Series Solutions of the Wave Equation The spherically symmetrical wave equation in N space dimensions and 1 time dimension is Suppose a solution of this equation has the form where the fixed integers p and q represent the lowest powers of r and t (respectively) appearing in any term with a non-zero coefficient. Hence we have amq ¹ 0 for some index m, and we have apn ¹ 0 for some index n. The first and second partial derivatives of this wave function with respect to r are Likewise the second partial derivative of the wave function with respect to t is Substituting for these partials into equation (1) gives Notice that we have absorbed the factor 1/r in the second term into the summation. Combining all the summations, we have In order to collect terms by powers of r and t we must take the right-most terms with the indices m-2 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 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 To enhance the symmetry, we can replace n with n-2, allowing us to write the conditions in the form We know that ajk = 0 if j