Wave equation boundary conditions at infinity

Are there general boundary conditions for the wave equation PDE at infinity? If there is, could someone suggest a book/monograph that deals with these boundary conditions?

More specifically, if we have the following wave equation:

$$$\nabla ^2 p = A\frac{{\partial ^2 p}}{{\partial t^2 }} + B\frac{{\partial p}}{{\partial t}}$$$

how would I select boundary conditions at infinity to ensure that the wave will continue to propagate approaching infinity?

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Are there general boundary conditions for the wave equation PDE at infinity? If there is, could someone suggest a book/monograph that deals with these boundary conditions?

More specifically, if we have the following wave equation:

$$$\nabla ^2 p = A\frac{{\partial ^2 p}}{{\partial t^2 }} + B\frac{{\partial p}}{{\partial t}}$$$

how would I select boundary conditions at infinity to ensure that the wave will continue to propagate approaching infinity?
Often with guided waves, the field and the gradient of the field go to zero at infinity. This results in waves that do not lose energy as a radiation field.

Offhand, I don't know a good general book, but optics and electromagnetic books often deal with guided wave propagating modes For example, I remember reading a book "Light Transmission Optics" by D. Marcuse, years ago.

stevenb: Thank you for your response! So I think that this would imply that:

$$${\bf{p}} = 0$$$

$$$\nabla {\bf{p}} = 0$$$

Does this also apply to waves in free-field conditions? What about acoustic waves?

Might this also imply the following at infinity?

$$$\frac{{\partial p}}{{\partial t}} = 0$$$

$$$\frac{{\partial ^2 p}}{{\partial t^2 }} = 0$$$

If the field and the gradient of the field go to zero as we approach infinity, then would it be reasonable to suggest that

$$$0 = A\frac{{\partial ^2 p}}{{\partial t^2 }} + B\frac{{\partial p}}{{\partial t}}$$$

Born2bwire
Gold Member
Often with guided waves, the field and the gradient of the field go to zero at infinity. This results in waves that do not lose energy as a radiation field.

Offhand, I don't know a good general book, but optics and electromagnetic books often deal with guided wave propagating modes For example, I remember reading a book "Light Transmission Optics" by D. Marcuse, years ago.
Most often though requiring uniqueness to the solution imposes infinite boundary conditions. The Sommerfeld radiation condition in electromagnetics dictates that at infinity the point source field must go to zero for the uniqueness of the solution to hold. This is different for an appropriately confined wave as you stated.

If the field and the gradient of the field go to zero as we approach infinity, then would it be reasonable to suggest that

$$$0 = A\frac{{\partial ^2 p}}{{\partial t^2 }} + B\frac{{\partial p}}{{\partial t}}$$$
I think you need to take into more consideration the physics of the problem. First, I don't think it is very useful to arrange a PDE of the limiting behavior. Rather, it would probably be more useful to first solve the original PDE and then apply the boundary conditions to the solution. For example, many general wave equations will afford solutions that represent both incoming and outgoing waves. However, appropriate boundary conditions would require you to reject the incoming waves for example. This is something that you would apply after you have derived your general solution. Without knowing what this wave equation is and how it should behave, I would guess that the only thing you can set boundary conditions for is uniqueness as I mentioned above.

Chew's "Waves and Fields in Inhomogeneous Media" discusses solving wave equations for electromagnetics and addresses such topics in his first chapter, specifically 1.5 and he also talks about deriving the boundary conditions at a 1D planar inhomogeneity in 2.1. While these are specific to electromagnetics, he works directly from the PDE and uses mathematical arguments to derive these boundary conditions and solutions. So the techniques should be useful learning tools even for wave equations of different phenomenon.

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Most often though requiring uniqueness to the solution imposes infinite boundary conditions. The Sommerfeld radiation condition in electromagnetics dictates that at infinity the point source field must go to zero for the uniqueness of the solution to hold. This is different for an appropriately confined wave as you stated.

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Yes, very good point. I first interpreted his question to mean "guided solutions" because of his statement "how would I select boundary conditions at infinity to ensure that the wave will continue to propagate approaching infinity?". However, rereading this and considering the follow on questions, I'm not now sure if this interpretation is correct.

Marcuse's book does a good job of discussing radiation solutions as well. It's an old book, but it's a classic. I highly recommend it to the OP if he understands electromagnetics well enough.

stevenb and Born2bwire: These are extremely good comments, and I will take a look at these books that have been suggested. Thank you very much for helping to point me in the right direction!