In mathematics, the Helmholtz equation is the eigenvalue problem for the Laplace operator. It corresponds to the linear partial differential equation:

where ∇² is the Laplace operator, k² is the eigenvalue, and f is the (eigen)function. When the equation is applied to waves, k is known as the wave number. The Helmholtz equation has a variety of applications in physics and other sciences, including the wave equation, the diffusion equation, and the Schrödinger equation for a free particle.

In optics, the Helmholtz equation is the wave equation for the electric field.^[1]

The equation is named after Hermann von Helmholtz, who studied it in 1860.^[2]

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 a time-independent form of the wave 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 that the expression on the left side depends only on r, whereas the right 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 the same constant value. This argument is key in the technique of solving linear partial differential equations by separation of variables. From this observation, we obtain two equations, one for A(r), the other for T(t):

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 (homogeneous) Helmholtz equation:

Likewise, after making the substitution ω = kc, where k is the wave number, and ω is the angular frequency (assuming a monochromatic field), the second equation becomes

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, whose exact form is determined by initial conditions, 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.

Solving the Helmholtz equation using separation of variables

The solution to the spatial Helmholtz equation:

can be obtained for simple geometries using separation of variables.

Vibrating membrane

The two-dimensional analogue of the vibrating string is the vibrating membrane, with the edges clamped to be motionless. The Helmholtz equation was solved for many basic shapes in the 19th century: the rectangular membrane by Siméon Denis Poisson in 1829, the equilateral triangle by Gabriel Lamé in 1852, and the circular membrane by Alfred Clebsch in 1862. The elliptical drumhead was studied by Émile Mathieu, leading to Mathieu's differential equation.

If the edges of a shape are straight line segments, then a solution is integrable or knowable in closed-form only if it is expressible as a finite linear combination of plane waves that satisfy the boundary conditions (zero at the boundary, i.e., membrane clamped).

If the domain is a circle of radius a, then it is appropriate to introduce polar coordinates r and θ. The Helmholtz equation takes the form

We may impose the boundary condition that A vanishes if r = a; thus

the method of separation of variables leads to trial solutions of the form

where Θ must be periodic of period 2π. This leads to

It follows from the periodicity condition that

and that n must be an integer. The radial component R has the form where the Bessel function J_n(ρ) satisfies Bessel's equation and ρ = kr. The radial function J_n has infinitely many roots for each value of n, denoted by ρ_m,n. The boundary condition that A vanishes where r = a will be satisfied if the corresponding wavenumbers are given by