I'm trying to follow a derivation in a book on marine acoustics for finding the amplitude along a sound ray, as a function of the arc length, s, of the ray.(adsbygoogle = window.adsbygoogle || []).push({});

The book gives the following equation:

[tex]

2\frac{dA_0}{ds}+ \left[ \frac{c}{J}\frac{d}{ds}\left(\frac{J}{c}\right) \right]A_0 = 0

[/tex]

The book then says that by integrating the above equation, we get:

[tex]

A_0(s)= A_0(0)\left| \frac{c(s)J(0)}{c(0)J(s)} \right|^{1/2}

[/tex]

The book doesn't give any intermediate steps, and I'm not really sure how the integration is actually done. I gather that the limits are from 0 tos, but I don't know how you deal with something like this where J, c and A_0 all seem to depend ons.

In these equations,cis the sound speed at a given arc lengthsalong the ray, and J is the Jacobian determinant, given as:

[tex]

J = r\left[\frac{dr}{ds}\frac{dz}{d\theta} - \frac{dz}{ds}\frac{dr}{d\theta}\right]

[/tex]

or alternatively:

[tex]

J = r\left[\left(\frac{dz}{d\theta}\right)^2 + \left(\frac{dr}{d\theta}\right)^2\right]^{1/2}

[/tex]

So...can anyone explain to me the steps involved in going from the first equation to the second? Any help would be appreciated.

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# Integration question

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