Suppose we are dealing with wave propagation in the frequency domain, that is, wave propagation governed by the Helmholtz equation. How do reflection and transmission coefficients for waves correspond to Neumann and Dirichlet boundary conditions at the interface between two mediums?

Can the reflection and transmission coefficients be derived from knowing that the wave propagation is governed by the Helmholtz equation and that certain Neumann or Dirichlet boundary conditions have been specified at an interface?

  • $\begingroup$ The boundary condition is just that the wave and its derivatives be continuous at the boundary. Unless there is a charge layer, $\endgroup$ Commented Dec 9, 2016 at 10:09
  • $\begingroup$ I am asking specifically about how the transmission and reflection coefficients relate to Dirichlet or Neumann boundary conditions. For example, is there a formula that relates them? $\endgroup$
    – sonicboom
    Commented Dec 9, 2016 at 12:25
  • $\begingroup$ One simple example would be in derivation of surface waves from seismic wave equation , free slip is assumed leading to normal and shear stress being equal along boundaries . This constitutes Dirichlet condition based on which Rayleigh waves are described . This is further used to derive transmission and reflection coefficients . $\endgroup$
    – shrey
    Commented Dec 9, 2016 at 12:35

1 Answer 1


On an interface, for normal incident waves, you would use a Cauchy boundary condition, which is both a Neumann and Dirichlet boundary condition. In the one dimension case, this insures there will be a unique continuation and thus a unique full solution. The higher dimensional cases may not be as nice, but usually everything works out.

Now, as per what you added in the comments, the reflection and transmission coefficients are determined by the boundary conditions, but those boundary conditions are not specified independently. They need to match with the specified Helmholtz wave equation.

In the general case, you would suppose continuity and use the wave equation together with any other physical constraints in your model (e.g. in electromagnetism) to determine the exact boundary conditions on the interface. After that, you would solve the Helmholtz equation on the volumes of the different mediums, obtaining some kind of harmonic functions. If separated in cartesian coordinates, then this leads to plane waves. Finally match those different solutions using the boundary conditions as above determined will lead you to the coefficients.


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