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The Helmholtz equation in Cartesian coordinates is: \[ \left(\frac{\partial^2}{\partial x^2} + \frac{\partial^2}{\partial y^2} + \frac{\partial^2}{\partial z^2} \right) f(x,y,z) + \kappa^2 f(x,y,z) = 0 \] ### Step 1: Apply Separation of Variables We assume a separable solution of the form: \[ f(x, y, z) = X(x) Y(y) Z(z) ____ _____ _______ ________ _________ _________ _______ ________.
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