Find the solution of wave equation for a fixed string

with the boundary conditions

*u* (0,*t*) = *u* (*L,t*) = 0
(the string is fixed at the endponts). The initial displacement and velocity are given by

where

*f *(*x*) and

*g *(*x*) are some functions defined by the user, such that

Solution.

We will look for all periodic solutions in which the variables

*x* and

*t* are separated, i.e. in the form

Then

Substituting this into the wave equation, we obtain

Here the function on the left-hand side depends only on

*x*, whereas the function on the right-hand side depends only on

*t*.
This can happen if both sides of the equation are constant. Hence,

If the constant

*α* is positive, we can put

to get

with the general solution

Such solution cannot produce periodic functions in

*t*. Therefore, we get that the constant

*α* is negative:

.
Then our wave equation can be split into two ODEs:

Solving the first equation, we find that

where

*C*_{1} and

*C*_{2} are constants of integration.

Considering the boundary conditions for the fixed string, we set

Then

By setting

*C*_{2} ≠ 0 (otherwise we would get the trivial solution

*X* ≡ 0), we find that

*λL = πn* (

*n* is an integer).

Thus, the so-called

*eigenvalues* are

The corresponding

*eigenfunctions* are written as

For

*λ = λ*_{n} the second equation yields

Thus, we can write:

Here

*n* is a positive integer,

*A*_{n} and

*B*_{n} are arbitrary constants depending on the initial conditions.

Now we can combine the general solution of the wave equation as a linear combination of the particular solutions:

Assuming that the series is differentiable, we find that

Determine the constants

*A*_{n} and

*B*_{n} using the initial conditions:

As seen, we should expand the functions

*f* (*x*) and

*f* (*x*)
into the series based on the orthogonal system

.

By the formulas for Fourier coefficients,

Thus, the solution to the wave equation with the given boundary and initial conditions is given by the infinite series

where the coefficients

*A*_{n} and

*B*_{n} are defined by the formulas given above.

The first term

*u*_{1}(*x,t*) of the series is called
the

*fundamental tone*, the other terms

*u*_{n}(*x,t*)
are called

*harmonics*. The

*period* and

*frequency* of a harmonic are given by the formulas