Series Solutions of the Wave Equation 

The spherically symmetrical wave equation in N space dimensions and 1 time dimension is

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Suppose a solution of this equation has the form 

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where the fixed integers p and q represent the lowest powers of r and t (respectively) appearing in any term with a nonzero coefficient. Hence we have a_{mq} ¹ 0 for some index m, and we have a_{pn} ¹ 0 for some index n. The first and second partial derivatives of this wave function with respect to r are 

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Likewise the second partial derivative of the wave function with respect to t is 

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Substituting for these partials into equation (1) gives 

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Notice that we have absorbed the factor 1/r in the second term into the summation. Combining all the summations, we have 

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In order to collect terms by powers of r and t we must take the rightmost terms with the indices m2 and n+2, which means the factor becomes (n+2)(n+1). Hence, setting the coefficients of every term in the overall expansion to zero gives the condition 

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If, instead of numbering the indices m,n from p,q to infinity, we number them from 0 to infinity, then we must replace each m with p+m and each n with q+n respectively. On this basis the conditions are expressed as 

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To enhance the symmetry, we can replace n with n2, allowing us to write the conditions in the form 

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We know that a_{jk} = 0 if j<p or k<q, so the left sides of these conditions vanish when n = 0 or 1, and the right sides vanish and when m = 0 or 1. Consequently we have the four conditions 

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We also know that a_{jq} is nonzero for some j, and a_{pk} is nonzero for some k. It follows from the above conditions for n = 0 and m = 0 that q(q1) = 0 and p(N+p2) = 0. Consequently, we must have q = 0 or 1, and p = 0 or 2N, so there are four possible combinations of least indices. 

Incidentally, this implies that it's possible for y(0,t) to be an odd or an even function, depending on whether q is 0 or 1, and y(r,0) can also be an odd or even function if the number N of spatial dimensions is odd. The general solution can be expressed as a sum of separable terms of the form f(t)g(r), where f and g have one of the four combinations of possible q and p values, so if N is odd we a complete basis of odd and even functions for the solution. This is why, in an odd number of spatial dimensions, we can propagate arbitrary waveforms, and the strong form of Huygens' Principle applies. However, if the number N of spatial dimensions is even, both possible indices p = 0 and p = 2N for the radial exponent are even. Consequently we cannot have a purely odd function of r, so we do not have a complete basis for solutions. Only a restricted set of radial waveforms can propagate in a space with an even number of dimensions, and these imply disturbances inside the future null cone of a pulse. Therefore, the strong form of Huygen's Principle is not satisfied in an evendimensional space. 

In the case p = q = 0, the above conditions do not constrain the values of a_{j,q}, a_{j,q+1}, and a_{p,j} for any j. However, they do require that all the coefficients a_{1,j} vanish (assuming N is not equal to 1). The recursive relation in this case reduces to 

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We can make an array of the coefficients as shown below. 



The arrow indicates the pairs of coefficients that are related by the recurrence formula. One consequence is that, since all the a_{1j} coefficients are zero, all the a_{3j} coefficients must also vanish, and in general all the a_{mj} coefficients for odd indices m must vanish. Therefore the array of coefficients looks like this 



The first row represents the coefficients of the wave function at the spatial origin as a function of time, i.e., the coefficients of y(0,t). Likewise the first column represents the coefficients of the wave function at the initial instant as a function of radial position, i.e., the coefficients of y(r,0). The second column is essentially the rate of change of the wave function at the initial instant as a function of radial position. 

The form of the recursive relation implies that we can fully specify the entire wave function (for the case p = q = 0) simply by specifying the value at the origin for all time. In other words, we are free to specify the top row of coefficients a_{00}, a_{01}, a_{02},..., and these are sufficient to compute all the remaining coefficients, as indicates by the arrows. On the other hand, the first column of coefficients, by itself, is not sufficient to determine the wave function, because those values determine a_{0j} only for even indices j. In order to fix the values of a_{0j} for odd j we must specify the second column of coefficients, i.e., the rate of change of the wave function at the initial instant. 

Notice that if the initial rate of change of the wave function is zero, then all the coefficients of odd powers of t in y(0,t) to vanish, which implies that y is an even function of time, i.e., symmetrical with respect to time about any instant in which the wave function is stationary. 

To illustrate, suppose the wave function at the spatial origin is y(0,t) = cos(t). This implies that the first row of coefficients are 

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Using the recurrence formula, the next row of nonzero coefficients is 

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These two rows, and the subsequent rows, can be summarized as follows 

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This shows that the overall wave function is separable, i.e., we can factor cos(t) out of these coefficients, and we have y(r,t) = cos(t) f(r) where 

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If N = 1 this function is simply cos(r), and if N = 3 this function is sin(r)/r. If N = 2 this function is the zerothorder Bessel function, often denoted as J_{0}(r). (For more on this, see Huygens' Principle.) 

We can also derive the family of series solutions of the wave equation based on inverse powers of time, i.e., we can assume a wave function of the form 

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Using the same method as above, we arrive at the following conditions on the coefficients: 

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Since a_{0,n} is nonzero for some n, we must have p = 0 or else p = 2N. Also, these conditions imply that a_{j,0} = a_{j,1} = a_{j,2} = 0 for all j greater than 0, and they also imply that a_{m,n} = 0 for all odd indices m. Hence we have an array of coefficients of the form 

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and so on. If we take q = 2, the upper row of coefficients represents the wave function at the spatial origin of the form 

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It's interesting to consider the response to a unit "impulse" occurring at the spatial origin at time t = 0. An impulse can be represented by a scaled Cauchy function in the limit as the scale factor e goes to zero, i.e., 

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Expanding this into a power series in e gives 

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Taking this as the wave function at the spatial origin, the coefficient array has the form 

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Collecting by powers of e, the first two orders give the wave function 

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in the limit as e approaches zero. Notice that if N = 3, the quantity in the square brackets is just a geometric series in (r/t)^{2}, so (formally) we have 

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For any nonzero value of t, the first term vanishes in the limit as e goes to zero, and we are left with 

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This shows that, in the limit e ® 0, the wave function vanishes except on the "null cone" where r^{2}  t^{2} = 0. It also shows that if the ratio of e^{3} to r^{2}  t^{2} goes to one on the null cone itself as e goes to zero, then the magnitude of the wave function drops in proportion to 1/t^{2} = 1/r^{2}. 
