who have started exploring the exciting world of coding and modelling physical processes, especially those involving the flow of water and movement of contaminants in the natural and built environments.
- 24. 5 DOI: 10.1201/9780429288579-2 Non-Linear and Simultaneous Equations This chapter introduces a variety of physical processes – sometimes called as systems – in hydraulics, and water resources engineering and are described equiva- lently in the mathematical form as a single or a set of equations. If an equation is possible to be rearranged in terms of the unknown variable explicitly, we may obtain an answer without resorting to advanced computational methods. However, if the equation is non-linear in terms of the unknown variable, then it may not be possible to write an explicit expression for evaluating the variable. In such cases, root-finding algorithms – like the Newton–Raphson Method – may have to be used. An extension of the single equation involving one unknown variable is the system of say, n, equa- tions in terms of n independent variables. Here too, the n equations may be solved for the n unknowns by standard methods of linear algebra if the system of equation is linear, that is if the variables are not arranged in powers greater than one. Otherwise, techniques such as the Newton–Raphson may be used by extending it for n unknowns. This chapter demonstrates the solutions for some problems of these kinds and pro- vides computer programs in Python for implementing the solution algorithms. 2.1 EXAMPLES OF NON-LINEAR FUNCTIONS Assuming that finding solutions to linear equations in one-variable does not require the knowledge of any special technique, we start with a few examples of non-linear equations (in one-variable) for problems encountered in hydraulic engineering. 2.1.1 Normal Depth of Flow in a Trapezoidal Channel An example of a non-linear equation encountered in the field of hydraulics is the finding of normal depth (yn) of the flow taking place through a long trapezoidal chan- nel (Figure 2.1). Such a flow depth occurs at the uniform flow region, much upstream from the influence of any cause for non-uniformity, such as a weir. The normal depth yn may be obtained by solving the following equation (Chaudhry, 2008): Q n AR S = 1 2 3 0 1 2 / / (2.1) where Q is the steady-state discharge passing through the channel, n is the Manning’s roughness coefficient, A and R are the cross-sectional area and wetted perimeter, respectively, both of which are functions of the depth of water, and S0 is the 2
- 25. Discovering Diverse Content Through Random Scribd Documents
- 26. There now remains the very interesting question as to whether the effect of long ether waves induce any variation of growth. The results given below show that growing plants not only perceive but respond to the stimulus of electric waves. The effects to be presently described are exhibited by all plants. I shall, however, content myself in describing a typical experiment carried with the seedling of wheat. The specimen was mounted on the Balanced Crescograph, and the growth exactly balanced. This gives a horizontal record; an acceleration of growth above the normal is, in the following records, represented by a down curve, and a retardation by an up-curve. Effect of feeble stimulus: Experiment 161.—I first studied the effect of feeble stimulus. This was secured by decreasing the energy of sparks of the radiator. The response was an acceleration of rate of growth as seen in figure 156a. The analogy of this with the accelerating effect of sub-minimal intensity of light (p. 224) is very remarkable. Fig. 156.—Record of responses to electric wave by the Balanced Crescograph (a) response to feeble stimulus by acceleration of growth, (b) response to strong stimulus by retardation, (c) responses to medium stimulation—retardation followed by recovery. Down-curve represents acceleration, and up-curve retardation of growth: (Seedling of wheat.) Effect of strong stimulus: Experiment 162.—The maximum energy radiated by my transmitter, as stated before, was only moderate. In spite of this its effect on plants was exhibited in a very striking manner. The balance was immediately upset, indicating a retardation of the rate of growth. The latent period, i.e., the interval between the incident wave and the response, was only a few seconds (Fig. 156b). The record given in the figure was obtained with the moderate magnification of 2,000 times only. But with my Magnetic Crescograph, the magnification can easily be raised ten million times; and the response of plant to the space signalling can be exalted in the same proportion. Under an intensity of stimulus slightly above the sub-minimal, the responses exhibit retardation of growth followed by quick recovery, as seen in the series of records given in Fig. 156c. A remarkable peculiarity in the response was noticed during the course of the experiments. Strong stimulation by ether waves gives rise, as we have seen, to a very marked retardation of the rate of growth. Repeated stimulation induces fatigue, and temporary insensitiveness of the organ. Under moderate fatigue the effect is a prolongation of the latent period. Thus in a particular experiment the plant failed to give any response to a short signal. But after an interval of five minutes a marked response occurred to the wireless stimulus that had been received previously. The plant had perceived
- 27. the stimulus but on account of fatigue the latent period was prolonged, from the normal 5 seconds to as many minutes. SUMMARY. Plants not only perceive, but also respond to long ether waves employed in signalling through space. Mechanical response to wireless stimulation is exhibited by the leaf of Mimosa pudica. All plants give electric response to the stimulus of long ether waves. Growing plants exhibit response to electric waves by modification of rate of growth. Feeble stimulus induces an acceleration, while strong stimulus causes a retardation of the rate of growth. The perceptive range of the plant is far greater than ours; it not only perceives but responds to the different rays of the vast ethereal spectrum. [26] Pfeffer—Vol. II, p. 104. [27] Plant Response—p. 618 (1905). [28] Comparative Electro-Physiology—p. 149.
- 28. XXXIX.—GEOTROPISM By Sir J. C. Bose. No phenomenon of tropic movement appears so inexplicable as that of geotropism. There are two diametrically opposite effects induced by the same stimulus of gravity, in the root a movement downwards, and in the shoot a movement upwards. The seeming impossibility of explaining effects so divergent by the fundamental reaction of stimulus, has led to the assumption that the irritability of stem and root are of opposite character. I shall, however, be able to show that this assumption is unnecessary. The difficulty of relating geotropic curvature to a definite reaction to stimulus is accentuated by the fact that the direction of the incident stimulus, and the side which responds effectively to it are not clearly understood; nor is it known, whether the reaction to this stimulus is a contraction, or its very opposite, an expansion. Taking the simple case of a horizontally laid shoot, the geotropic up-curvature is evidently due to differential effect of the stimulus on upper and lower sides of the organ. The up-curvature may be explained by one or the other of two suppositions: (1) that the stimulus of gravity induces contraction of the upper side; or (2) that the fundamental reaction is not a contraction but an expansion and this of the lower side. The second of these two assumptions has found a more general acceptance. Tropic curvatures in general are brought about by the differential effect of stimulus on two sides of the organ. Thus light falling on one side of a shoot induces local contraction, the rays being cut off from acting on the further side by the opacity of the intervening tissue. But there is no opaque screen to cut off the vertical lines of gravity,[29] which enter the upper side of a horizontally laid shoot and leave it by the lower side. Though lines of force of gravity are transmitted without hindrance, yet a differential action is found to take place, for the upper side, where the lines of force enter, becomes concave, while the lower side where they emerge becomes convex. Why should there be this difference? For the removal of various obscurities connected with geotropism it is therefore necessary to elucidate the following: 1. The sign of excitation is, as we found, a contraction and concomitant galvanometric negativity. Does gravitational stimulus, like stimulus in general, induce this excitatory reaction? 2. What is the effective direction of geotropic stimulus? In the case of light, we are able to trace the rays of light which is incident on the proximal side and measure the angle of inclination. In the case of gravity, the invisible lines of force enter by one side of the organ and leave by the other side. Assuming that the direction of stimulus is coincident with the vertical lines of gravity, is it the upper or the lower side of the organ that undergoes effective stimulation? 3. What is the law relating to the 'directive angle' and the resulting geotropic curvature? By the directive angle (sometimes referred to as the angle of inclination) is meant, as previously explained, the angle which direction of stimulus makes with the responding surface. 4. We have finally to investigate, whether the assumption of opposite irritabilities of the root and the shoot is at all justifiable. If not, we have to find the true explanation of the opposite curvatures exhibited by the two types of organs.
- 29. Of these the first three are inter-related. They will, however, be investigated separately; and each by more than one method of inquiry. The results will be found to be in complete harmony with each other. I propose in this and in the following chapters to carry out the investigations sketched above, employing two independent methods of enquiry, namely, of mechanical and of electrical response. I shall first describe the automatic method I have been able to devise, for an accurate and magnified record of geotropic movement and its time relations. THE GEOTROPIC RECORDER. The recorder shown in figure 157 is very convenient for study of geotropic movement. The apparatus is four-sided and it is thus possible to obtain four simultaneous records with different specimens under identical conditions. The recording levers are free from contact with the recording surface. By an appropriate clock-work mechanism, the levers are pressed for a fraction of a second against the recording surfaces. The successive dots in the record may, according to different requirements, be at intervals varying from 5 to 20 seconds. The records therefore not only give the characteristic curves of geotropic movements of different plants, but also their time durations. For high magnification, I employ an Oscillating Recorder, the short arm of the lever being 2·5 mm., and the long arm 250 mm., the magnification being a hundredfold; half that magnification is, however, sufficient for general purposes. Fig. 157.—The Quadruplex Geotropic Recorder. DETERMINATION OF THE CHARACTER OF GEOTROPIC REACTION. The observed geotropic concavity of the upper side of a horizontally laid shoot may be due to excitatory contraction of that side, or it may result from passive yielding to the active responsive expansion of the lower side. The crucial test of excitatory reaction under geotropic stimulus is furnished by investigations on geo-electric response. When a shoot is displaced from vertical to horizontal position, the upper side of the organ is found to undergo an excitatory electric change of galvanometric negativity indicative of diminution of turgor and contraction. The electric change induced on the lower
- 30. side is one of galvanometric positivity, which indicates an increase of turgor and expansion. The tropic effect of geotropic stimulus is thus similar to that of any other mode of stimulation, i.e., a contraction of the upper (which in the present case is the proximal) and expansion of the lower or the distal side. The vertical lines of gravity impinge on the upper side of the organ which undergoes effective stimulation. In order to show that the concavity of the upper side is not due to the passive yielding to the expansion of the lower half, I restrained the organ from any movement. I have explained that excitatory electric response is manifested even in the absence of mechanical expression of excitation; and under geotropic stimulus, the securely held shoot gave the response of galvanometric negativity of the upper side. Hence the fundamental reaction under geotropic stimulus is excitatory contraction as under other modes of stimulation. Finally, I employed the additional test of induced paralysis by application of intense cold. Excitatory physiological reaction is, as we know, abolished temporarily by the action of excessive cold. Experiment 163.—I obtained records of mechanical response to determine the side which undergoes excitation under geotropic stimulus, the method of discrimination being local paralysis induced by cold. I took the flower-scapes of Amarayllis and of Uriclis, and holding them vertical applied fragments of ice on one of the two sides. I then laid the scape horizontal, first with cooled side below, the record showed that this did not affect the geotropic movement. But on cooling the upper side, the geotropic movement became arrested, and it was not till the plant had assumed the temperature of the surroundings that the geotropic movement became renewed. Figure 158 shows the effect of alternate application of cold, on the upper and lower sides of the organ.[30] In the shoot, therefore, it is the upper side of the organ that becomes effectively stimulated. Before proceeding further I shall make brief reference to the highly suggestive statolithic theory of gravi-perception. Fig. 158.—Effect of alternate application of cold on the upper and lower sides of the organ. Application of cold on upper side (down-pointing arrow) induces arrest of geotropic movement. Application below (up-pointing arrow) causes no arrest. THEORY OF STATOLITHS. With regard to the perception of geotropic stimulus there can be no doubt that this must be due to the effect of weight of cell contents, whether of the sap itself, or of the heavy particles contained in the cells, exerting pressure on the sensitive plasma. The theory of statoliths advocated by Noll, Haberlandt and Nemec (in spite of certain difficulties which further work may remove) is the only rational explanation hitherto offered for gravi-perception. The sensitive plasma is the ectoplasm of the entire cell, and statoliths are relatively heavy bodies, such as crystals and starch grains. Haberlandt has found
- 31. statoliths in the apo-geotropic organs like stems.[31] When the cell is laid horizontal, it is the lower tangential wall which has to support the greater weight, and thus undergo excitation. In the case of multicellular plants laid horizontally, the excitation on the upper side is, as we have seen, the more effective than on the lower side. This inequality, it has been suggested, is probably due to this difference that the statoliths on the upper side press on the inner tangential walls of the cells while those on the lower side rest on the outer tangential walls. When the organ is held erect, the action of statoliths would be symmetrical on the two sides. But when it is laid horizontal a complete rearrangement of the statoliths will take place, and the differential effects on the upper and lower sides will thus induce geotropic reaction. This period of migration must necessarily be very short; but the reaction time, or the latent period, is found to be of considerable duration. Even in rapidly reacting organs there is always an interval of about one to one and a half hours, before the horizontally placed organ shows a noticeable curvature, and this latent period may in other cases be extended to several hours.[32] This great difference between the period of migration and the latent period offers a serious difficulty in the acceptance of the theory of statoliths. But it may be urged that the latent period has hitherto been obtained by relatively crude methods, and I therefore undertook a fresh determination of its value by a sensitive and accurate means of record. DETERMINATION OF THE LATENT PERIOD. As regards the interpretation of the record of geotropic movement, it should be borne in mind that after the perception of stimulus a certain time must elapse before the induced growth-variation will result in curvature. There is again another factor which causes delay in the exhibition of true geotropic movement; for the up-movement of stems, in response to the stimulus of gravity, has to overcome the opposite down movement, caused by weight, before it becomes at all perceptible. On account of the bending due to weight there is a greater tension on the upper side, which as we have seen (p. 193), enhances the rate of growth, and thus tends to make that side convex. The exhibition of geotropic response by induced contraction of the excited upper side thus becomes greatly delayed. In these circumstances I tried to discover specimens in which the geotropic action would be quick, and in which the retarding effect of weight could be considerably reduced. Geotropic response of flower stalk of Tuberose: Experiment 164.—For this I took a short length of flower stalk of tuberose in a state of active growth; the flower head itself was cut off in order to remove unnecessary weight. After a suitable period of rest for recovery from the shock of operation, the specimen was placed in a horizontal position, and its record taken. The successive dots in the curve are at intervals of 20 seconds, and the geotropic up-movement is seen to be initiated (Fig. 159) after the tenth dot, the latent period being thus 3 minutes and 20 seconds, the greater part of which was spent in overcoming the down-movement caused by the weight of the organ. Fig. 159. Fig. 160.
- 32. Fig. 159.—Geotropic response of flower stalk of tube rose: preliminary down-movement is due to weight. Fig. 160.—Geotropic response of petiole of Tropæolum: latent period shorter than 20 seconds. Geotropic response of petiole of Tropæolum: Experiment 165.—I expected to obtain still shorter latent period by choosing thinner specimens with less weight. I therefore took a cut specimen of the petiole of Tropæolum, and held it at one end. The lamina was also cut off in order to reduce the considerable leverage exerted by it. The response did not now exhibit any preliminary down-movement, and the geotropic up-movement was commenced within a few seconds after placing the petiole in a horizontal position (Fig. 160). The successive dots in the record are at intervals of 20 seconds and the second dot already exhibited an up-movement; the latent period is therefore shorter than 20 seconds. It will thus be seen that the latent period in this case is of the same order as the hypothetical period of migration of the statoliths. I may state here that I have been successful in devising an electric method for the determination of the latent period, in which the disturbing effect of the weight of the organ is completely eliminated. Applying this perfect method, I found that the latent period was in some cases as short as a second. The experiment will be found fully described in a later chapter. THE COMPLETE GEOTROPIC CURVE. The characteristics of the geotropic curve are similar to those of other tropic curves. That is to say the susceptibility for excitation is at first feeble; it then increases at a rapid rate; in the third stage the rate becomes uniform; and finally the curvature attains a maximum value and the organ attains a state of geotropic equilibrium (cf. page 353). The period of completion of the curve varies in different specimens from a few to many hours. Experiment 166.—The following record was obtained with a bud of Crinum, the successive dots being at intervals of 10 minutes. After overcoming the effect of weight (which took an hour), the curve rose at first slowly, then rapidly. The period of uniformity of movement is seen to be attained after three hours and continued for nearly 90 minutes. The final equilibrium was reached after a period of 8 hours (Fig. 161). Fig. 161.—The Complete Geotropic curve (Crinum). For studying the effect of an external agent on geotropic action, the period of uniform movement is the most suitable. Acceleration of the normal rate (with enhanced steepness of curve) indicates that the
- 33. external agent acts with geotropism in a concordant manner; depression of the rate with resulting flattening of the curve shows, on the other hand, the antagonistic effect of the outside agent. DETERMINATION OF EFFECTIVE DIRECTION OF STIMULUS. The experiments which have been described show that it is the upper side (on which the vertical lines of gravity impinge) that undergoes excitation. The vertical lines of gravity must therefore be the direction of incident stimulus. This conclusion is supported by results of three independent lines of inquiry: (1) the algebraical summation of effect with that of a different stimulus whose direction is known, (2) the relation between the directive angle and geotropic reaction, and (3) the torsional response under geotropic stimulus. Fig. 162. Fig. 163. Fig. 162.—Stimulus of light or gravity, represented by arrow, induces up curvature as seen in dotted figure. Fig. 163.—The effect of super-imposition of photic stimulus. The first, third, and fifth parts of the curve, give normal r geotropic stimulus. Rate of up-movement enhanced under light L. EFFECT OF ALGEBRAICAL SUMMATION. Experiment 167.—A flower bud of Crinum is laid horizontally, and record taken of its geotropic movement. On application of light on the upper side at L, the responsive movement is enhanced, proving that gravity and light are inducing similar effects. On the cessation of light, the original rate of geotropic movement is restored (Fig. 163). Application of light of increasing intensity from below induces, on the other hand, a diminution, neutralisation, or reversal of geotropic movement. Light acting vertically from above induces a concavity of the excited upper side in consequence of which the organ moves, as it were, to meet the stimulus. The geotropic response is precisely similar. In figure 162 the arrow represents the direction of stimulus which may be rays of light or vertical lines of gravity. ANALOGY BETWEEN THE EFFECTS OF STIMULUS OF LIGHT AND OF GRAVITY.
- 34. In geotropic curvature we may for all practical purposes regard the direction of stimulus as coinciding with the vertical lines of gravity. The analogy between the effects of light and of gravity is very close[33]; in both the induced curvature is such that the organ moves so as to meet the stimulus. This will be made still more evident in the investigations on torsional geotropic response described in a subsequent chapter. The tropic curve under geotropic stimulus is similar to that under photic stimulus. The tropic reaction, both under the stimulus of light and of gravity, increases similarly with the 'directive' angle. These real analogies are unfortunately obscured by the use of arbitrary terminology used in description of the geotropic curvature of the shoot. In figure 163 records are given of the effects of vertical light and of vertical stimulus of gravity, on the responses of the horizontally laid bud of Crinum. In both, the upper side undergoes contraction and the movement of response carries the organ upwards so as to place it parallel to the incident stimulus. Though the reactions are similar in the two cases, yet the effect of light is termed positive phototropism, that of gravity negative geotropism. I would draw the attention of plant-physiologists to the anomalous character of the existing nomenclature. Geotropism of the shoot should, for reasons given above, be termed positive instead of negative, and it is unfortunate that long usage has given currency to terms which are misleading, and which certainly has the effect of obscuring analogous phenomena. Until the existing terminology is revised, it would perhaps be advisable to distinguish the geotropism of the shoot as Zenithotropism and of the root as Nadirotropism. RELATION BETWEEN THE DIRECTIVE ANGLE AND GEOTROPIC REACTION. When the main axis of the shoot is held vertical, the angle made by the surface of the organ with lines of force of gravity is zero, and there is no geotropic effect. The geotropic reaction increases with the directive angle; theoretically the geotropic effect should vary as the sine of the angle. I shall in the next chapter describe the very accurate electrical method, which I have been able to devise for determination of relative intensities of geotropic action at various angles. Under perfect conditions of symmetry, the intensity of effect is found to vary as the sine of the directive angle. This quantitative relation fully demonstrates that geotropic stimulus acts in a definite direction which coincides with the vertical lines of gravity. The conditions of perfect symmetry for study of geotropic action at various angles will be fully described in the next chapter. In the ordinary method of experimentation with mechanical response the organ is rotated in a vertical plane. The geotropic movement is found increased as the directive angle is increased from zero to 90°. DIFFERENTIAL GEOTROPIC EXCITABILITY. It has been shown that geotropic stimulus acts more effectively on the upper side of the organ. The intensity of geotropic reaction is, moreover, modified by the excitability of the responding tissue. It is easy to demonstrate this by application of depressing agents on the more effective side of the organ. The rate of geotropic up-movement will be found reduced, or even abolished by the local application of cold, anæsthetics like chloroform, and of poisonous potassium cyanide solution. The different sides of a dorsiventral organ are unequally excitable to different forms of stimuli. I have already shown (p. 85) that the lower side of the pulvinus of Mimosa, is about 80 times more excitable to electric stimulus than the upper side. Since the effect of geotropic stimulus is similar to that of other forms of stimuli, the lower side of the pulvinus should prove to be geotropically more excitable than the upper side. This I have been able to demonstrate by different methods of investigation which will be described in the following chapters. Under ordinary circumstances, the upper half of the pulvinus is, on account of its favourable position, more effectively stimulated by geotropic stimulus; in consequence of this the leaf assume a more or less horizontal position of dia-geotropic equilibrium. But when the plant is inverted the more excitable lower half of the organ now occupies the favourable position for geotropic excitation. The leaf
- 35. now erects itself till it becomes almost parallel to the stem. The response of the same pulvinus which was formerly dia-geotropic now becomes negatively geotropic; but an identical organ cannot be supposed to possess two different specific sensibilities. The normal horizontal position assumed by the leaf is, therefore, due to differential geotropic excitabilities of the two sides of a dorsiventral organ. I have explained (p. 401) that when the pulvinus of Mimosa is subjected to lateral stimulation of any kind, it undergoes a torsion, in virtue of which the less excitable half of the organ is made to face the stimulus. Experiments will be described in a subsequent chapter which show that geotropic stimulus also induces similar torsional response. The results obtained from this method of enquiry give independent proof: (1) that the lower half of the pulvinus is geotropically the more excitable, and (2) that the direction of incident geotropic stimulus is the vertical line of gravity which impinges on the upper surface of the organ. SUMMARY. The stimulus of gravity is shown to induce an excitatory reaction which is similar to that induced by other forms of stimulation. The direct effect of geotropic stimulus is an incipient contraction and retardation of rate of growth. The upper side of a horizontally laid shoot is more effectively stimulated than the lower side, the excited upper side becoming concave. Electrical investigation also shows that it is the upper side that undergoes direct stimulation. Tropic reactions are said to be positive, when the directly stimulated side undergoes contraction with the result that the organ moves to meet the stimulus. According to this test, the geotropic response of the stem is positive. The geotropic response is delayed by the bending down of the horizontally laid shoot. Reduction of weight is found to shorten the latent period; in the case of the petiole of Tropæolum this is shorter than 20 seconds. The latent period of geotropic response is found to be of the same order as the migration period of the hypothetical statoliths. The complete geotropic curve shows characteristics which are similar to tropic curves in general. In a dorsiventral organ the geotropic excitabilities of the upper and lower sides are different. In the pulvinus of Mimosa the geotropic excitability of the lower half is greater than that of the upper half. The differential excitabilities of a dorsiventral organ modifies its position of geotropic equilibrium. [29] I shall in what follows take the direction of vertical lines of gravity as that of movement of falling bodies, from above towards the centre of the earth. [30] Plant Response—p. 505. [31] Haberlandt—Physiological Plant Anatomy—p. 603. [32] Jost—Ibid, p. 437. [33] Exception to this will be found in page 336, where explanation is offered for the difference.
- 36. XL.—GEO-ELECTRIC RESPONSE OF SHOOT By Sir J. C. Bose, Assisted by Satyendra Chandra Guha, M.Sc. The experiments that have been described in the preceding chapter show that the upper side of a horizontally laid shoot undergoes excitatory contraction, in consequence of which the organ bends upwards. The fundamental geotropic reaction is, therefore, not expansion, but contraction which results from all modes of stimulation. In confirmation of the above, I wished to discover and employ new means of detecting excitatory reaction under geotropic stimulus. In regard to this, I would refer to the fact which I have fully established that the state of excitation can be detected by the induced electromotive change of galvanometric negativity. This electrical indication of excitation may be observed even in plants physically restrained from exhibiting response by mechanical movement.[34] ELECTRIC RESPONSE TO STIMULUS. Before giving account of the results of investigations on the detection of geotropic excitation by means of electric response, I shall describe a few typical experiments which will fully explain the method of the electrical investigation, and show the correspondence of mechanical and electric responses. I have explained how tropic curvatures are brought about by the joint effects, of contraction of the directly excited proximal side A, and the expansion of the distal side B. In the diagram of mechanical response to stimulus (Fig. 164a) the excitatory contraction is indicated by - sign, and the expansion, by + sign. The resulting movement is, therefore, towards the stimulus as shown by the curved arrow. I shall now describe the corresponding electric effects in response to unilateral stimulus. We have to determine the induced electrical variation at the proximal side A, and at the distal side B.
- 37. Fig. 164.—Diagrammatic representation of the mechanical and electrical response to direct unilateral stimulation indicated by arrow:— (a) Positive mechanical response (curved arrow) due to contraction of directly stimulated A, and expansion of indirectly stimulated B. (b) Electric response of induced galvanometric negativity of A under direct stimulation. (c) Electric response of induced galvanometric positivity at the distal point B. (d) Additive effects of direct and indirect stimulations; galvanometric negativity of the directly stimulated proximal A, and galvanometric positivity of the indirectly stimulated distal point B. Electric response to direct stimulation: Experiment 168.—For the determination of electric response at the directly excited proximal side A, we take a shoot with a lateral leaf. The point A, which is to undergo stimulation, is connected with one terminal of the galvanometer, the other terminal being led to an indifferent or neutral point N on the leaf. Application of any form stimulus at A, gives rise to an electric current which flows through the galvanometer from the neutral to the excited point A (Fig. 164b). The directly stimulated point A thus becomes galvanometrically negative. The action current lasts during the application of stimulus and disappears on its cessation. Electric response to indirect stimulation: Experiment 169.—We have also seen that application of stimulus at A causes indirect stimulation of the distal point B resulting in an increase of turgor and expansion. The corresponding electric change of the indirectly stimulated point B is found in the responsive current, which flows now through the galvanometer from the indirectly stimulated B to the neutral point N (Fig. 164c). The indirectly stimulated point thus becomes galvanometrically positive. Having thus obtained the separate effects at A and B, we next modify the experiment for obtaining the joint effects. For this purpose the neutral point N is discarded and A and B connected directly with the indicating galvanometer. On stimulation of A that point becomes negative and B positive, and the current of response flows through the galvanometer from B to A. The deflection is increased by the joint electrical reactions at A and B (Fig. 164d). The results may thus be summarised:— TABLE XXXIII.—ELECTRIC RESPONSE TO DIRECT UNILATERAL STIMULUS. Electrical change at the proximal side A. Electrical change at the distal side B. Galvanometric negativity indicative of contraction and diminution of turgor. Galvanometric positivity indicative of expansion and increase of turgor. The corresponding tropic curvature is positive movement towards stimulus.
- 38. Galvanometric negativity is thus seen to indicate the effect of direct stimulus, and galvanometric positivity that of indirect stimulus. We thus see the possibility of electric detection of the effects of geotropic stimulation. This method would, moreover, enable us to discriminate the side of the organ which undergoes greater excitation. EXPERIMENTAL ARRANGEMENTS FOR OBTAINING GEO-ELECTRIC RESPONSE. Returning to the investigation on electric response to geotropic stimulus, the specimen of plant is at first held erect; two electrodes connected with a sensitive galvanometer are applied, one to an indifferent point, and the other to one side of the shoot. The sensitiveness of the galvanometer was such that a current of one millionth of an ampere produced a deflection of the reflected spot of light through 1,000 divisions of the scale. An action current is produced on displacement of the plant from vertical to horizontal position. Non-polarisable electrodes.—The electrical connections with the plant are usually made by means of non-polarisable electrodes (amalgamated zinc rod in zinc-sulphate solution and kaolin paste with normal saline). I at first used this method and obtained all the results which will be presently described. But the employment of the usual non-polarisable electrodes with liquid electrolyte is, for our present purpose, extremely inconvenient in practice; for the plant-holder with the electrodes has to be rotated from vertical to horizontal through 90°. The reliability of the non-polarisable electrode, moreover, is not above criticism. The zinc-sulphate solution percolates through the kaolin paste and ultimately comes in contact with the plant, and seriously affects its excitability. The name non-polarisable electrode is in reality a misnomēr; for the action current (whose polarising effect is to be guarded against) is excessively feeble, being of the order of a millionth of an ampere or even less; the counter polarisation induced by such a feeble current is practically negligible. The idea that non-polarisable electrodes are meant to get rid of polarisation is not thus justified by the facts of the case. The real reason for its use is very different; the electrical connections with the plant has to be made ultimately by means of two metal contacts. If we take two pieces of metal even from the same sheet, and put them in connection with the plant, a voltaic couple is produced owing to slight physical differences between the two electrodes. Amalgamation of the two zinc rods with mercury reduces the electric difference but cannot altogether eliminate it. I have been able to wipe off the difference of potential between two pieces of the same metal, say of platinum, and by immersing them in dilute salt solution from a voltaic couple. The circuit is kept complete for 24 hours, and the potential of the two electrodes by this process is nearly equalised. A perfect equality is secured by repeated warming and cooling of the solution and by sending through the circuit, alternating current which is gradually reduced to zero. I have by this means been able to obtain two electrodes which are iso-electric. The specially prepared electrodes (made of gold or platinum wire) are put in connection with the plant through kaolin paste moistened with normal saline solution. Care should be taken to use opaque cover over the plant-holder, so as to guard against any possible photo- electric action; moistened blotting paper maintains the closed chamber in a uniform humid condition. The direct method of contact described above is extremely convenient in practice; the resistance of contact is considerably reduced, and there is no possibility of its variation during the necessary process of rotation of the plant for subjecting it to geotropic action.
- 39. Fig. 165.—Diagrammatic representation of geo-electric response. The middle figure represents vertical position. In figure to the right rotation through +90° has placed A above with induced electric change of galvanometric negativity of A. In the figure to the left, rotation is through -90° A being below; the electric response is by induced galvanometric positivity of A. For simplification of diagram, vertical position of sepal is not always shown in the figure. GEO-ELECTRIC RESPONSE OF THE UPPER AND LOWER SIDES OF THE ORGAN. We have next to discover the electric change induced by geotropic stimulus on the upper and lower sides of the organ. For this purpose it is necessary to find a neutral point which is not affected by the inclination of the organ from vertical to horizontal position. For the present experiment, I employed the flower of the water lily Nymphæa, the peduncle of which is sensitive to geotropic action. One electrical contact is made with a sepal, which is always kept vertical; the other electric contact is made at the point A, on one side of the flower stalk (Fig. 165). On making connections with a sensitive galvanometer a very feeble current was found, which was due to slight physiological difference between the neutral point, N, and A. This natural current may be allowed to remain, the action current due to geotropism being superposed on it; or the natural current may be neutralised by means of a potentiometer and the reflected spot of light brought to zero of the scale. Induced electric variation on upper side of the organ: Experiment 170.—While the sepal is held vertical, the stalk is displaced through +90° so that the point A is above. Geotropic stimulation is at once followed by a responsive current which flows through the galvanometer from N to A, the upper side of the organ thus exhibiting excitatory reaction of galvanometric negativity (Right-hand figure of 166). When the stalk is brought back to vertical position geotropic stimulation disappears, and with it the responsive current. Electric response of the lower side: Experiment 171.—The stalk is now displaced through -90°; the point A, which under rotation through +90° pointed upwards, is now made to point downwards. The direction of the current of response is now found to have undergone a reversal; it now flows from A on the lower side to the neutral point N; thus under geotropic action the lower side of the organ exhibits galvanometric positivity indicative of increase of turgor and expansion (Left-hand figure 166).[35] Having thus found that the upper side of the organ under geotropic stimulus becomes galvanometrically negative, and the lower side, galvanometrically positive, we make electric connections with two diametrically opposite points of the shoot A and B, and subject the organ to alternate rotation through +90° and -90°. The electro-motive changes induced at the two sides now became algebraically summated. I employ two methods for geotropic stimulation: that (1) of Axial Rotation, and (2) of Vertical Rotation.
- 40. Fig. 166.—Diagrammatic representation of the Method of Axial Rotation H, and of Vertical rotation V (see text). METHOD OF AXIAL ROTATION. In the method of Axial Rotation, the organ is held with its long axis horizontal (Fig. 166 H). We have seen that the geotropic action increases with the angle which the responding surface of the organ makes with the vertical lines of gravity. When the organ is held with its length horizontal, the angle made by its two sides, A and B, with the vertical is zero and there is thus no geotropic effect. There is, moreover, no differential effect, since the two sides are symmetrically placed as regards the vertical lines of force. The plant is next rotated round its long axis, the angle of rotation being indicated in the circular scale. When the rotation is through +90°, A is above and B below; this induces a differential geotropic effect, the upper side exhibiting excitatory electric change of galvanometric negativity. Experiment 172.—I shall, as a typical example, give a detailed account of experiments with the petiole of Tropæolum which was found so highly excitable to geotropic stimulus (p. 434). The specimen was held horizontal with two symmetrical contacts at the two sides, the electrodes being connected in the usual manner with the indicating galvanometer. When the plant is rotated through +90° there is an immediate current of response, the upper side becoming galvanometrically negative. This excitatory reaction on the upper side finds, as we have seen, mechanical expression by contraction and concavity, with positive or up-curvature. Fig. 167.—Diagrammatic representation of the geo-electric response of the shoot (see text).
- 41. The differential stimulation of A and B disappears on rotation of the axis back to zero position, and the induced electro-motive response also disappears at the same time. If now the axis be rotated through -90°, A will become the lower, and B the upper and the excited side. The electro-motive change is now found to have undergone a reversal, B becoming galvanometrically negative. This induced electro-motive variation under geotropic stimulus is of considerable intensity often exceeding 15 millivolts. The characteristic electric change is shown diagrammatically in figure 167 in which the middle figure shows the symmetrical or zero position. On rotation through +90° (figure to the right) A occupies the upper and B the lower position. A is seen to exhibit induced change of galvanometric negativity. Rotation through -90° reverses the current of response, as B now occupies the upper and A the lower position. CHARACTERISTICS OF GEO-ELECTRIC RESPONSE. There are certain phenomena connected with the electric response under geotropic stimulus which appear to be highly significant. According to statolithic theory Geotropic response begins as soon as an organ is deflected from its stable position, so that a few starch-grains press upon the ectoplasts occupying the walls which are underneath in the new position; an actual rearrangement of the starch-grains is therefore not an essential condition of stimulation. As a matter of fact, the starch-grains do very soon migrate on to the physically lower walls, when a positively or negatively geotropic organ is placed horizontally, with the result that the intensity of stimulation gradually increases attaining its maximum value when all the falling starch-grains have moved on to the lower region of the ectoplast. The time required for the complete rearrangement of the statoliths may be termed the period of migration; its average length varies from five to twenty minutes in different organs.[36] Stimulation, according to the statolithic theory, is induced by the displacement of the particles. The diameter of the geotropically sensitive cells is considerably less than 0·1 mm.; and the stimulus will be perceived after the very short interval taken by the statoliths to fall through a space shorter than 0·1 mm. This may be somewhat delayed by the viscous nature of the plasma, but in any case the period for perceptible displacement of the statoliths should be very short, about a second or so, and the latent period of perception of stimulus should be of this order. The mechanical indication of response to stimulus is delayed by a period which is somewhat indefinite; for the initiation of responsive growth variation will necessarily lag behind the perception of stimulus.
- 42. Fig. 168.—Geo-electric response of the petiole of Tropæolum. Experiment 173.—The mechanical response with its drawbacks is thus incapable of giving an accurate value of the latent period. The electrical method of investigation labours under no such disadvantage, since the excitation is here detected even in the absence of movement. The perception of stimulus will thus be followed by response without undue delay. I shall in this connection give a record of electric response of the quickly reacting petiole of Tropæolum, when the angle of inclination is increased from zero to 90°. The responsive movement of the galvanometer spot of light was initiated in less than 5 seconds and the maximum deflection was reached in the course of 90 seconds. The angle was next reduced to zero, and the deflection practically disappeared in the further course of a minute and a half (Fig. 168). There was a small excitation remainder. But with vigorous specimens the recovery is complete.
- 43. Fig. 169.—Geo-electric response of the scape of Uriclis. The latent period of quickly reacting petiole of Tropæolum is thus about 5 seconds, a value which is more consonant with the idea of particles inducing excitation by their fall through an exceedingly short distance. In very sluggish organs latent period may be as long as a minute (Fig. 169), which is considerably shorter than an hour, the generally accepted value. Further even in the electric response, the latent period will be delayed beyond the period of perception. For this perception takes place in some unknown sensitive layer in the interior of the tissue, while electric contact is made with the epidermis outside. It is obvious that certain time must elapse before the excitation, initiated at the sensitive layer, should reach the epidermis. Under ideal conditions of experiment which will be described in a subsequent chapter, the latent period for geotropic excitation, I find, to be sometimes as short as a second. PHYSIOLOGICAL CHARACTER OF GEO-ELECTRIC RESPONSE. The intensity of the electro-motive variation is found to depend on the physiological vigour of the specimen. The Tropæolum plant, used for most of the above experiments, are at the best condition of growth in Calcutta in February; after this the plants begin to decline in March and die off by the end of April. Experiment 174.—In February the intensity of electric response was nearly double of that in March; it was only in March that I made quantitative determination of the induced electro-motive force between the upper and lower contacts on rotation of the specimen from zero to 90°. The E. M. F. was determined by the potentiometer method. I give below the following typical values obtained with two different specimens:— Specimen Induced E. M. F. (1) 12 millivolts. (2) 15 In the most favourable season the induced electro-motive force is likely to exceed the above value very considerably. Effect of Age.—While a young petiole gave the above value, an old specimen from the same plant exhibited no response. The plants were in a dying condition in April and all indications of electrical reaction were found abolished. The physiological character of the response was also demonstrated by first obtaining the normal electric response in a vigorous specimen; after death, by immersion in boiling water, the plant gave no electric response to geotropic stimulus. EFFECT OF DIFFERENTIAL EXCITABILITY OF THE ORGAN. I have hitherto described the geo-electric effect of radial and isotropic organs. The induced E. M. F. at 90° was found practically the same whether A was above and B below, and vice versâ. In the mechanical response of the pulvinus of Mimosa, the geotropic excitability was, however, found to be greater in the lower half than in the upper (p. 440). I wished to investigate the question of differential geotropic excitability anew, by means of electric response. Experiment 175.—Electric connections with the galvanometer were made with the upper and lower halves of the pulvinus, the organ being placed in the vertical or neutral position. The angle of inclination was then increased to 90° in the positive and negative directions alternately. TABLE XXXIV.—DIFFERENCE OF GEO-ELECTRIC RESPONSE OF UPPER AND LOWER HALVES OF THE PULVINUS OF Mimosa. Specimen. Position of particular half of pulvinus. Induced E. M. F.
- 44. (1) Upper half above Lower half above 23 millivolts. 30 (2) Upper half above Lower half above 16 29 In the former case the upper half of the pulvinus occupied the up-position; in the second case the up-position was occupied by the lower half of the pulvinus. In both cases strong electric responses were obtained, the upper point of contact being always galvanometrically negative. There was, however, a difference between the two responses, the excitatory electro-motive variation was invariably greater when the lower half of the organ occupied the favourable up-position. This will be seen from the results of two typical experiments in table given above. The electrical mode of investigation thus leads to confirm the result obtained with mechanical method that the lower half of the pulvinus of Mimosa is geotropically more excitable than the upper half. RELATION BETWEEN ANGLE OF INCLINATION AND GEOTROPIC EFFECT. In the Method of Axial Rotation, the condition of the experiment is ideally perfect; in the neutral position the sides A and B are both parallel to the vertical lines of gravity, and are little affected by geotropic reaction. As the specimen is rotated on its long axis the vertical component of the force of gravity increases with the angle of inclination. The hypothetical statolithic particles will become displaced all along the cell, and the vertical pressure exerted by them will also increase with the angle. The geo-electric response will then afford us a measure of the intensity of excitation induced at various angles of inclination. The mechanical response on account of its inherent defects does not afford us the true relation between the angle of inclination and intensity of geotropic reaction. But the electric method of inquiry is free from the defects of the mechanical method. Experiment 176.—The specimen was rotated so that the angle of rotation was 45°, and the maximum electric response observed. The angle was next increased to 90° and the reading for the enhanced response taken. The ratio of the geo-electric response at 90° and 45°, thus affords us a measure of the effective stimulations at the two angles. I give below a table which gives results obtained with 24 different specimens. TABLE XXXV.—RELATION BETWEEN ANGLE OF INCLINATION AND GEOTROPIC EFFECT. No. of specimen. Galvanometric deflection. Ratio b ⁄a. (a) at 45° (b) at 90° 1 70 divisions 110 divisions 1·5 2 30 45 1·5 3 90 126 1·4 4 70 100 1·4 5 21 33 1·6 6 30 50 1·6 7 12 20 1·6 8 14 20 1·4 9 10 16 1·6 10 45 75 1·5 11 25 40 1·6 12 14 20 1·4 13 13 20 1·5
- 45. 14 30 50 1·5 15 38 54 1·4 16 50 75 1·5 17 55 90 1·5 18 13 20 1·5 19 17 25 1·4 20 80 130 1·5 21 15 22 1·4 22 45 75 1·5 23 135 220 1·6 24 55 93 1·5 Mean ratio = 1·49 The mean ratio 1·49 may thus be regarded as the relative geotropic effects at 90° and 45°; this is practically the same as Sin 90° ⁄Sin 45° = 1·4. Hence we arrive at the following law: The intensity on geotropic action varies as the sine of the directive angle. METHOD OF VERTICAL ROTATION. I have hitherto described results obtained with the Method of Axial Rotation; I shall now take up the second method, that of Vertical Rotation, diagrammatic representation of which is given in figure 166V. The specimen is held vertical and two electrical contacts, A and B, made with the two lateral sides; it is then rotated round a horizontal axis perpendicular to the length of the specimen. Rotation may be carried in a right-handed direction with increasing angle with the vertical. The point A is thus subjected to enhanced geotropic stimulation and exhibits increasing electric change of galvanometric negativity; continuous decrease of angle of inclination to zero by rotation in the reverse direction causes a disappearance of the induced electric change. The rotation is next continued in the negative direction by which the point B is increasingly subjected to geotropic action. B is now found to exhibit excitatory reaction, the current of response having undergone a reversal. Rotation to the right and left will be distinguished by plus and minus signs. ELECTRIC RESPONSE THROUGH AN ENTIRE CYCLE. Experiment 177.—When the specimen is vigorous, characteristic response with its changing sign may be obtained through an entire cycle from 0° to +45° to +90°; then back to 45° to 0° to -45° to -90°. With less vigorous specimens the responses becomes enfeebled under fatigue. I give below the results of a typical experiment carried out with a vigorous specimen, the response being distinguished as - when A is above, and + when A is below, the inversion bringing about a reversal direction of the responsive current.
- 46. Angle of inclination +45°+90°+45°0°-45°-90° Galvanometer deflection-19 -35 -18 0 +14 +25 RELATION BETWEEN ANGLE OF VERTICAL ROTATION AND INTENSITY OF GEOTROPIC REACTION. The relation between the angle of inclination and the resulting geotropic action has already been determined by the Method of Axial Rotation. The ratio between the geotropic effects at 90° and 45° was thus found to be 1·49, which is nearly the same as Sin 90° ⁄Sin 45°. I was next desirous of determining the relative excitations at the two angles by the Method of Vertical Rotation. It is necessary here to refer to certain differences of condition in the two methods. In the Axial Method, the hypothetical statoliths are distributed uniformly through the length of the cell, and rotation round the long axis causes displacement of the statoliths, the resulting pressure thus increasing with the sine of the angle of inclination. But in the case of vertical rotation through 45° to the right, the statoliths originally at the base of the cell accumulate to the right hand corner of the cell; a portion of the basal side of the cell is thus subjected to pressure. When the angle is increased to 90° the statoliths pass along the whole length including the basal and apical sides of the cell; but the excitability of the apical half may prove to be greater than that of the basal half. Hence excitatory geotropic effect is not likely to vary strictly as in sine of angle of inclination. Whatever the reason may be, I find as a result of experiments with 12 different specimens that the mean ratio of the effects at 90° and 45°, obtained by the Method of Vertical Rotation, is 1·8:1 which is greater than 1·49:1 obtained by the Method of Axial Rotation, this latter value being practically the same as Sin 90° ⁄Sin 45°. SUMMARY. It is shown that the state of excitation under direct stimulus is exhibited by an electrical change of galvanometric negativity; the effect of indirect stimulus induces, on the other hand, an electrical change of galvanometric positivity. The negative electric change corresponds to contraction and diminution of turgor; the positive electric change indicates, on the other hand, an expansion and increase of turgor. The electric response to geotropic stimulus is studied by the two methods of Axial and Vertical Rotation. The upper side of a horizontally laid shoot is found to undergo an excitatory change of galvanometric negativity. In quick reacting organs the latent period of geo-electric response is about 5 seconds, and the maximum excitation is induced in the course of 2 minutes. The geo-electric response is due to physiological reaction. The intensity of response declines with age and is abolished at the death of the plant. Under symmetrical conditions, the intensity of geotropic reaction is found proportional to the sine of the angle of inclination. Electric investigation shows that the lower half of the pulvinus of Mimosa is geotropically more excitable than the upper half. [34] Comparative Electro-Physiology, p. 20. [35] For detailed account cf. Chapter XLIII. [36] Haberlandt—Ibid—p. 598.
- 48. XLI.—THE MECHANICAL AND ELECTRICAL RESPONSE OF ROOT TO VARIOUS STIMULI By Sir J. C. Bose. In the last chapter we studied the electric response of the shoot to the stimulus of gravity, and found that the excitatory effect of that stimulus is similar to that of other forms of stimulation. Before taking up the subject of the geo-electric response of the root to gravitational stimulus, I shall describe the effects of other forms of stimuli on the mechanical and electrical response of the root. In connection with this subject, it should be borne in mind that the responsive curvature in the root takes place in the sub-apical growing zone which is separated by a certain distance from the tip. The stimulus is therefore direct when applied at the responding growing region; it is indirect when applied at the tip of the root. The intervening distance between the root-tip and the responsive zone of growth is semi-conducting or non-conducting. I shall proceed to give an account of my investigations on the response of the root to direct and indirect unilateral stimulation. We shall study:— (1) The Mechanical response to Direct unilateral stimulus. (2) The Electrical response to Direct unilateral stimulus. (3) The Mechanical response to Indirect unilateral stimulus. (4) The Electrical response to Indirect unilateral stimulus. MECHANICAL RESPONSE TO DIRECT STIMULUS. As the geotropic responses of the shoot and the root are opposed to each other, the object of the investigation is to find out; whether the response of the root to various stimuli is specifically different from that of the shoot. We have seen that tissues in general respond to direct unilateral stimulus by contraction of the proximal and expansion of the distal side, the tropic curvature being thus positive. We shall now determine whether direct unilateral stimulation of the root induces a tropic movement which is similar or dissimilar to that exhibited by the shoot. Experiment 178.—In experimenting with roots of various plants I obtained results which are precisely similar to that of the shoot. The movement of the root was observed by means of a reading microscope focussed on the tip of the organ. I employed various forms of stimuli, mechanical, thermal, and chemical. Unilateral application of these on one side of the growing region gave rise to a positive tropic curvature, resulting in a movement towards the stimulus. These experiments confirm Sachs' observation that unilateral application of stimulus in the region of growth induces positive curvature of the root. ELECTRICAL RESPONSE TO DIRECT STIMULATION.
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