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Let's say I have an infinite line source in 3d wave medium, like a pulsating cylinder, emitting a broadband noisy signal. I know that the pressure falls off at 3 dB per doubling in distance, unlike a point source which falls off at 6 dB per doubling in distance, because the cylindrical wavefront is only spreading out along one direction.

But I am unclear on whether this has a frequency-dependent effect, and on the relationship of this situation to the http://en.wikipedia.org/wiki/Stationary_phase_approximation.

Is the signal received by a listener some distance from the line frequency-dependent? Does it roll-off at high frequencies?? And am I posting at the wrong site?

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migrated from math.stackexchange.com Apr 7 '11 at 3:17


Probably needs transferring to physics stack exchange (which loves problems like this). –  Carl Brannen Apr 7 '11 at 0:04
In physic.se one would like to know what kind of wave and medium Matt talks about. Electromagnetic? Sound? Medium homogenous? Isotropic? –  Georg Apr 7 '11 at 8:42

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If you let me interpret 'broadband, noisy', 'pressure', and 'pulsating cylinder' creatively enough, sure, you can get behavior that deviates from $1/r$, at least for a little while.

One can construct a carefully rigged source such that the waves it emits fall off in funny ways. For example, suppose your waves are light, your wave medium is vacuum, and your line source emits two short pulses of light simultaneously. One pulse propagates away perpendicular to the line source; the other pulse propagates at a small angle $\theta$ away from the line source.

The two pulses will both look like expanding cylindrical shells, but they will have different velocities in the radial direction, since one pulse is also carrying energy in the axial direction. Each individual pulse will fall off like $~1/r$, but they will eventually cease to overlap, so their summed intensity won't fall off like $1/r$.

If your broadband, noisy source is quasi-cw and incoherent, and your wave medium is not absorptive or reflective, however, I think it will have to emit waves with a $1/r$ intensity falloff to conserve energy. I don't see how this would depend on frequency, and I don't see what this has to do with the stationary phase approximation. Is this the specific case you're interested in? If so, I can try to justify these statements.

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