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Part V: Solutions of the Helmholtz wave equation in three dimensions

2.11. Rectangular coordinates

In the steady state, Eq. (2.20b) for the three-dimensional wave equation in rectangular coordinates can be written

(\nabla^{2} + k^{2}) \tilde{p} (x, y, z) = 0,

$(\nabla^{2} + k^{2}) \tilde{p} (x, y, z) = 0,$

(2.146)

where the Laplace operator is given by

\nabla^{2} = \frac{\partial^{2}}{\partial x^{2}} + \frac{\partial^{2}}{\partial y^{2}} + \frac{\partial^{2}}{\partial z^{2}}

$\nabla^{2} = \frac{\partial^{2}}{\partial x^{2}} + \frac{\partial^{2}}{\partial y^{2}} + \frac{\partial^{2}}{\partial z^{2}}$

(2.147)

and k = ω/c = 2π/λ. Let the solution to Eq. (2.146) be of the form

\tilde{p} (x, y, z) = {\tilde{p}}_{0} X (x) Y (y) Z (z) .

$\tilde{p} (x, y, z) = {\tilde{p}}_{0} X (x) Y (y) Z (z) .$

(2.148)

Substituting this in Eq. (2.146) and dividing through by X(x)Y(y)Z(z) yields

(\frac{1}{X} \frac{\partial^{2} X}{\partial x^{2}} + k_{x}^{2}) + (\frac{1}{Y} \frac{\partial^{2} Y}{\partial y^{2}} + k_{y}^{2}) + (\frac{1}{Z} \frac{\partial^{2} Z}{\partial z^{2}} + k_{z}^{2}) = 0,

$(\frac{1}{X} \frac{\partial^{2} X}{\partial x^{2}} + k_{x}^{2}) + (\frac{1}{Y} \frac{\partial^{2} Y}{\partial y^{2}} + k_{y}^{2}) + (\frac{1}{Z} \frac{\partial^{2} Z}{\partial z^{2}} + k_{z}^{2}) = 0,$

(2.149)

where

k^{2} = k_{x}^{2} + k_{y}^{2} + k_{z}^{2} .

$k^{2} = k_{x}^{2} + k_{y}^{2} + k_{z}^{2} .$

(2.150)

For example, in the case of a plane wave with a direction of travel in the zx plane at an angle θ to the z axis, we have k _z = k cos θ, k _x = k sin θ, and k _y = 0. The first bracketed term of Eq. (2.149) depends on x only, whereas the second term depends on y only and the third term z only. However, they must all add up to zero, which means that either they all have constant values, the combination of which is zero, or they are all zero. We shall assume the latter, in which case Eq. (2.149) can be separated into three equations, one for each ordinate as follows.