Problem X10
The movie below shows an animation of a plot of a standing wave given by the equation: \(y(x,t) = A\sin(kx)\cos(\omega t)\) where \(A = 3.0\), \(k = \pi/2\) and \(\omega = \pi/4\). The movie shows the wave from time \(t = 0\) to \(t = 16\).- Click on the play button a few times and watch the evolving wave until you are comfortable
with the idea of this as a “standing wave.”
- Determine the wavelength of the standing wave from the movie. Does your value agree with what you would expect from the equation for \(y(x,t)\)?
- Click here to go to Wolfram Alpha where you can manipulate the plots, like you did for problems X1 and X2 last week. Vary the times for the plots, and determine the period of the standing wave. Does your value agree with what you would expect from the equation for \(y(x,t)\)?
- Write an equation for a standing wave in the same system (with the same wave speed parameter) with twice the wavelength as that from parts (a) - (c).
- A standing wave doesn't have to be a pure, single normal mode. There can be more than one normal mode superposed. The movie
below shows a standing wave composed of the above standing wave mode and the next lowest frequency mode.
You should notice that the ends at \(x = -4\) and \(x = +4\) are still nodes, even with two waves superposed. Also, there are moments during the movie where you can see the pure modes peeking through.
Click here to get to Wolfram Alpha to see the command that made these plots. Play around with the time to get an appreciation for what this superposition state is doing.
We will see superpositions like this much more in the next unit on quantum mechanics.