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The solution of a partial differential equation can only be found if the boundary conditions and the initial conditions are given. Therefore, for every problem statement the boundary conditions are very important. However, in many textbooks the boundary condition of the wave equation (or the Helmholtz equation) is not considered when free space propagation is discussed.

Question 1: What is the correct boundary condition (of the wave or helmholtz equation) for a propagation of a wave in free space? I have read that the eigenmodes of the wave equation in free space are plane waves. However, for finding the eigenmodes, there need to be known boundary conditions as far as I know.

Question 2: In many textbook examples the solution of the Helmholtz equation is given by a superposition integral (Example 1). However, I always wonder if this solution set is really complete. In most cases, an ansatz respectively an assumption is used to find this solution of the PDE. This is a contradiction to my undergraduate math lectures. In these lectures, the solution of a PDE was always found by a) finding the eigenfunctions of the PDE from the boundary conditions and b) calculating the eigenfunction coefficients from the initial condition. There was no 'guess' involved in the process of solving the given PDEs.

Qmechanic
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cakelover
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2 Answers2

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The boundary condition is plane waves fill all space. The initial condition is they have existed from the infinite past.

In practice, this usually means plane waves extend far enough that you don't have to worry about the edge, they are bigger than your system. They exist and fill space at the start of your problem.

For example, light from a distant star is indistinguishable from a plane wave. Yes, it is really a big spherical wave and started a few billion years ago. Even though the Sun has a visible disk, light from the Sun often close enough to a plane wave that it is treated as one.

Light from a collimating lens might be treated as a plane wave, unless the aperture is small enough that you need to worry about the edge or diffraction. Usually lens aberrations are a bigger problem than diffraction.

mmesser314
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Perhaps this discussion could help: Wave function boundary condition in scattering problem

In scattering problems one usually assumes a form of an outgoing/incoming wave. These are also often reformulated as initial value problem with initial conditions far in the past.

I do agree that basic QM and E&M texts do poor job differentiating eigenvalue and scattering problems.

Roger V.
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