derivation of Wave equation
- 1. ASSINGMENT NO 3: “wave in one dimension, wave motion” Section/Class BS PHYSICS Subject (Electrodynamics II) Semester 7TH AMEER HAMZA Roll no B1F17BSPH0003 Submitted to PROF. NUSARA KHAN Department of Physics University of Central Punjab Lahore, Sub- Campus Bahawalpur
- 2. WAVE EQUATION: Wave equation is the prototype hyperbolic partial differential equation. Wave equation is written as, 𝜕2 ψ 𝜕𝑥2 = 1 𝑣2 𝜕2 ψ 𝜕𝑡2 Solution of wave equation. Let us considered the solution of the wave equation is in this form. Y(x,t) = f(x± vt) ψ(x,t) = f(x ± vt) … …1) let 𝑧 = (x ± vt) the the equation 1) become, ψ(x,t) = f(z)…… … . 𝐴) Take double derivative of equation A) with respect to position x and time t Differentiate the equation A) with respect to ‘x’ 𝜕𝜓 𝜕𝑥 = 𝜕𝑓 𝜕𝑥 By chain rule the above equation can be written as. 𝜕𝜓 𝜕𝑥 = 𝜕𝑓 𝜕𝑧 . 𝜕𝑧 𝜕𝑥 … … …2) As we know that 𝑧 = (x ± vt) 𝜕𝑧 𝜕𝑥 = 𝜕 𝜕𝑥 (x ± vt) 𝜕𝑧 𝜕𝑥 = ( 𝜕 𝜕𝑥 x ± 𝜕 𝜕𝑥 vt) 𝜕𝑧 𝜕𝑥 = (1 ± 0) 𝜕𝑧 𝜕𝑥 = 1… … . . 𝑎) Put the value of equation a) in equation 2) then we get.
- 3. 𝜕𝜓 𝜕𝑥 = 𝜕𝑓 𝜕𝑧 . 1 𝜕𝜓 𝜕𝑥 = 𝜕𝑓 𝜕𝑧 Take again derivative of above equation with respect to x. 𝜕 𝜕𝑥 ( 𝜕𝜓 𝜕𝑥 ) = 𝜕 𝜕𝑥 𝜕𝑓 𝜕𝑧 By chain rule be can write the above equation in this form. 𝜕2 ψ 𝜕𝑥2 = ( 𝜕 𝜕𝑧 𝜕𝑧 𝜕𝑥 ) 𝜕𝑓 𝜕𝑧 𝜕2 ψ 𝜕𝑥2 = 𝜕 𝜕𝑧 ( 𝜕𝑧 𝜕𝑥 𝜕𝑓 𝜕𝑧 )… … 3) As we know that 𝜕𝑧 𝜕𝑥 = 1… … . . 𝑎) So, the equation no 3) can be written as. 𝜕2 ψ 𝜕𝑥2 = 𝜕 𝜕𝑧 (1. 𝜕𝑓 𝜕𝑧 ) 𝜕2 ψ 𝜕𝑥2 = 𝜕 𝜕𝑧 ( 𝜕𝑓 𝜕𝑧 ) 𝜕2 ψ 𝜕𝑥2 = 𝜕2 f 𝜕𝑧2 … … . .𝐵 Now take again the equation no A) ψ(x,t) = f(z)…… … . 𝐴) Differentiate this equation with respect to time ‘t’. 𝜕 𝜕𝑡 𝜓 = 𝜕𝑓 𝜕𝑡 𝜕𝜓 𝜕𝑡 = 𝜕𝑓 𝜕𝑧 . 𝜕𝑧 𝜕𝑡 … …4) As we know that 𝑧 = (x ± vt) 𝜕𝑧 𝜕𝑡 = 𝜕 𝜕𝑡 (x ± vt)
- 4. 𝜕𝑧 𝜕𝑡 = ( 𝜕 𝜕𝑡 x ± 𝜕 𝜕𝑡 vt) 𝜕𝑧 𝜕𝑡 = (0 ± 𝑣) 𝜕𝑧 𝜕𝑡 = (±𝑣)… …. . 𝑏) Put the value of equation b) in equation 4) then we get. 𝜕𝜓 𝜕𝑡 = 𝜕𝑓 𝜕𝑧 . ±𝑣 𝜕𝜓 𝜕𝑡 = ±𝑣 𝜕𝑓 𝜕𝑧 Take again derivative of above equation with respect to ‘t’. 𝜕 𝜕𝑡 ( 𝜕𝜓 𝜕𝑡 ) = ±𝑣 𝜕 𝜕𝑡 𝜕𝑓 𝜕𝑧 By chain rule be can write the above equation in this form. 𝜕2 ψ 𝜕𝑡2 = ±𝑣 ( 𝜕 𝜕𝑧 𝜕𝑧 𝜕𝑡 ) 𝜕𝑓 𝜕𝑧 𝜕2 ψ 𝜕𝑡2 = ±𝑣 𝜕 𝜕𝑧 ( 𝜕𝑧 𝜕𝑡 𝜕𝑓 𝜕𝑧 ) … …5) As we know that 𝜕𝑧 𝜕𝑡 = (±𝑣)… … .. 𝑏) So, the equation no 5) can be written as. 𝜕2 ψ 𝜕𝑡2 = ±𝑣 𝜕 𝜕𝑧 (±𝑣. 𝜕𝑓 𝜕𝑧 ) 𝜕2 ψ 𝜕𝑡2 = (±𝑣)(±𝑣) 𝜕 𝜕𝑧 ( 𝜕𝑓 𝜕𝑧 ) 𝜕2 ψ 𝜕𝑡2 = 𝑣2 𝜕2 f 𝜕𝑧2 1 𝑣2 𝜕2 ψ 𝜕𝑡2 = 𝜕2 f 𝜕𝑧2 …… … . . 𝐶)
- 5. The right side of equation C) and B) are equal the the left side should also be equal so can can write, 1 𝑣2 𝜕2 ψ 𝜕𝑡2 = 𝜕2 ψ 𝜕𝑥2 By rearranging 𝜕2 ψ 𝜕𝑥2 = 1 𝑣2 𝜕2 ψ 𝜕𝑡2 This the wave equation of a particle/atom/body in one dimension. For three dimension this equation can be written as, 𝜕2 ψ 𝜕𝑥2 + 𝜕2 ψ 𝜕𝑦2 + 𝜕2 ψ 𝜕𝑧2 = 1 𝑣2 𝜕2 ψ 𝜕𝑡2 In simple form it can be written as. ∇2 Ψ = 1 𝑣2 𝜕2 ψ 𝜕𝑡2 Where ∇2 = 𝜕2 ψ 𝜕𝑥2 + 𝜕2 ψ 𝜕𝑦2 + 𝜕2 ψ 𝜕𝑧2 The mathematical description of the one- dimensional waves (both traveling and standing) can be expressed as 𝜕2 ψ 𝜕𝑥2 + 𝜕2 ψ 𝜕𝑦2 + 𝜕2 ψ 𝜕𝑧2 = 1 𝑣2 𝜕2 ψ 𝜕𝑡2 with ψ is the amplitude of the wave at position x and time t, and v is the velocity of th e wave (Figure 2.1) is called the classical wave equation in one dimension and is a linear partial differenti al equation. It tells us how the displacement ψ can change as a function of position a nd time and the function. The waves that do must satisfy both the initial conditions and the boundary condi tions, i.e., on how the wave is produced and what is happening on the ends of the st ring. For example, for a standing wave of string with length L held taut at two ends boundary conditions are
- 6. ψ (0, t) =0 and u (L, t) =0 for all values of t. As expected, different system will have different boundary conditio ns and hence different solutions. The One-Dimensional Wave Equation: What is Wave definition? Waves involve the transport of energy without the transport of matter. Learning Objectives To introduce the wave equation including time and position dependence In the most general sense, waves are particles or other media with wavelike properti es and structure (presence of crests and troughs). Figure 2.1 Translational (transverse) wave: n physics, a transverse wave is a wave that vibrates perpendicular to the direction of the wave or path of propagation. The simplest wave is the (spatially) one. dimensional sine wave with a varing amplitude A described by the equation: A (x, t) =Aosin(kx−ωt+ϕ)
- 7. Where; Ao is the maximum amplitude of the wave, maximum distance from the highest point of the disturbance in the medium (the crest) to the equilibrium point during one wave cycle. In Figure 2.1, this is the maximum vertical distance between the baseline and the wave. x is the space coordinate t is the time coordinate k is the wavenumber ω is the angular frequency ϕ is the phase constant. One can categorize “waves” into two different groups: traveling waves and stationary waves. Traveling Waves: a wave in which the particles of the medium move progressively in the direction of the wave propagation is called traveling wave Traveling waves, such as ocean waves or electromagnetic radiation, are waves that “move,” meaning that they have a frequency and are propagated through time and s pace. Another way of describing this property of “wave movement” is in terms of ener gy transmission – a wave travels, or transmits energy, over a set distance. The most important kinds of traveling waves in everyday life are electromagnetic waves, sound waves, and perhaps water waves, depending on where you live. It is difficult to anal yze waves spreading out in three dimensions, reflecting off objects, etc., so we begin with the simplest interesting examples of waves, those restricted to move along a lin e. Let’s start with a rope, like a clothesline, stretched between two hooks. You take o ne end off the hook, holding the rope, and, keeping it stretched fairly tight, wave you hand up and back once. If you do it fast enough, you’ll see a single bump travel alon g the rope:
- 8. Figure 2.2 A one-dimensional traveling wave at one instance of time t. This is the simplest example of a traveling wave. You can make waves of different sh apes by moving your hand up and down in different patterns, for example an upward bump followed by a dip, or two bumps. You’ll find that the traveling wave keeps the s ame shape as it moves down the rope. Taking the rope to be stretched tightly enoug h that we can take it to be horizontal, we’ll use its rest position as our xaxis (Figure 2. 2). The yaxis is taken vertically upwards, and we only wave the rope in an up-and- down way, so actually y(x,t)will be how far the rope is from its rest position at x at tim e t: that is, Figure 2.2 shows where the rope is at a single time t. We can now express the observation that the wave “keeps the same shape” more pr ecisely. Taking for convenience time t=0to be the moment when the peak of the wav e passes x=0, we graph here the rope’s position at t = 0 and some later times tt as a movie (Figure 2.3 Denoting the first function by y(x,0) =f(x), then the second y(x,t)=f(x−vt): it is the same function with the “same shape,” but just moved over by v t, where v is the velocity of the wave. Figure 2.3: A one-dimensional traveling wave at as a function of time. Traveling waves propagate energy from one spot to another with a fixed velocity v.
- 9. To summarize: on sending a traveling wave down a rope by jerking the end up and d own, from observation the wave travels at constant speed and keeps its shape, so th e displacement y of the rope at any horizontal position at x at time t has the form y(x,t)=f(x−vt) We are neglecting frictional effects in a real rope, the bump gradually gets smaller as it moves along. Standing Waves: In contrast to traveling waves, standing waves, or stationary waves, remain in a cons tant position with crests and troughs in fixed intervals. One way of producing a variet y of standing waves is by plucking a melody on a set of guitar or violin strings. When placing one’s finger on a part of the string and then plucking it with another, one has created a standing wave. The solutions to this problem involve the string oscillating i n a sine- wave pattern (Figure 2.4) with no vibration at the ends. There is also no vibration at a series of equally- spaced points between the ends; these "quiet" places are nodes. The places of maxi mum oscillation are antinodes. Figure 2.4: Animation of standing wave in the stationary medium What is the difference between a traveling wave and a standing wave? Travelling waves transport energy from one area of space to another, whereas standing waves do not transport energy. Also, points on a standing wave oscillate in phase, whereas on a travelling wave only points a wavelength apart oscillate in phase - the rest on that wavelength oscillate out of phase with the original point. Standing waves occur in confined spaces – in a microwave oven, on a violin string, they do not
- 10. transport energy from one place to another because they are confined. The waves interfere as they move about within the space to set up a series of nodes, or points of minimum vibration, and antinodes or points of maximum vibration. The most striking feature of standing waves is that they only occur for certain frequencies. Travelling waves on the other hand actually move from place to place, transporting energy. Travelling waves can have any frequency, and can undergo interference just like standing waves. A travelling wave could be set up by a bomb exploding in the open air. Summary: Waves which exhibit movement and are propagated through time and space. The tw o basic types of waves are traveling and stationary. Both exhibit wavelike properties and structure (presence of crests and troughs) which can be mathematically describ ed by a wavefunction or amplitude function. Both wave types display movement (up and down displacement), but in different ways. Traveling waves have crests and troughs which are constantly moving from one poin t to another as they travel over a length or distance. In this way, energy is transmitte d along the length of a traveling wave. In contrast, standing waves have nodes at fix ed positions; this means that the wave’s crests and troughs are also located at fixed i ntervals. Therefore, standing waves only experience vibrational movement (up and d own displacement) on these set intervals - no movement or energy travels along the length of a standing wave.