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## Green's Function via Dimensional Reduction

To find the Green's function whose domain dimension is two or higher, introduce a technique whose virtue is that it reduces the problem to a Green's function problem in just one dimension. The potency of this technique is a consequence of the fact that it leads to success whenever the Helmholtz equation is separable relative to the curvilinear coordinate system54 induced by the boundarie(s) of the given domain.

Polar coordinates is a case in point. It is illustrated by the following

Problem (Green's Function for Radiation in the Euclidean Plane) The equation for the Green's function relative to polar coordinates is

 (564)

Let the homogeneous boundary conditions for be
(i)
Sommerfeld's outgoing radiation condition

(ii)
is finite at , where is undefined.

Solution

This problem is solved by expanding the Green's function as a Fourier series on :

 (565)

with to-be-determined Fourier coefficients. The method of dimensional reduction consists of establishing that each of them satisfies a 1-dimensional Green's function problem. Next one constructs its solution using formula Eq.(4.19) on page . Finally one introduces this solution into the Fourier series expansion. This yields the desired 2-dimensional Green's function. As an additional benefit one finds that this expansion can be summed into a closed form expression given by a familiar function.

The details of this four step procedure are as follows: Introduce the Fourier expansion into the 2-dimensional Green's function equation and obtain

 (566)

To isolate the equation obeyed by the each of the coefficient functions introduce the Fourier representation of the Dirac delta function restricted to :

and make use of the linear independence of the functions . Alternatively, multiply both sides of Eq.(5.66) by , integrate , make use of orthogonality, and finally drop the prime to obtain

the equation for the 1-dimensional Green's function. The boundary conditions for imply that the solution

satisfies

This is a set of 1-dimensional Green's function problems whose solutions yield the 2-d Green's funtion, Eq.(5.65). The two functions which satisfy the homogeneous differential equation and the respective boundary conditions are

 and

while their Wronskian is

Consequently, the 1-dimensional Green's function is

The 2-dimensional Green's function, Eq.(5.65), for outgoing radiation in the Euclidean plane is therefore

This expression can be simplified by means of the displacement formula for cylinder modes, Property 19, on page ,

Set , compare the Green's function with the right hand side of the displacement formala, and conclude that

 (567)

in other words,

 (568)

Thus one has obtained an expression for the 2-dimensional Green's function which exhibits the rotational and translational symmetry of the linear system. It represents an asymtotically (large !) outgoing wave whose source is located at . This is the amplitude profile of a wave that you make when you stick your wiggling finger at into an otherise motionless pond.

#### Footnotes

... system54
In three dimensions the Helmholtz equation is separable in eleven coordinate systems. They are listed and depicted at the end of chapter five of reference [#!Morse_SPMamp_Feshbach5!#]

Next: Green's Function: 2-D Laplace Up: Boundary Value Problems in Previous: Solution via Green's Function   Contents   Index
Ulrich Gerlach 2010-12-09