The Green's functions for a 3d wave are like δ(r - ct)/r -- so if you have static source at the origin that is turned on at t=0, you get an expanding ball around it of radius ct, with strength 1/r. If you look just at the XY plane, you see an expanding disc of value 1/r.(adsbygoogle = window.adsbygoogle || []).push({});

Similarly, if you turned the source on at t = 0 and off at t=1, you would get a an expanding spherical shell of radius ct and thickness c, or an expanding annulus in the XY plane.

However, for 2d waves, there is an "afterglow" -- I don't recall the exact Green function, but rather than a δ-function, it's a lograthmic or exponential decay, and there's something similar for 1-D waves -- e.g. If someone turns a light on for one second, you will still see lit for longer than 1s, though it will get dimmer as time goes on.

This is often discussed around Huygens' principle -- IIRC the principle is the "no afterglow" rule for 3D, which holds in odd dimensions > 1.

However, because of symmetry, the spherical wave equation satisfies the one-dimensional wave equation: (rV),tt = (rV),rr -- where r is the distance from the origin (√xx+yy+zz), V(r, t) is the amplitude a distance r from the origin, and t is the time.

So, then, shouldn't our point source at the origin - which is obviously spherically-symmetrical -- exhibit the afterglow,and therefore *not* give the simple constant 1/r dependence? E.g. Since there is an afterglow, the value at r is not just affected by the source at t=r/c, but also all previous times, leading to (apparently) V -> infinity as t-> infinity?

Similarly, it would seem th XY plane (or any plane through the origin) would satisfy a 2D wave equation, by symmetry...but I'm not so sure there.

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# 3D spherical vs 2D radial waves

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