PDE notes: Wave Equation

Michael Liu

4 years ago

2.1. Introduction

The one-dimensional wave equation for small displacement of a perfectly elastic string of length with no frictional forces and no restoring forces is

for , , where . We assume both tension and linear density to be constant. Tension has units of and linear density has units of , so has the units of , which is also the unit of speed. Thus, turns out to be the velocity of wave propagation along the string. The one-dimensional wave equation models sound waves, water waves, vibrations in solids, and longitudinal or torsional vibrations in a rod, among other things.

Since the PDE in (89) contains the second time derivative, two initial conditions are required. The initial conditions usually take the follwing form:

Typical boundary conditions are of the same form as thus given in the discussion of one-dimensional heat equation. For homogeneous Dirichlet conditions,

for . The end ends of the vibrating strings are fixed.

For homogeneous Neumann conditions,

for . These conditions are be achieved, for example, by attaching the ends of the string to a frictionless sleeve that moves vertically.

2.2. Solution by separation of variables

We now solve the one-dimensional wave equation with homogeneous Dirichlet boundary conditions. The following problem is defined for and :

Solution Since both the PDE and the boundary conditions are linear and homogeneous, the method of separation of variables is attempted. We look for special product solutions of the form:

Substitute (99) into (94) yields

Divide both sides by separates the variables:

where is the separation constant. This implies that

We consider (103) first since it has a complete set of boundary conditions. Letting , then . So . Using results from the heat equation, we know that and yields trivial solutions. For , the general solution is

Apply and we arrive at . Consider , we arrive at . If , we get a trivial solution. Therefore, we get a nontrivial solution if and only if

This means that

The eigenvalues are

The eigenfunctions corresponding to the eigenvalues are

The time-dependent part of the solution is

for . Thus, for , each of the functions

is a solution to the PDE and satisfies the boundary condition. By superposition principle, we can solve the initial value problem by considering a linear combinations of all product solutions:

The initial conditions in (97) and (98) are satisfied if,

We can consider the fact that satisfies the following orthogonality relation:

Multiply (112) by and integrating from to yields

Solving for yields,

Multiply (113) by and integrating from to yields

Solving for yields,

Therefore, the PDE with homogeneous Dirichlet boundary conditions has a simple explicit solution.

The product solutions are also called the normal modes of vibration. The coefficients of inside the product solutions, namely , are called the frequencies. The fundamental mode of the string has a frequency of . The th overtone is just the th integral multiple of the fundamental.

We now consider the motion of a vibrating string governed by the homogeneous Neumann boundary conditions for the wave equation:

Solution Again, we use the method of separation of variables. We look for production solutions of the form:

with

The general solution for (126) is

We also need

Apply and we arrive at . Apply and we arrive at . Since yields a trivial solution, we get a nontrivial solution if and only if