In the article Spherical Numbers I presented what boils down to the generalisation of complex numbers. These are multi-dimensional numbers well suited to for a multi-dimensional solution to the wave equation. The new solution is found by using the one-dimensional form of the equation
where is any partial increment in space; is a partial increment in time; and is the propagation velocity defined by the characteristic of the space.
We begin with the assumptions that the -dimensional function exists, and that is simultaneously a function of position and a function of time . Consequently, a third simultaneous function can be formulated as the product of the square roots of the time and position functions
where is a function of which is a point or position in space, and a function of . Any variations in and can be independently looked at. Now, by taking the second partial derivatives of , once with respect to and once with respect to and introducing the results of these partial derivatives of into (1), we find that
where the third term is introduced in anticipation of the desired result. The derivatives are total as and are independent of one another, but also equal to each other if and only if The solutions of the two second order differential equations are
where is an arbitrary constant, independent of time or position and characterises the function . All that remains is to square the above functions, to write the solution of (1) by introducing the results into (2)
The function now has multi-dimensional properties. The first order solution could describe a propeller like function propagating along its axis of rotation, its tips tracing the path of a helix.
The order of the solution can be increased to describe spherical wave objects, whose centre propagates with velocity and the object intensity does not dilute with time or distance.
This solution differs from the plain-wave solution normally derived to describe, say, the spherical expansion of a compression/rarefaction in a media.
The solution (3) is a spherical object. As per its definition it also is an eigenfunction of the linear operator or such that
is so dimensioned that it is ensured that the path followed from any point on the spherical curve, defined by , always repeats itself exactly in time. Therefore any point on the spherical curve defined by can be used as a boundary. This only happens when the ratios
are integer ratios.