follows the well-known governing wave equation for a homogeneous medium: (Note: Instead of vectorial only scalar can be used, i.e. waves are travelling only along the axis, and the scalar wave equation follows as .) The above vectorial partial differential equation of the 2nd order delivers two mutually independent solutions. From the quadratic velocity term can be seen that there are two waves travelling in opposite directions and are possible, hence results the designation “two-way wave equation”. It can be shown for plane longitudinal wave propagation that the synthesis of two one-way wave equations leads to a general two-way wave equation. For special two-wave equation with the d'Alembert operator results: For this simplifies to Therefore, the vectorial 1st-order one-way wave equation with waves travelling in a pre-defined propagation direction results as Scalar wave equation in three space dimensions 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. Spherical waves 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 so we get This is the Helmholtz equation and can be solved using separation of variables. If spherical coordinates are used to describe a problem, then the solution to the angular part of the Helmholtz equation is given by spherical harmonics, and the radial equation now becomes Here , and the complete solution is now given by where and are the spherical Hankel functions. Example To gain a better understanding of the nature of these spherical waves, let us go back and look at the case when . In this case, there is no angular dependence, and the amplitude depends only on the radial distance i.e. . In this case, the wave equation reduces to or This equation can be rewritten as where the quantity satisfies the one-dimensional wave equation. Therefore, there are solutions in the form where and are general solutions to the one-dimensional wave equation and can be interpreted as respectively an outgoing and incoming spherical waves. The outgoing wave can be generated by a point source, and they make possible sharp signals whose form is altered only by a decrease in amplitude as increases (see an