From: tao@sonia.math.ucla.edu (Terence Tao) Subject: Re: Question on the Wave Equation Date: 30 Nov 1999 07:11:09 GMT Newsgroups: sci.math.research Keywords: Huygens' principle (no diffusion in odd dimensions) In article <09920fb9.990310f6@usw-ex0107-042.remarq.com>, Enrico Bernardini wrote: >I am reading Charles Nash's "Differential Topology and Quantum Field >Theory", and found a rather enigmatic passage in the second chapter. >Here the author discusses the wave equation with N space and 1 time >dimensions. He writes that for N = 3, 5, 7, etc. the support and the >non-singular support of the Green's function G(x,t) associated with >this problem coincide, and are thus light cones. Since the Green's >function is responsible for the propagation of the wave, this means >that waves do not diffuse in odd space dimensions - a fact which is >referred to as Huygens' principle. >What does the author mean by "diffuse" and how does this relate to >what physicists refer to as Huygens' principle? What happens in even >space dimensions? The sharp Huygens principle states that the support of a wave propagates at the speed of light and never below the speed of light in odd dimensions. Thus for instance if a wave is supported in the sphere { |x| < 1 } at time 0, then it will be supported in the annulus { T-1 < |x| < T+1 } at later times T > 1, with the wave vanishing inside the sphere { |x| < T-1 }. In even dimensions this is not true; the wave does pick up a non-zero component in the interior of the light cone, although this component tends to decay rather rapidly and is very smooth. So one only has the weak Huygens principle, which states that the support of a wave propgates at or below the speed of light. (Water waves, for instance, have this property: drop a pebble in water and you keep getting ripples well inside the expanding sphere given by the speed of water waves. This doesn't happen on a string or in air because these are odd-dimensional situations). What physicists generally refer to as Huygen's principle asserts that wave fronts (or singularities of a wave) propagate at c in the direction of the normal of the surface. This can be made rigorous mathematically using wave front sets and bicharacteristic flows in phase space, but I don't know if the principle has a formal name in mathematics. Terry ============================================================================== From: Stephen Montgomery-Smith Subject: Re: Question on Huygens' principle Date: Thu, 25 Nov 1999 21:54:26 -0600 Newsgroups: sci.math Enrico Bernardini wrote: [quoted as above --djr] Yes, I came across this a year or two ago, and it surprized me also. Here is an example: d^2u/dt^2 = c^2 sum_{i=1}^n d^2 u/dxi^2 u(0,x) = 0 d/dt u(0,x) = delta(x) where delta(x) is the Dirac delta function. In 3 dimensions, the solution is u(t,.) is a measure supported on the sphere of radius c t (multiplied by a suitable factor so as to preserve the total measure). That is, if there is a big explosion at the origin, the wave propagates as a shell moving outwards at velocity c t. In two dimensions, rather surpizingly, this is not true. The answer is someone more complicated. This means that if there is an instantaneous explosion at the origin in two dimensions, and I happen to be at the point (1,0) on the plane, I will not feel the effect of the explosion at one time (at time 1/c), but rather a will feel the effect of the explosion over a period of time "spread out." If you don't believe me, just try solving the equation and see what you get. (I think you gotta use Bessel functions or something like that.) This is rather like the heat equation du/dt = c^2 sum_{i=1}^n d^2 u/dxi^2 u(0,x) = delta(x) Here, in any number of dimension, you see that the solution is not a shell emanating from the origin, but that the heat spreads out, and gets "diffused." Hope this helps. -- Stephen Montgomery-Smith stephen@math.missouri.edu 307 Math Science Building stephen@showme.missouri.edu Department of Mathematics stephen@missouri.edu University of Missouri-Columbia Columbia, MO 65211 USA Phone (573) 882 4540 Fax (573) 882 1869 http://www.math.missouri.edu/~stephen