The Wave Equation applies to any uniform medium that provides a restoring force on every particle in the medium proportional to the displacement of that particle from its equilibrium position. In one spatial dimension the medium can be thought of as an ideal string, of uniform density and with no stiffness. In two dimensions the medium will be a membrane with similar properties -- an idealized drum head.
The equation describes a highly symmetrical relationship between the way such displacements vary with time and the way they vary with position inside the medium. The uniform speed of sound in the medium defines the relative scale of that symmetry.
Solutions of the wave equation must fulfill two types of constraints in order to describe any particular situation. Boundary conditions reflect the symmetrical constraints imposed by the shape of the medium and the nature of its boundaries (i.e. the nature of reflections that occur there). Initial conditions define the displacement and velocity of every particle of the medium at some specified instant in time.
The solutions take the form of expressions for the displacement of a particle from its equilibrium position; thus they must be functions of both position and time. Solutions to the wave equation, however, turn out to be products of two terms, one dependent only on time and the other dependent only on position within the medium. This allows wave equation problems to reduce to Eigenvalue problems characterized by families of discrete solutions [a harmonic series of solutions, for instance].
In one dimension, the wave equation may be written
where 
and 
are operators
acting on the variable 
, which represents
displacement from equilibrium at any point x along the string and any time t.
These operators, respectively, perform second partial derivatives with respect
to x and t . [In a partial derivative with respect to one
variable, all other variables are held constant.] The speed of sound is the
constant c relating the temporal and spatial variations in displacement.
It is given by
, where 
is the density of
the one-dimensional medium (mass/length) and T is its tension.
We expect solutions of the form
where f(x) is independent of t and 
is an
angular frequency (typically measured in radians/sec). Inserting this form into
the wave equation gives us
since the factor involving the other variable is just a constant multiplier for the result of each partial derivative operator. Performing the operation on the time-dependent factor, we obtain
with two factors of and one sign change coming
from the two differentiations with respect to time. Dividing both sides of the
equation by the common cosine factor and by c2 then yields
,and our task has reduced to solving a simpler equation in a single variable
x, with no partial derivatives to worry about. If we define a wave number
k such that 
, we can rewrite the remaining equation as
.Notice that we have here a result very similar to the one we just described
for two successive differentiations -- two factors of a constant and a change in
sign -- only this time the derivatives are with respect to x rather than
t and the constant is k rather than c. f(x)
must involve either a sine or cosine of something proportional to x. If
we assume our string has a finite length, i.e.
, and impose boundary conditions appropriate
to a string fixed at both ends, i.e. 
, it becomes
clear that the trigonometric function in f(x) must be a sine rather than
a cosine. [You may want to consider also the effects of different boundary
conditions: both ends free to be displaced, or one end fixed and the other
free. Both those cases are used in describing the longitudinal displacement of
the air in organ pipes.]
This has become an Eigenvalue problem, with a whole familly of discrete solutions for f defining a set of normal vibrational modes for an ideal string:
where n can be any positive integer. Notice that each n defines its
own wave number 
, and angular frequency
. The frequencies of these solutions are harmonically
related: 
. Notice how the fundamental frequency
depends on the length, density, and tension of the string. The full set of
coefficients An define the initial conditions: they can be
chosen for any desired shape of the string at t = 0 and describe how
much energy is stored in each of the normal modes of the string.
The full solution for our wave equation in one dimension, then -- combining the spatial and temporal factors and summing over all the normal modes -- becomes:
.Although there have been square drums, our primary purpose in this section is to emphasize how directly solutions to the one-dimensional wave equation can be generalized to two spatial dimensions. A rectangular symmetry allows all references to position to be in terms of two orthogonal axes that can be dealt with identically. Note the close parallels between what follows and the discussion of the previous section. In rectangular symmetry, the wave equation can be written
. is again the speed of sound in the medium. The
tension T providing the restoring force proportional to the displacement
is now surface tension, however -- force per unit length, measured across any
line in the two-dimensional membrane's plane -- and the density
now has units of mass per unit area. The "ideal"
medium again is assumed to have no stiffness. The single spatial operator
of the one-dimensional case has become a combination of two orthogonal operators
.Again we expect solutions of the form
,In that event, in close analogy to the one-dimensional case, the wave equation may be written
.Performing the partial derivatives with respect to time we obtain
,and dividing both sides by the common cosine factor and by c2 we have
,where we again have defined a wave number k as .
Let the dimensions of the membrane be given by 
and
. Then the boundary conditions for fixed
edges can be written
.In a simple extension of the one dimensional case, this again yields an Eigenvalue prblem with solutions
.The coefficients containing the initial conditions now carry two indices, one for each spatial dimension, as do the wave numbers
.Each of the indices -- l and m -- can take on any positive integer value. The angular frequencies of each normal mode are given by
,so the full solution may be written
.In this case it is easiest to use a polar coordinate system with its origin
in the center of the membrane. Any point then can be specified in terms of the
orthogonal variables 
, distance from the center
where R is the maximum radius of the membrane, and
, an angle measured from an arbitarary reference
radial line. In terms of these cylindrical coordinates, the wave
equation becomes
where again is the speed of sound in the
membrane, with surface tension T having dimensions of force/distance
and density mass/surface. The spatial derivatives
operator becomes, in polar coordinates,
and we again expect solutions of the form
where this time for the sake of convenience we have adopted a complex exponential notation for the time dependence, rather than a cosine function. In close analogy to our earlier cases, such a solution form causes the wave equation to take the form
or, taking the partial derivatives with respect to time,
which can be written
where wave number 
. The solutions to this
Eigenvalue problem are called Bessel functions:
,with separate factors for the radial and angular parts of the solution, and
with the boundary conditions requiring 
.
This limits the solutions to discrete values of kR and, hence, k;
to wit.:
where j can be any positive integer. Normal mode frequencies correspond to values of knjR for which Jn(knjR) = 0. Since
,the frequencies are simply
where c' is defined as the speed of sound in circumferences per unit time. To tabulate normal mode frequency ratios we need only compute
.The full solution can be written
where the first of the two terms in the factor in square brackets describes
the initial position of each particle in the membrane and the second
term its initial velocity. One circular node, at radius R, is
common to all the normal modes. For j > 1 there will be additional
circular nodes, at certain distances from the center of the membrane. For j
= 2, for example, the additional circular node is at 
,
while for j = 3 there are two additional circular nodes, at
and at 
.
Two primary ways in which real media may differ from the ideal models we have been using are nonzero stiffness in a string or membrane and dissipation. of energy.
Stiffness in a string or membrane provides an additional restoring force, one not proportional to displacement from equilibrium position. This will increase the frequencies of normal modes, an effect that will be relatively stronger for higher order modes because of their shorter wavelengths and, thus, shorter bending radii within the medium. The effect of stiff strings, for instance, is most noticeable in the relative sharpness of higher harmonics.
Energy dissipation causes the amplitude of oscillation to decrease with time. Mechanisms that can cause such dissipation include radiation of sound into adjacent media, and conversion of vibrational energy into heat (due to friction within a real string or membrane, for instance, or viscous heating within a fluid medium).
The wave equation in the three dimensions x, y, and z is a straightforward generalization of the rectangular membrane case discussed above. As with two dimensions, different choices of coordinates can simplify the equations for problems with different symmetries. For sound radiating equally in all directions from a point source, for instance, the use of spherical coordinates allows simplification to a wave equation with the single spatial variable r.