10 - One-Dimensional Wave Equation

Summary

10.1 Homogeneous Equation: D’Alembert's Formula

The Cauchy problem or the initial value problem (IVP) for the one-dimensional wave equation is

Here, c > 0 is a constant, called the speed of propagation.

This equation models many real-world problems: small transversal vibrations of a string, longitudinal vibrations of a rod, electrical oscillations in a wire, torsional oscillations of shafts, oscillations in gases and so on. The wave equation is a prototype of hyperbolic equations. It has two real and distinct characteristics, given by x ± ct = constant. Using the characteristic variables ξ = x + ct and τ = x − ct, we have and hence,. Its general solution is therefore given by

Using the initial conditions in equation (10.1.1) to determine F and G results in the D’Alembert's formula for the solution u:

More generally, data can be prescribed on a non-characteristic curve: t = φ(x), with some conditions on φ, but an explicit formula for the solution may not be found. We state the foregoing in the following theorem:

Theorem 10.1. Suppose the initial conditions satisfy that u₀ ∈ C²(ℝ) and u₁ ∈ C¹(ℝ). Then, the function u given by the D’Alembert's formula (10.1.2) is a C² function in ℝ × [0, ∞) and satisfies equation (10.1.1) and hence unique.

10.2 Domain of Dependence and Other Concepts

The following observations based on the D’Alembert's formula (10.1.2) may be made. The value of the solution u at (x, t), t > 0 depends on the values of the initial data only in the interval [x − ct, x + ct] on the initial line t = 0, that is, the x-axis. This is referred to as the domain of dependence (of the solution) at (x, t). Similarly, a point y on the initial line can influence the value of u for some t > 0, only in a line segment. This is referred to as the range of influence of the point (y, 0). These are illustrated in Figure 10.1.