Boundary Conditions

The standing wave solution to the wave equation is of the form

y = A sin (k x) cos (w t) .

Consider a wave on a string (or a sound in a tube) of fixed length L. By altering the "boundary conditions" at the ends, we can obtain specific "natural" or "resonant" frequencies which are quantized. We do this by either fixing or leaving loose one or both ends of the string (closing or opening each end of the tube). We will examine each type of boundary condition in turn. Note that y is zero at x = 0; this is a boundary condition chosen implicitly by choosing sine for the x dependence.

By forcing y to be 0 at x = L, we obtain the "closed - closed" boundary condition: both ends of the string are fixed (both ends of the tube are closed). But if y must be zero at x = L for any value of t, the wave number will be restricted to only the values

k = n p / L

(for integers n). Each value of n is a "mode"; n = 1 is called the "fundamental" or "first harmonic", n = 2 is the "first overtone" or "second harmonic", etc. Values of x for which y is always zero are called "nodes". The frequencies of the various harmonics are

n n = n c / 2 L .

If we let the string be free at x = L (leave the tube open at one end), we have the "closed - open" boundary conditions. Now the only allowed values are

k = n p / 2 L,

but only for odd integers n. The harmonic frequencies for the closed - open boundary condition are

n n = n c / 4 L.

Finally, we have the "open - open" boundary conditions. For this we must choose cosine dependence on x, so that

y = A cos (k x) cos (w t).

The allowable wave numbers and harmonics are formally the same as the closed - closed case.

Use the "Sound Modes" Mathematica notebook or the "Waves" Java program to experiment with the effects of different boundary conditions, as well as wave superposition and phase differences.

Boundary conditions also affect travelling waves when the medium changes in some way. For example, a wave pulse which is travelling along a string will be reflected back along the string when it reaches the end. If the end is fixed, the reflection will be inverted, while if the end is free, the reflection will maintain its former amplitude. Similarly, if two strings of different mass per unit length or stiffness are joined and a wave hits the boundary, there will be both a transmitted and a reflected wave. If the second string is denser, the reflection will be inverted.

Problems

The next section is about the Doppler Effect.



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1996, Kenneth R. Koehler. All Rights Reserved. This document may be freely reproduced provided that this copyright notice is included.

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