Problem X8
The movie below shows an animation of a plot of a traveling wave given by the equation: \(y(z,t) = A\cos(kz+\omega t)\) where \(A = 5.0\), \(k = \pi/2\) and \(\omega = \pi/4\). The movie shows the wave from time \(t = 0\) to \(t = 4.0\).- Click on the play button a few times and watch the evolving wave until
you are comfortable with the idea of this as a "traveling wave'' -- i.e.,
that the wave is propagting smoothly to the left.
- The graphs contained in the movie were generated by "Wolfram Alpha'' (www.wolframalplha.com),
which is a really nice web site that enables you to solve a wide range of mathematical equations and
also make plots. It is like the program Mathematica, except that you don't actually have to buy
Mathematica -- you can do everything online.
Click here to open up a separate tab (or window) that will take you to Wolfram Alpha. Once there, it should automatically plot the wave at the time \(t = 0.0\).
Change the time \(t\) to 1.0 (click in the yellow box and change "t = 0.0" to "where t = 1.0" and hit enter. It should re-plot the wave at the new time. Do this for \(t = 2.0\) as well.
Sketch these plots on your paper, and then use Wolfram Alpha to answer the questions below.
- Choose one of the peaks of the wave. Where is this peak (i.e., at what value of \(z\)) at \(t = 0.0\)? Where is the same peak at \(t = 1.0\)? Where is the same peak at \(t = 2.0\)?
- In which direction is the wave traveling? Does your answer agree with what you know from looking at the equation for the traveling wave?
- Use your answers to part (c) to calculate the speed at which the wave is propagating. Does your answer agree with the speed determined from \(\omega/k\)?