# Wave equation

The wave equation is an important partial differential equation which generally describes all kinds of waves, such as sound waves, light waves and water waves. It arises in many different fields, such as acoustics, electromagnetics, and fluid dynamics. Variations of the wave equation are also found in quantum mechanics and general relativity.

Historically, the problem of a vibrating string such as that of a musical instrument was studied by Jean le Rond d'Alembert, Leonhard Euler, Daniel Bernoulli, and Joseph-Louis Lagrange.

The general form of the wave equation for a scalar quantity $\displaystyle u$ is:

$\displaystyle { \partial^2 u \over \partial t^2 } = c^2 \nabla^2u$

Here c is usually a fixed constant, the speed of the wave's propagation (for a sound wave in air this is about 330 m/s, see speed of sound). For the vibration of string this can vary widely: on a spiral spring (a slinky) it can be as slow as a meter per second. If c however is changing in function of the wavelength, it shall be replaced by the phase velocity:

$\displaystyle v_\mathrm{p} = \frac{\omega}{k}.$

Also note that a wave may be superimposed onto another movement (for instance sound propagation in a moving medium like a gas flow). In that case the scalar u will contain a Mach factor (which is positive for the wave moving along the flow and negative for the reflected wave).

u = u(x,t), is the amplitude, a measure of the intensity of the wave at a particular location x and time t. For a sound wave in air u is the local air pressure, for a vibrating string it is the physical displacement of the string from its rest position. $\displaystyle \nabla^2$ is the Laplace operator with respect to the location variable(s) x. Note that u may be a scalar or vector quantity.

The general solution to the one dimensional scalar wave equation was derived by d'Alembert as: $\displaystyle u(x,t) = F(x-ct) + G(x+ct)$ where F and G are arbitrary functions, corresponding to a forward-traveling wave, and a back-ward traveling wave, respectively. To determine F and G one must consider the two initial conditions:

$\displaystyle u(x,0)=f(x)$
$\displaystyle u_t(x,0)=g(x)$

Thus the d´Alembert formula becomes:

$\displaystyle u(x,t) = \frac{f(x-ct) + f(x+ct)}{2} + \frac{1}{2c} \int_{x-ct}^{x+ct} g(s) ds$

In the classical sense if $\displaystyle f(x) \in C^k$ and $\displaystyle g(x) \in C^{k-1}$ then $\displaystyle u(t,x) \in C^k$ .

The wave equation in the one dimensional case can be derived in the following way: Imagine an array of little weights of mass m interconnected with slinkies of length h . The slinkies have a stiffness of k :

File:Array of masses.png

Here u (x) measures the distance from the equilibrium of the mass situated at x. The equation of motion for the weight at the location x+h is:

$\displaystyle m{\partial^2u(x+h,t) \over \partial t^2}= k[u(x+2h,t)-u(x+h,t)-u(x+h,t)+u(x,t)]$

where the time-dependence of u(x) has been made explicit.

If the array of weights consists of N weights spaced evenly over the length L = N h of total Mass M = N m, and the total stiffness of the array K = k/N we can write the above equation as:

$\displaystyle {\partial^2u(x+h,t) \over \partial t^2}={KL^2 \over M}{u(x+2h,t)-2u(x+h,t)+u(x,t) \over h^2}$

Taking the limit N $\displaystyle \rightarrow \infty$ , h$\displaystyle \rightarrow 0$ one gets:

$\displaystyle {\partial^2 u(x,t) \over \partial t^2}={KL^2 \over M}{ \partial^2 u(x,t) \over \partial x^2 }$

(K $\displaystyle L^2$ )/M is the square of the propagation speed in this particular case.

The basic wave equation is a linear differential equation which means that the amplitude of two waves interacting is simply the sum of the waves. This means also that a behavior of a wave can be analyzed by breaking up the wave into components. The Fourier transform breaks up a wave into sinusoidal components and is useful for analyzing the wave equation.

The one-dimensional form can be derived from considering a flexible string, stretched between two points on a x-axis. It is

$\displaystyle { \partial^2 u \over \partial t^2 } = c^2 { \partial^2 u \over \partial x^2 }$

The general solution to this is a Fourier series: an infinite sum of sine and cosine waves. If the domain of the equation is infinite with no boundary conditions, then D'Alembert's method can be used to solve it.

In two dimensions, expanding the Laplacian gives:

$\displaystyle { \partial^2 u \over \partial t^2 } = c^2 \left ({ \partial^2 u \over \partial x^2 } + { \partial^2 u \over \partial y^2 } \right )$

An example of the solution to the 2-D wave equation is the motion of a tightly-stretched drumhead. In this case, rather than sinusoids, the solutions are combinations of Bessel functions.

The wave equation is the prototypical example of a hyperbolic partial differential equation.

More realistic differential equations for waves allow for the speed of wave propagation to vary with the frequency of the wave, a phenomenon known as dispersion. Another common correction is that, in realistic systems, the speed also can depend on the amplitude of the wave, leading to a nonlinear wave equation:

$\displaystyle { \partial^2 u \over \partial t^2 } = c(u)^2 \left ({ \partial^2 u \over \partial x^2 } + { \partial^2 u \over \partial y^2 } \right )$

In three dimensions, for instance to study the propagation of sound in a space:

$\displaystyle { \partial^2 u \over \partial t^2 } = c^2 \left ({ \partial^2 u \over \partial x^2 } + { \partial^2 u \over \partial y^2 } + { \partial^2 u \over \partial z^2 }\right )$

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:

$\displaystyle \rho{ \ddot \bold{u}} = \bold{f} + ( \lambda + 2\mu )\nabla(\nabla \cdot \bold{u}) - \mu\nabla \times (\nabla \times \bold{u})$

where:

• $\displaystyle \lambda$ and $\displaystyle \mu$ are the so-called Lamé moduli describing the elastic properties of the medium,
• $\displaystyle \rho$ is density,
• $\displaystyle \bold{f}$ is the source function (driving force),
• and $\displaystyle \bold{u}$ is displacement.

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