Notice the expression on the left-hand side depends only on r, whereas the right-hand expression depends only on t. Substituting this form into the wave equation, and then simplifying, we obtain the following equation:
Separation of variables begins by assuming that the wave function u(r, t) is in fact separable: 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. Motivation and uses The Helmholtz equation often arises in the study of physical problems involving partial differential equations (PDEs) in both space and time.
Where ∇2 is the Laplacian, k is the wavenumber, and A is the amplitude.ġ Motivation and uses o 1.1 Harmonic solutions 2 Solving the Helmholtz equation using separation of variables o 2.1 Vibrating membrane o 2.2 Three-dimensional solutions 3 Paraxial approximationĤ Inhomogeneous Helmholtz equation 5 References 6 External links The Helmholtz equation, named for Hermann von Helmholtz, is the elliptic partial differential equation The real part of the resulting field A, A is the solution to the inhomogeneous Helmholtz equation Two sources of radiation in the plane, given mathematically by a function ƒ which is zero in the blue region. Helmholtz equation From Wikipedia, the free encyclopedia
