NetNews Usenet Archive 1992 #31/NN_1992_31.iso/spool/sci/math/17421

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From: [e-mail] (RING, DAVID WAYNE)
Newsgroups: sci.math,sci.physics
Subject: Re: More on Huygens' principle
Date: 25 Dec 1992 22:03 CST
Organization: Texas A&M University, Academic Computing Services
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[e-mail] (Michael Weiss) writes...
>If one assumes that Huygens' principle holds in 3 spatial dimensions, one
>can deduce that it should fail in 2.  Hadamard called this argument the
>"method of descent".  Briefly: consider a pulse disturbance at t=0 along
>the entire z-axis.  It creates spreading waves that obviously will have no
>dependence on z, and hence satisfy the 2-dimensional wave equation.  Now we
>apply Huygens' principle in 3 dimensions--- that is, we add up spherical
>waves spreading from each (0,0,z) (i.e., integrate over z).  An observer
>positioned at (x,y,0) will of course "hear" the pulse at t = sqrt(x^2+y^2),
>due to the spherical wave spreading out from (0,0,0) but will also hear
>something at t = sqrt(x^2 + y^2 + z^2), thanks to the spherical wave from
>(0,0,z). QED (I was hoping there was a similarly intuitive argument for why
>Huygens' should work for 3 dimensions, but I guess not.)

I feel like something is missing. Wouldn't a similar line of reasoning
show that HP in 5 dimensions implies no HP in 3?

Dave Ring
[e-mail]