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Re: Wave reflection.

If 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:
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.


At 8:20 PM +0200 8/10/06, reinifrosch@xxxxxxxxxx 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

(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@xxxxxxxxxx .