constant, namely , where the sign is chosen appropriately. Using this, we can get the relation , again choosing the right sign: And similarly for the final boundary segment: Adding the three results together and putting them back in the original integral: Solving for we arrive at In the last equation of the sequence, the bounds of the integral over the source function have been made explicit. Looking at this solution, which is valid for all choices compatible with the wave equation, it is clear that the first two terms are simply d'Alembert's formula, as stated above as the solution of the homogeneous wave equation in one dimension. The difference is in the third term, the integral over the source. Wave equation for inhomogeneous media, three-dimensional case For one-way wave propagation, i.e. wave are travelling in a pre-defined wave direction ( or ) in inhomogeneous media, wave propagation can also be calculated with a tensorial one-way wave equation (resulting from factorization of the vectorial two way wave equation) and an analytical solution can be derived. Other coordinate systems In three dimensions, the wave equation, when written in elliptic cylindrical coordinates, may be solved by separation of variables, leading to the Mathieu differential equation. Further generalizations Elastic waves The elastic wave equation (also known as the Navier–Cauchy 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: where: and are the so-called Lamé parameters describing the elastic properties of the medium, is the density, is the source function (driving force), and is the displacement vector. By using the elastic wave equation can be rewritten into the more common form of the Navier–Cauchy equation. Note that in the elastic wave equation, both force and displacement are vector quantities. Thus, this equation is sometimes known as the vector wave equation. As an aid to understanding, the reader will observe that if and are set to zero, this becomes (effectively) Maxwell's equation for the propagation of the electric field , which has only transverse waves. Dispersion relation In dispersive wave phenomena, the speed of wave propagation varies with the