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Study Guide

Now we've seen waves that pack up their bags and go, but what about waves that are a little more unadventurous? Waves that stay in one place are called **standing waves**. They're perfectly content to sit at home and watch TV. Standing waves still oscillate up and down, but they do it in one place, like a person jumping rope.

In a standing wave, there are set points that never move despite all the wiggling going on around them. These are called the wave's **nodes**. Halfway between the nodes are the points that move the most—swinging from the wave's crest to its trough and back again. These points are **antinodes**.

This may seem like a strange phenomena at first—waves that don't move!—but standing waves are all around you, especially if you've ever played a musical instrument or listened to a band. While the screaming crowd in an arena may start a traveling wave, the guitars played by the band are using standing waves to make music: the guitar's strings are fastened to the instrument on each end (nodes), and plucked in the middle, creating a wave that wiggles the string back and forth. The faster the string vibrates, the higher the frequency of the wave, the higher the note made by the guitar, and the more nodes and antinodes that appear in the string.

We can find the position of any point along the standing wave at any time by combining two trig functions:

Here, *y* gives the displacement from equilibrium, at point *x* and time *t*. The sine wave represents the shape of the standing wave at *t* = 0. The cosine wave represents how that wave shape moves up and down with time, just as in the case of traveling waves. The product of the two waves gives us something that fully describes our standing wave.

We can find the nodes of the standing wave by looking at this formula wherever *y* = 0 regardless of *t*—that is:

Dividing by the cosine term on each side gives:

Sine waves are equal to zero when their argument is equal to multiples of π (0, π, 2π, etc.), so we know our nodes must be located at *x* = 0, , *x* = λ, , and so on. This should look familiar—"nodes" are really just the points where sine crosses zero.

This idea of standing waves is actually a bit of a simplification. Standing waves are usually caused by two traveling waves moving in opposite directions—say a wave moving to the right on string (blue), and another one of the same size moving to the left (red). Since these waves are moving through the same string, they **interfere** with each other.

When waves interfere, we can find the resulting wave (whether it still travels or is standing) by simply adding the amplitudes of the interfering waves together. If the two waves are in phase with each other, their crests will match up, and the resulting wave is much bigger than either of its two components—this is called **constructive interference**, where the waves build on one another.

However, if the interfering waves are out of phase, the negative amplitude of one will try to cancel out the positive amplitude of the other, something called **destructive interference**. This will make the resulting wave much smaller than either of its components—or maybe cancel it out entirely. Think pass interference negating a big football play.

When the two interfering waves have exactly the same amplitude, frequency, and wavelength, they'll combine to make a standing wave—a bit of coincidental mathematical witchcraft responsible for everything from pipe organs to lasers.

**Brain Snack**

Perhaps the most impressive display of standing waves comes from the way air moves in the metal tubes of a pipe organ, a giant instrument that has to have a building built around it. Air resonates back and forth in the tube, creating standing waves that echo throughout the chamber that houses the organ.

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