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).

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.

Each value of n determines the **mode of vibration**:

n = 1 is called theThe 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.fundamentalor firstharmonic,n = 2 is the first

overtoneor second harmonic, etc.

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.

For each type of wave there is a characteristic velocity

c = λ ν = ω / kwhich is called the speed of the wave: the

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

νIf the boundary conditions are different at the ends of the medium, the frequency of a wave of mode n is_{n}= n c / 2 l.

νSince 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_{n}= n c / 4 l.

Since n is a free parameter in our discussion, it is obvious that a medium of a given length can oscillate at any of an infinite number of frequencies. Because our original equation for the harmonic oscillator is **linear**:

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:

yThe 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!_{n1,n2}(x,t) = sin (k_{1}x) cos (ω_{1}t) + sin (k_{2}x) cos (ω_{2}t).

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 (ωwhich with the trigonometric identity_{1}t) + sin (ω_{2}t),

sin (a) + sin (b) = 2 cos ((a - b) / 2) sin ((a + b) / 2)is equivalent to

y(t) = 2 cos ((ωThis 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_{1}- ω_{2}) t / 2) sin ((ω_{1}+ ω_{2}) t / 2).

This applet is a demonstration of the ideas we have discussed in this section, and should prepare you for our next set of problems. With it you can alter the boundary conditions of a string and superpose waves of differing modes and phase angles.

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

©2009, Kenneth R. Koehler. All Rights Reserved. This document may be freely reproduced provided that this copyright notice is included.

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