Subject: Re: Wave reflection. From: "Richard F. Lyon" <DickLyon@xxxxxxxx> Date: Thu, 10 Aug 2006 17:07:44 -0700If this WKB validity issue matters to you, you better check out the appendix to Lloyd Watts's JASA paper (L. Watts, "The Mode-Coupling Liouville-Green Approximation for a two-dimensional Cochlear Model", Journal of the Acoustical Society of America, vol. 108, no. 5, pp. 2266-2271, Nov., 2000), which you can find linked here: http://www.lloydwatts.com/cochlea.shtml He says the usual criterion is only "first order" correct, and that in fact the WKB solution remains valid to much smaller k values (longer wavelengths) than it suggests; so the long-wave solution is OK near the base, where this criterion says it should not be. He also shows what goes wrong past resonance, and how to fix it. Dick At 8:20 PM +0200 8/10/06, reinifrosch@xxxxxxxx wrote: >Hello again ! > >I just found a good introductory treatment on nearly >reflection-free waves, in the book "Physics of Waves" >by W. C. Elmore and M. A. Heald (Dover, New York, 1969). >In their section 9.1, they show that the WKB (Wentzel, Kramers, >Brillouin) approximation is reflection-free, and that it is >accurate if the local wavelength lambda obeys the following >inequality: > >(d lambda / dx)^2 << 32 pi^2 . [their equation (9.1.15)] > >The corresponding inequality for the local wave number k is: > >k^-4 * (dk / dx)^2 << 8 . > >Reinhart Frosch. > > > > > >Reinhart Frosch, >Dr. phil. nat., >r. PSI and ETH Zurich, >Sommerhaldenstr. 5B, >CH-5200 Brugg. >Phone: 0041 56 441 77 72. >Mobile: 0041 79 754 30 32. >E-mail: reinifrosch@xxxxxxxx .