Re: Wave reflection.

["Richard F. Lyon" <DickLyon@xxxxxxx>]

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:
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@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
> 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@xxxxxxxxxx .