Liouville-Green approximation. ("reinifrosch@xxxxxxxx" )

Subject: Liouville-Green approximation.
From:    "reinifrosch@xxxxxxxx"  <reinifrosch@xxxxxxxx>
Date:    Fri, 18 Aug 2006 10:54:49 +0000

Dear colleagues,

A week ago I recommended the introduction into the
WKB (Wentzel, Kramers, Brillouin) [or Liouville-Green] 
approximation in section 9.1 of the book "Physics of 
Waves" by W. C. Elmore and M. A. Heald (Dover, 
New York, 1969).

While I still think that their treatment is recommendable 
for non-mathematicians (like me), I now find that some 
details are not perfect. 

For those who own the book:

After Equation (9.1.12) they wrote: 
"Substitution of (9.1.10) and (9.1.12) into (9.1.6) gives ..."

(9.1.10), however, is no longer valid if (9.1.12) is adopted.
Equation (9.1.13) must be replaced by an equation 
involving derivatives of the function epsilon(x).

A function similar to the right-hand side of (9.1.13), 
however, can be obtained in a simpler way, by 
requiring that the neglected second derivative A'' 
in (9.1.6) be small compared to the second term in 
(9.1.6), omega^2 * A / c^2. That new function (which
is required to be << 1) is twice the right-hand side 
of (9.1.13).

A second point:

After Equation (9.1.13), Elmore and Heald wrote:
"... the term containing the second derivative c'' will 
generally be negligible compared with that 
containing c'^2."

In the case of the often-used cochlear model 
c(x) = c(0) * e^[-x/(2d)],
where d is about 5 mm, the above statement is wrong. 

For a "stiffness-dominated" basilar-membrane 
impedance (featuring a real wave number k and so 
a real phase velocity c = omega / k ), I find the 
following WKB criterion:

? 2 c'' * c - c'^2 ? << 4 omega^2 ,

or equivalent inequalities involving the local wave 
number k or the local wavelength lambda.

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 .