In mathematics, the Helmholtz equation is the eigenvalue problem for the Laplace operator. It corresponds to the elliptic partial differential equation:
∇ 2 f = − k 2 f , {\displaystyle \nabla ^{2}f=-k^{2}f,}
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. The equation is named after Hermann von Helmholtz, who studied it in 1860.
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
( ∇ 2 − 1 c 2 ∂ 2 ∂ t 2 ) u ( r , t ) = 0. {\displaystyle \left(\nabla ^{2}-{\frac {1}{c^{2}}}{\frac {\partial ^{2}}{\partial t^{2}}}\right)u(\mathbf {r} ,t)=0.}
Separation of variables begins by assuming that the wave function u(r, t) is in fact separable:
u ( r , t ) = A ( r ) T ( t ) . {\displaystyle u(\mathbf {r} ,t)=A(\mathbf {r} )T(t).}
Substituting this form into the wave equation and then simplifying, we obtain the following equation:
∇ 2 A A = 1 c 2 T d 2 T d t 2 . {\displaystyle {\frac {\nabla ^{2}A}{A}}={\frac {1}{c^{2}T}}{\frac {\mathrm {d} ^{2}T}{\mathrm {d} t^{2}}}.}
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):
∇ 2 A A = − k 2 {\displaystyle {\frac {\nabla ^{2}A}{A}}=-k^{2}}
1 c 2 T d 2 T d t 2 = − k 2 , {\displaystyle {\frac {1}{c^{2}T}}{\frac {\mathrm {d} ^{2}T}{\mathrm {d} t^{2}}}=-k^{2},}
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:
∇ 2 A + k 2 A = ( ∇ 2 + k 2 ) A = 0. {\displaystyle \nabla ^{2}A+k^{2}A=(\nabla ^{2}+k^{2})A=0.}
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
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