Wave equation

A pulse traveling through a string with fixed endpoints as modeled by the wave equation.
1-d standing wave as a superposition of two waves traveling in opposite directions

In the case of a stress pulse propagating longitudinally through a bar, the bar acts much like an infinite number of springs in series and can be taken as an extension of the equation derived for Hooke's law. A uniform bar, i.e. of constant cross-section, made from a linear elastic material has a stiffness K given by

Another way to arrive at this result is to factor the wave equation into two one-way wave equations:

The total wave function for this eigenmode is then the linear combination

A solution of the initial-value problem for the wave equation in three space dimensions can be obtained from the corresponding solution for a spherical wave. The result can then be also used to obtain the same solution in two space dimensions.

The wave equation can be solved using the technique of separation of variables. To obtain a solution with constant frequencies, let us first Fourier-transform the wave equation in time as

For physical examples of non-spherical wave solutions to the 3D wave equation that do possess angular dependence, see dipole radiation.

Cut-away of spherical wavefronts, with a wavelength of 10 units, propagating from a point source.

From this we can observe that the peak intensity of the spherical wave oscillation, characterized as the squared wave amplitude

We can use the three-dimensional theory to solve this problem if we regard u as a function in three dimensions that is independent of the third dimension. If

Approximating the continuous string with a finite number of equidistant mass points one gets the following physical model:

Figure 1: Three consecutive mass points of the discrete model for a string

Taking the sum of these two forces and dividing with the mass m one gets for the vertical motion:

using an 8th order multistep method the 6 states displayed in figure 2 are found:

Figure 2: The string at 6 consecutive epochs, the first (red) corresponding to the initial time with the string in rest
A solution of the wave equation in two dimensions with a zero-displacement boundary condition along the entire outer edge.

To simplify this greatly, we can use Green's theorem to simplify the left side to get the following:

The left side is now the sum of three line integrals along the bounds of the causality region. These turn out to be fairly easy to compute

Adding the three results together and putting them back in the original integral: