Derivation of Helmholtz Equation from the Wave Equation

where

• c is the velocity of propagation,
• is a prescribed source
• and

is the wave equation.

It is a typical hyperbolic equation, and is sometimes written as

where

In the literature, derivations of the Helmholtz equation consider the wave equation in cases where all data are simple-harmonic (or zero) :
For instance, we might have

where omega is frequency.

Assuming data of this type we can look for simple-harmonic solutions of the form

Substituting equations (1) and (2) into the wave equation gives the following :

Note that

Dividing through by the factor gives

which is the Helmholtz equation with

Boundary Conditions