Friday, March 20, 2020

Traveling Waves and Standing Waves

Section 13.2 of Intermediate Physics for Medicine and Biology discusses waves. Russ Hobbie and I note that two solutions to the wave equation exist: traveling waves and standing waves.

Traveling Waves

We write that the pressure distribution p(x,t) = f(xct), where f is any function,
obeys the wave equation... It is called a traveling wave. A point on f(xct), for instance its maximum value, corresponds to a particular value of the argument xct. To travel with the maximum value of f(xct), as t increases, x must also increase in such a way as to keep xct constant. This means that the pressure distribution propagates to the right with speed c... 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.

Standing Waves

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) = sin a 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.

Visualizations

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).

Two traveling waves adding to form a standing wave.

I like the next video because it shows a traveling wave turning into a standing wave when it reflects off a boundary.

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.


Standing waves created experimentally on a string fastened at both ends.

Finally, here’s a Flipping Physics video comparing standing and traveling waves. It’s a little corny, but I like it that way.

A lecture about waves from Flipping Physics.

Enjoy!

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