Helmholtz Equation - Motivation and Uses

Motivation and Uses

The Helmholtz equation often arises in the study of physical problems involving partial differential equations (PDEs) in both space and time. The Helmholtz equation, which represents the time-independent form of the original equation, results from applying the technique of separation of variables to reduce the complexity of the analysis.

For example, consider the wave equation

Separation of variables begins by assuming that the wave function u(r, t) is in fact separable:

Substituting this form into the wave equation, and then simplifying, we obtain the following equation:

Notice the expression on the left-hand side depends only on r, whereas the right-hand expression depends only on t. As a result, this equation is valid in the general case if and only if both sides of the equation are equal to a constant value. From this observation, we obtain two equations, one for A(r), the other for T(t):

and

where we have chosen, without loss of generality, the expression −k2 for the value of the constant. (It is equally valid to use any constant k as the separation constant; −k2 is chosen only for convenience in the resulting solutions.)

Rearranging the first equation, we obtain the Helmholtz equation:

Likewise, after making the substitution

the second equation becomes

where k is the wave vector and ω is the angular frequency.

We now have Helmholtz's equation for the spatial variable r and a second-order ordinary differential equation in time. The solution in time will be a linear combination of sine and cosine functions, with angular frequency of ω, while the form of the solution in space will depend on the boundary conditions. Alternatively, integral transforms, such as the Laplace or Fourier transform, are often used to transform a hyperbolic PDE into a form of the Helmholtz equation.

Because of its relationship to the wave equation, the Helmholtz equation arises in problems in such areas of physics as the study of electromagnetic radiation, seismology, and acoustics.