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The Wave Equation

Introduction to Wave Equations

The wave equation is a hyperbolic partial differential equation (PDE). Many physical phenomena can be modeled with wave equations, including acoustics, structures, gravity waves, along with countless others. A ripple on a pond, for example the one seen in Figure 1↓, is well-described by the wave equation.

In general, the wave equation may be written as shown in Eqn. 1↓.

(1) (∂²u(x, t))/(∂t²) = C²∇²u

Figure 1 The waves on the surface of a pond can be described by the wave equation.

Here, C is a (usually known) constant that represents the speed of the propagation of a wave. A unique solution requires initial conditions and boundary conditions.

It is possible to examine the wave equation in one spatial dimension. For instance motion of a pulse in a string or the torsional deformation of a beam can be modeled with one dimensional wave equation:

(2) (∂²u(x, t))/(∂t²) = C²(∂²u(x, t))/(∂x²)

If this equation models the motion of a pulse in a string then u is displacement,t is time, x is axial coordinate, (Fig. 2↓). One could tell that C is a velocity term by examining the dimensions of the equation, and realizing that they must balance:

(3) (L²)/(t²) = C²*1

C has units of velocity, (L)/(t). Clearly the analysis does not change if one were to examine more than one dimension, since ∇²u is dimensionless.

C is the speed at which a disturbance propagates through a domain, and therefore C is effectively the “speed of sound”. Indeed, when used to model acoustics, C is the speed of sound.

Figure 2 An example of the displacement of a string at a given time.