Waves in Ascending and Descending Dimensions 

There are some remarkable relationships between elementary spherically symmetrical solutions of the wave equation in spaces of different dimensions. As discussed in the note on spherical waves in higher dimensions, by the separation of variables we can split the wave equation into spatial and temporal parts, and the spatial part f(r) satisfies the differential equation 

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where n is the number of spatial dimensions. (We’ve chosen units such that the wave length equals unity.) Letting f_{n} denote a solution of (1) in a space of n dimensions (and one time dimension), the function given by 

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is a solution of (1) in space of n+2 dimensions. To prove this, note that the definition of g(r) implies (after multiplying through by –n/r and differentiating) 

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where dots signify derivatives with respect to r. Also, differentiating equation (1) gives 

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Substituting for the derivatives of fn from the preceeding expressions, we get 

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Simplifying this expression, we get 

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which proves that g(r) is a solution in space of n+2 dimensions. Thus we have a sequence of solutions in ascending dimensions 

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Interestingly, this can also be expressed as a sequence of (scaled) solutions in descending dimensions. To see this, use equation (1) to replace the left side of (2), and then multiply through by –r/n to give 

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Integrating both sides, we get (up to a constant of integration) 

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Now we multiply through by r^{n}^{1} to give 

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The expression inside the square brackets is simply the derivative of r^{n}f_{n+2}, so this can be written as 

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Thus if we define the scaled wave function F_{n}(r) = r^{n}^{2 }f_{n}(r), we have 

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In other words, F_{n} is simply a scaled derivative of F_{n+2}, so this gives a sequence of solutions in descending space dimensions, by a formula very similar to equation (2). 

At this point we should note that the negative sign in (2) is potentially misleading, because the negation can be accomplished either by multiplying by negative 1 or by applying a phase shift to the periodic parts of the function, specifically by subtracting p from the phase. As discussed in the note on fractional calculus, a generalized differentiation operator applied to the sine or cosine function amounts to a simple phase shift, and this just happens to give negation when the phase shift is p. Therefore, instead of writing equation (2) with a leading factor of 1, we will write it with a leading factor of F^{}^{1}, where F is a linear phase shift operator defined by 

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where g is a periodic function. Linearity means that F(a+b) = F(a) + F(b). For example, 

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Using this operator, we will take as the general form of equation (2) the expression 

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We can make use of either (3) or (5) to derive simple explicit expressions for elementary wave solutions in n dimensions, for both odd and even values of n. First, notice that in view of the identity 

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we can express equations (3) and (5) in the form 

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Given the elementary onedimensional solution F_{1}(r), the left hand relation allow us to determine a sequence of solutions by repeated integration as follows 

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and so on. Now, in onedimensional space we have the elementary solution f_{1}(r) = cos(r), which in scaled form is F_{1}(r) = cos(r)/r. Therefore we have 

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and hence f_{3}(r) = sin(r)/r as expected. Integrating again (with the appropriate scale factor) on F_{3} gives F_{5}, and so on. Thus, using Cauchy’s formula for repeated integration (see the note on fractional calculus for a derivation of this useful formula), we can express F_{2n}_{+3} by performing the integration n+1 times. The product of the leading scale factors is 

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so the repeated integrations give the formula 

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Factoring an r^{2n} out of the integrand and making the substitution x = u/r, this can be rewritten as 

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As discussed in the note on fractional calculus, the natural generalization of this (n+1)th order integral is to allow n to be a noninteger n, and replacing the factorial of n with the gamma function of n+1. Thus we arrive at 

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For space of two dimensions we set n = 1/2 in the above formula to give the solution 

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This agrees with the result found in the note on Huygens’ Principle by means of a series evaluation, and we note that it is the lowestorder Bessel function, i.e., f_{2}(r) = J_{0}(r). 

Incidentally, with regard to equation (6), it’s worth noting the identity 

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Thus equation (6) can be written in the form 

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which shows that the wave function is the mean of cos(xr) on the interval from x = 0 to 1 with the density proportional to (1x^{2})^{n}. 

An alternative approach to developing wave solutions in n dimensions is to use the ascending sequence of solutions given by equation (5), beginning with the elementary onedimensional solution f_{1}(r) = cos(r), as follows 

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and so on. Thus the ascent from n dimensions to n+2 dimensions is given by applying the differential operator 

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Applying this operator n times to a onedimensional solution gives an expression for a 2n+1 dimensional solution. So, on this basis, our expression for higherdimensional solutions is 

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Since each differentiation (with respect to r^{2}) raises the number of space dimensions by two, we need to apply a halforder differentiation to produce solutions for spaces with even numbers of dimensions. Using the Left Hand Rule for fractional differentiation, we first perform a halfintegration on sin(r) with respect to r^{2} by means of Cauchy’s formula, which gives 

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Dividing out the r^{2}, and making the substitution x = u/r, and noting that du = r dx, this can be written as 

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We now need to perform one whole differentiation with respect to r^{2}, and we can do this inside the integration. Noting that 

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we finally arrive at the halfderivative of cos(r) 

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Inserting this into equation (7) with n = 1/2 gives 

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This again equals the Bessel function J_{0}(r), although the equivalence is nontrivial. Thus we can arrive at consistent results using the sequence of solutions related by differentiation in either the ascending or descending space dimensions. 

The correspondences between solutions in higher and lower dimensions is essentially onetoone, and all the solutions in higher dimensions can be mapped down to (or up from) a solution in one dimension. Using the series solution technique described in the note on Huygens’ Principle, it can be shown that all onedimensional solutions are of the form 

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for some constants c_{0} and c_{1}. So, returning to equation (7), generalizing to noninteger orders by replacing n with the real value n and replacing the factorials with gamma functions, and inserting the general onedimensional solution in place of the special solutions cos(r), we have 

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The phase shift and differentiation operators are commutative, so we can apply the shift to the basic argument, and write this in the form 

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This shows that for any given number of space dimensions there is a twoparameter family of solutions. Also, for any choice of those two parameters we can vary the number of dimensions continuously. 
