Traveling Waves and Standing Waves
We write that the pressure distribution p(x,t) = f(x − ct), where f is any function,
obeys the wave equation… It is called a traveling wave. A point onf(x − ct), for instance its maximum value, corresponds to a particular value of the argumentx − ct. To travel with the maximum value off(x − ct), as t increases,x must also increase in such a way as to keep x − ct constant. This means that the pressure distribution propagates to the right with speedc… Solutions p(x,t) = g(x + ct), where g is any function, also are solutions to the wave equation, corresponding to a wave propagating to the left.
We then discuss standing waves.
Standing waves such as p(x,t) = p cos(ωt) sin(kx) are also solutions to the wave equation… [This] standing wave… has nodes fixed in space where sin(kx) is zero… A standing wave can also be written as the sum of two sinusoidal traveling waves, one to the left and one to the right. Conversely, two standing waves can be combined to give a traveling wave.
Converting Traveling Waves to Standing Waves
IPMB includes a homework problem asking the reader to show analytically that two traveling waves combine to make a standing wave, and vice versa.
Problem 8. Use the trigonometric identity sin(a ± b) = sina cos b ± cos a sin b to show that a traveling wave can be written as the sum of two out-of phase standing waves, and that a standing wave can be written as the sum of two oppositely-propagating traveling waves.
Russ and I also include figures illustrating the difference between a traveling wave (our Fig. 13.4) and a standing wave (Fig. 13.5). To gain insight, however, nothing can replace a dynamic visualization. Fortunately, the internet is full of such visualizations. One appears in the Wikipedia article about standing waves. The Physics Hypertextbook also has traveling and standing wave animations.
This Youtube video shows trigonometry in action: the sum of two oppositely going traveling waves (blue wave propagating right, and green left) add to form a single standing wave (red).
I like the next video because it shows a traveling wave turning into a standing wave when it reflects off a boundary.
Here’s a nice video showing how standing waves can be created experimentally.
Finally, here’s a Flipping Physics video comparing standing and traveling waves. It’s a little corny, but I like it that way.
Originally published at http://hobbieroth.blogspot.com.