Position and time plots learning objective
- 2. This learning object aims to explore the relationship between position plots and time plots. I initially struggled to comprehend the distinction between the two plots. However, I was able to reach an understanding after some research and practice. Hopefully my learning objective can help other students gain a better understanding of this topic as well!
- 3. Sine wave pattern which continues to move in an uninterrupted fashion until it encounters another wave along the medium or a boundary with another medium http://teal-blog.s3.amazonaws.com/2014/04/water_ripples_in_black_and_white_by_jeremyfe-d548l72.jpg
- 4. A stone thrown into a pond will form concentric sequence of circular waves. Travelling waves are produced in all directions from the source (stone). If a duck is floating on the water, it will move up and down as the wave passes, exhibiting simple harmonic motion.
- 5. For travelling harmonic wave in increasing x direction: For travelling harmonic wave in decreasing x direction: Where … • D(x,t) = displacement (m) • A= amplitude (m) • K= wave number (rad/m) • X=distance travelled (m) • ω = angular frequency (rad/s) • t= time (s) • Ф= phase constant (rad) • λ= wavelength (m) • T= period (s)
- 6. Looking at the formulas listed on the previous slide we understand that plotting D(x,t) would result in a three dimensional graph like this one… Instead, we choose to plot just one variable at a time, simplifying wave analysis
- 7. Here we keep time fixed at t=t0 to plot D(x, t=t0) as function of x only This graph depicts the displacement of every section of the medium at a given time This plot can be used to determine amplitude and wavelength of a wave
- 8. Here we keep position fixed at x=x0 to plot D(x=x0, t) as function of t only This graph demonstrates how the displacement of one section of the medium varies with time This plot can be used to determine amplitude, period and frequency of a wave
- 9. Imagine the duck was resting 1.0m from the point where the pebble hit the water so that the time plot below represents the displacement of the duck as a function of time at x=1.0m (wave is travelling along x-axis in positive x direction). The position plot below represents a cross section of the concentric sequence of circular waves formed by the pebble at t=1.0s. Using this information, determine the a) wavelength, b) speed, c) phase constant, and d) displacement equation for the wave. To answer these questions you may have to draw upon previous knowledge of simple harmonic waves.
- 10. x(m) 1.0m Used for position plot Used for time plot
- 12. To determine wavelength we look at the position plot. Looking at the position plot we can clearly see that the wavelength is λ=4m. λ=4m
- 13. To determine the speed of the wave we will first have to determine the frequency. ◦ Period and frequency are determined using the time plot. ◦ Looking at the time plot we see that the period is T=6s, so f=1/T=1/6s=(1/6)Hz Speed is v=fλ=(1/6s)(4m)= 0.6667m/s T= 6s-0s=6s
- 14. We have already determined that λ=4m and T=6s. From both the position and time plots we see that amplitude is A=0.10m. From here it would be most appropriate to use the equation for the displacement of a travelling harmonic wave in the positive x direction: Plugging in the known values of A, λ, and T we get: ◦ D(x,t)=(0.10m)sin(2πx/4-2πt/6+Ф) To solve for Ф, we need to substitute known values for x or t into the equation. Form the time plot we know that D(1,0)=0.025m so… ◦ D(1,0)=(0.10m)sin(2πx/4-2πt/6+Ф) ◦ 0.025=(0.10m)sin(2π/4+Ф) ◦ 0.25=sin(π/2+Ф) Since the sine graph differs from the cosine graph by π/2rad… ◦ 0.25=sin(π/2+Ф)=cos(Ф) ◦ Ф=1.3rad, Ф=5.0rad Plug in values to confirm that Ф=5.0rad will result in an equation corresponding to the graphs provided -->>>> MORE DETAIL
- 15. In steps a),b), and c) we determined all the information needed to fill in the equation for the displacement of the wave as function of time and position. Knowing that A= 0.10m, λ=4m, T=6s, and Ф=5.0rad we can solve for the displacement equation ◦ D(x,t)=(0.10m)sin(πx/2-πt/3+5.0)