Wave Equation--1-Dimensional

The one-dimensional wave equation is given by

In order to specify a wave, the equation is subject to boundary conditions

and initial conditions

The one-dimensional wave equation can be solved exactly by d'Alembert's solution, using a Fourier transform method, or via separation of variables.

d'Alembert devised his solution in 1746, and Euler subsequently expanded the method in 1748. Let

By the chain rule,

The wave equation then becomes

Any solution of this equation is of the form

where and are any functions. They represent two waveforms traveling in opposite directions, in the negative direction and in the positive direction.

The one-dimensional wave equation can also be solved by applying a Fourier transform to each side,

which is given, with the help of the Fourier transform derivative identity, by

where

This has solution

Taking the inverse Fourier transform gives

where

This solution is still subject to all other initial and boundary conditions.

The one-dimensional wave equation can be solved by separation of variables using a trial solution

This gives

So the solution for is

Rewriting (25) gives

so the solution for is

where . Applying the boundary conditions to (◇) gives

where is an integer. Plugging (◇), (◇) and (29) back in for in (◇) gives, for a particular value of ,

The initial condition then gives , so (31) becomes

The general solution is a sum over all possible values of , so

Using orthogonality of sines again,

where is the Kronecker delta defined by

gives

so we have

The computation of s for specific initial distortions is derived in the Fourier sine series section. We already have found that , so the equation of motion for the string (◇), with

is

where the coefficients are given by (◇).

A damped one-dimensional wave

given boundary conditions

initial conditions

and the additional constraint

can also be solved as a Fourier series.

where