# Standing Waves

Waves are oscillations in an elastic medium:

• your own vocal cords (the medium) vibrating as air is forced over them by your lungs;
• a stretched string (the medium) on a musical instrument vibrating as it is bowed, hammered or plucked;
• pressure oscillations in a column of air (the medium) in a wind instrument, organ pipe or your own oral and nasal cavities;
• radio waves whose electric and magnetic fields oscillate between buildings on a downtown street (here the medium is the air, although the radio waves could vibrate between the buildings even if there were no air).
In each case the medium has an equilibrium state, and when displaced or otherwise perturbed from that state, experiences a force which tends to restore it to equilibrium. For small perturbations, the restoring force is proportional to the displacement and the medium becomes a simple harmonic oscillator.

If the ends of the medium are fixed at both ends, as occurs in your vocal cords, the stringed instrument or the radio waves between buildings, the wave is described by

y(x,t) = A sin(kx) cos(ωt + δ),
where
k = n π / l,
n is an integer and l is the length of the medium. If the medium extends from x = 0 to x = l, the choice of sin (kx) for the x dependence guarantees that the displacement will be zero at the left end. The restriction on k does the same for the right end, by making sure that the argument of the sin function is a multiple of π at that end.

If one end of the medium is fixed but the other is allowed to freely vibrate, as in the wind instrument, organ pipe or your nasal cavity, the wave is described by the same equation, but with

k = n π / 2 l,
where now n is restricted to be an odd integer. As before, this guarantees that the left end is fixed, but now forces the sin (kx) to be 1 or -1 at the right end, so that the cosine function determines the displacement there: this allows the right end to oscillate freely.

If both ends of the medium are allowed to freely vibrate, such as in your oral cavity, the wave is described by

y(x,t) = A cos(kx) cos(ωt + δ),
where k and n are the same as when both ends are fixed. Here the choice of cos (kx) for the x dependence gives 1 at the left end, and the restriction on k means that the right end will be either 1 or -1, so that again the cos (ωt) determines the displacement and both ends may now vibrate freely.
n = 1 is called the fundamental or first harmonic,

n = 2 is the first overtone or second harmonic, etc.

The mode is equal to the number of half wavelengths which fit in the medium if the boundary conditions are the same at each end, or quarter wavelengths if they are different.

For each mode, there are a number of nodes: values of x for which y is always 0. These nodes are located at integer multiples of π / k for sine waves and odd integer multiples of π / 2 k for cosine waves. For fixed boundary conditions at both ends, a wave of mode n has n + 1 nodes (counting the nodes at each end). For fixed boundary conditions at one end and free at the other, a wave of mode n has (n + 1) / 2 nodes. If the boundary conditions are free at both ends, a wave of mode n has n nodes.

c = λ ν = ω / k
which is called the speed of the wave: the speed of sound (about 343 m/s in air at 20 degrees Celsius) or the speed of light (2.998 * 108 m/s in vacuum), which applies to radio waves as well.
For electromagnetic waves, c is 1 / Sqrt (μ ε). The relationship between the speed of light and the electrical permittivity and magnetic permeability was one of the first clues that light is an electromagnetic wave.
In the examples with which we began, the speed of the wave was quite variable: in your vocal cords and the stringed instrument, it depends on the tension and density of the medium and can be varied at will. The speed of vocal cord waves can vary from 3 to about 19 m/s; for guitar strings it is typically from 107 to 427 m/s.

The frequency of the wave is a function of this speed, the length of the medium, the mode number and the boundary conditions. If the boundary conditions are the same at both ends, the frequency of a wave of mode n is

νn = n c / 2 l.
If the boundary conditions are different at the ends of the medium, the frequency of a wave of mode n is
νn = n c / 4 l.
Since λ = c / ν, we see that the factor of 2 or 4 in the denominator corresponds to the half or quarter wavelengths which fit in the medium of length l. The units of ν are 1 / s, also called Hertz (abbreviated Hz).

### Superposition

m a = - k x,
(where this k is the spring constant,) any superposition of waves of different modes is given by simply adding the waves together:
yn1,n2(x,t) = sin (k1x) cos (ω1t) + sin (k2x) cos (ω2t).
The following applet will play a superposition of two frequencies. The default frequency is 440 Hz, which is "concert A" below middle C. The frequencies are adjustable from approximately four octaves below 440 (28 Hz) to four octaves about 440 (7040 Hz). Be sure to check your speaker or headphone volume before playing the 3 second sample!

You need a Java-capable browser to be able to use the applets. If they do not work with your Windows system, download the Java VM (Virtual Machine) for your version of Windows at the download section at java.sun.com.

Try leaving frequency 1 at 440 and setting frequency 2 to 880; 660; 587; 550; and 528. What do you notice about these particular superpositions? Do you recognize them? Why do you think they appeal (or not) to you as they do?
In this applet, the superposition as a function of time is
y(t) = sin (ω1 t) + sin (ω2 t),
which with the trigonometric identity
sin (a) + sin (b) = 2 cos ((a - b) / 2) sin ((a + b) / 2)
is equivalent to
y(t) = 2 cos ((ω1 - ω2) t / 2) sin ((ω1 + ω2) t / 2).
This superposed wave changes with two separate frequencies: the cosine function defines a sort of "envelope" inside of which the sine function oscillates. The envelope amplitude has two maxima for each cycle, and these make the sine oscillation appear to waver in "beats" with beat frequency Δ ω/2π. This waveform is a graph of a single beat:

You need a Java-capable browser to be able to use the applets. If they do not work with your Windows system, download the Java VM (Virtual Machine) for your version of Windows at the download section at java.sun.com.

You need a Java-capable browser to be able to use the applets. If they do not work with your Windows system, download the Java VM (Virtual Machine) for your version of Windows at the download section at java.sun.com.

The next section discusses the reflection, refraction and diffraction of traveling waves.