Wave Equation - Introduction

Introduction

Wave equations are examples of hyperbolic partial differential equations, but there are many variations.

In its simplest form, the wave equation concerns a time variable t, one or more spatial variables x₁, x₂, …, x_n, and a scalar function u = u (x₁, x₂, …, x_n; t), whose values could model the height of a wave. The wave equation for u is

where is the (spatial) Laplacian and where c is a fixed constant.

Solutions of this equation that are initially zero outside some restricted region propagate out from the region at a fixed speed in all spatial directions, as do physical waves from a localized disturbance; the constant c is identified with the propagation speed of the wave. This equation is linear, as the sum of any two solutions is again a solution: in physics this property is called the superposition principle.

The equation alone does not specify a solution; a unique solution is usually obtained by setting a problem with further conditions, such as initial conditions, which prescribe the value and velocity of the wave. Another important class of problems specifies boundary conditions, for which the solutions represent standing waves, or harmonics, analogous to the harmonics of musical instruments.

To model dispersive wave phenomena, those in which the speed of wave propagation varies with the frequency of the wave, the constant c is replaced by the phase velocity:

The elastic wave equation in three dimensions describes the propagation of waves in an isotropic homogeneous elastic medium. Most solid materials are elastic, so this equation describes such phenomena as seismic waves in the Earth and ultrasonic waves used to detect flaws in materials. While linear, this equation has a more complex form than the equations given above, as it must account for both longitudinal and transverse motion:

where:

• and are the so-called Lamé parameters describing the elastic properties of the medium,
• is the density,
• is the source function (driving force),
• and is the displacement vector.

Note that in this equation, both force and displacement are vector quantities. Thus, this equation is sometimes known as the vector wave equation.

Variations of the wave equation are also found in quantum mechanics, plasma physics and general relativity.