On this page we discuss some properties of the wave equation. To see how this equation is derived in the various cases, see the dedicated pages:

• Wave equation in elastic media
• Wave equation in a bar
• Wave equation in a chain of oscillators
• Wave equation in a string
• Wave equation in membranes
• Wave equation for sound
• Wave equation in water

Description of the equation

Consider the classical wave equation

${\displaystyle \frac{{\partial }^2}{\partial x^2}\psi (x,t)=\frac{1}{c^2}\frac{{\partial }^2}{\partial t^2}\psi (x,t)}$

Let us look at its elements:

1. The function ${\displaystyle \psi (x,t)}$ is called the wave function. It is the quantity that describes the wave in space (in this case only the coordinate x, but in general also y and z) and in time (coordinate t). It can stand for many different physical quantities depending on the type of wave. What all cases have in common is that it is a function of space and time.
2. A different symbol is used for the derivative and for infinitesimal increments: ${\displaystyle \partial \psi }$ instead of ${\displaystyle d\psi }$. This is precisely because ${\displaystyle \psi }$ is a function of several coordinates, and the new symbol indicates that the variation is taken with respect to one coordinate only, keeping the others fixed.

For example, the expression:

${\displaystyle \frac{\partial \psi (x,t)}{\partial x}}$

means that we calculate the variation of ${\displaystyle \psi (x,t)}$ with respect to the position x at a particular fixed value of the time t, while the expression

${\displaystyle \frac{\partial \psi (x,t)}{\partial t}}$

means that we calculate the variation of ${\displaystyle \psi (x,t)}$ with respect to the time t at a particular fixed point in space x.

Properties of the equation and consequences for its solutions

This equation, of which only the one-dimensional version is given here, describes many types of waves, such as: electromagnetic waves in vacuum, longitudinal waves in an elastic medium, sound waves in a gas, etc.

This equation, however, does not describe all types of waves. For example, it does not describe the more general elastic waves, such as surface waves, flexural waves in a bar, all non-linear phenomena (those in which waves interact with each other), etc.

• The equation is linear, so if ${\displaystyle f(x,t)}$ and ${\displaystyle g(x,t)}$ are solutions, so is ${\displaystyle af(x,t)+bg(x,t)}$, with a and b constants. In physical terms, this means that waves satisfying this equation do not interact with each other, and obey the superposition principle.

• The only external parameter appearing in the equation is the constant ${\displaystyle c^2}$, which has the dimensions of a velocity squared.

D'Alembert's solution

The following change of variables, due to D'Alembert, shows how to find the general solution. Indeed, substituting

${\displaystyle u=x+ct}$

${\displaystyle v=x-ct}$

the equation takes the particularly simple form

${\displaystyle \frac{{\partial }^2\psi }{\partial u\partial v}=0}$

whose general solution is obviously

${\displaystyle f(u)+g(v)}$

Going back to the original variables, we find that the general solution, up to constants set by the boundary conditions, takes the form

${\displaystyle f(x+ct)+g(x-ct)}$,

and can therefore always be broken down into a superposition of progressive and regressive waves, that is, waves travelling with velocity +c and -c.

Are there only oscillating solutions?

Absolutely not: waves and oscillations must not be confused, even though they are very often related.

An oscillatory motion is a motion that is periodic in time. A wave is just a disturbance that propagates. Of course, if the propagating disturbance is periodic in time, then the wave phenomenon is also oscillatory. The confusion is encouraged by the fact that, in most cases of practical interest, the source of the wave is periodic in time, and consequently so is the wave.

In the case of d'Alembert's solution it is immediately clear that the wave need not oscillate, since we said that f and g are arbitrary functions. However, even among standing-wave-like solutions it is easy to see that non-oscillating ones exist.

Consider the expression

${\displaystyle \psi (x,t)=\exp (\alpha x)\exp (\beta t)}$.

differentiating it twice with respect to x and to t we get, respectively,

${\displaystyle \frac{{\partial }^2\psi }{\partial x^2}={\alpha }^2\psi (x,t)}$

${\displaystyle \frac{{\partial }^2\psi }{\partial t^2}={\beta }^2\psi (x,t)}$,

and so we immediately see that if

${\displaystyle {\beta }^2=c^2{\alpha }^2}$

${\displaystyle \psi (x,t)}$ is a possible solution of the wave equation, non-oscillating (written as the product of a function of x and a function of t, like standing waves, although it is actually a function of ${\displaystyle x\pm ct}$).

Modifications of the basic equation

Telegraph equation

The most obvious modification of the wave equation is to add a "friction" term. This term is usually locally proportional to the rate at which the wave function changes in time. This is intuitive: in general we can expect oscillations to be damped more strongly the faster they are.

We therefore add a term proportional to the first derivative of ${\displaystyle \psi }$, and obtain the so-called telegraph equation, named after the fact that it describes the transmission of electromagnetic waves along electrical lines:

${\displaystyle \frac{{\partial }^2\psi }{\partial x^2}=\frac{1}{c^2}\{\frac{{\partial }^2\psi }{\partial t^2}+\gamma \frac{\partial \psi }{\partial t}\}}$

where ${\displaystyle \gamma }$ is the damping coefficient.

• If ${\displaystyle \gamma =0}$ the equation reduces to the classical wave equation.
• If ${\displaystyle \gamma \ll \omega }$, where ${\displaystyle \omega }$ is the angular frequency of the wave, the dissipative term cannot be neglected, but it is small, and the solution can be written approximately in the form

${\displaystyle \psi (x,t)=f(x-ct)\exp (-\gamma t/2)}$,

where f is an arbitrary function. In other words, it is a wave-like solution that decreases exponentially in time.

• If ${\displaystyle \gamma \gg \omega }$, the equation changes its nature completely, turning into the so-called "diffusion equation". Indeed, neglecting the second time derivative we obtain, with ${\displaystyle k=\gamma /c^2}$, an equation that describes, for example, the flow of heat.

${\displaystyle \frac{{\partial }^2\psi }{\partial x^2}=k\frac{\partial \psi }{\partial t}}$

Note that, although there can be oscillating systems in which the oscillating quantity is heat, the previous equation, which describes the propagation of heat, does not describe a wave phenomenon at all, but a diffusive one. Its properties are completely different from those of the wave equation.

Generalised telegraph equation

In the previous section we added a term proportional to the first derivative. The most general case, while remaining within linear equations (that is, keeping the superposition principle valid), is to add a term proportional to the wave function as well.

${\displaystyle \frac{{\partial }^2\psi }{\partial x^2}=\frac{1}{c^2}\{\frac{{\partial }^2\psi }{\partial t^2}+\gamma \frac{\partial \psi }{\partial t}+\delta \psi \}}$

Although the constant c appears in this equation, it can be shown that the solutions propagate at speeds that depend on the frequency. This is the phenomenon of dispersion.

Where does the source of the waves appear in the equation?

It does not appear, at least in the equation considered at the beginning. Indeed, this equation admits non-zero solutions in a region of space even without a source in that same region.

This feature is precisely what makes waves useful for communication: suppose there is a source of waves at point A. At point B there is a receiver, but no source. At a certain instant the source at point A switches on and generates waves. At that instant there is still no wave at point B. However, after a time ${\displaystyle \frac{\overline{AB}}{c}}$, waves appear at point B too. They are waves propagated from A to B. Note that waves appear at B even if A has been switched off in the meantime: even with the source switched off, the waves keep propagating.

This does not mean that we cannot write the wave equation for a region of space containing an active source. If the source is described by a function ${\displaystyle f(x,t)}$ (for example, for a string, the force applied per unit length, divided by the tension and with its sign changed), then the equation (called the "inhomogeneous wave equation") simply becomes

${\displaystyle \frac{{\partial }^2\psi }{\partial x^2}-\frac{1}{c^2}\frac{{\partial }^2\psi }{\partial t^2}=f(x,t)}$

Check your understanding