Standing Waves in a Clamped String

We now demonstrate how Eq. (1-20) can be used to predict the nature of standing waves in a string. Suppose that the string is clamped at v = 0 and L. This means that the string cannot oscillate at these points. Mathematically this means that 

Conditions such as these are called boundary conditions. Our question is, What functions ij/ satisfy Eq. (1-20) and also Eq. (1-23) We begin by trying to find the most general equation that can satisfy Eq. (1-20). We have already seen that A sin 2nx/X) is a solution, but it is easy to show that A cosi2nx/X) is also a solution. More general than either of these is the linear combination  

There are two remarks to be made at this point. First, some readers will have notieed that other funetions exist that satisfy Eq. (1-20). These are Aexp 27iix/X) and Aexp(—27rix/X), where i = The reason we have not included these in 

This means that any trigonometric function may be expressed in terms of such exponentials and vice versa. Hence, the set of trigonometric functions and the set of exponentials is redundant, and no additional flexibility would result by including exponentials in Eq. (1-24) (see Problem 1-1). The two sets of functions are linearly dependent. 

The second remark is that for a given A and B the function described by Eq. (1-24) is a single sinusoidal wave with wavelength X. By altering the ratio of to i , we cause the wave to shift to the left or right with respect to the origin. If = 1 and i = 0, the wave has a node at x = 0. If = 0 and 5 = 1, the wave has an antinode at x = 0. 

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Mathematically, this is precisely the same problem we have already solved in Chapter 1 for the standing waves in a clamped string. The solutions are... 

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