Not knowing much at all about sonar in particular, but something about signal transmission more in general, I feel we are mixing some concepts here. I speak as generally as I can here!
Some of the signal emitted above the DSC is certainly entering (or scattering into) the "acoustic waveguide" as it is very nicely described. This would be a function of angles, probably modeled as a point-wave source (Huygens-Fresnel approximation), or something like that. Within the layer, the signal smoothly curves up and, depth allowing, starts to rise again towards the surface.
What doesn't initially enter the DSC must essentially reflect back up. This would likely be detectable as "a first bounce" for a target above the DSC. What enters the DSC + what reflects back up = original signal - losses along the way.
The signal component that entered the DSC would start to smoothly turn up again.
Now, there are two phenomena pushing the wave that entered into the DSC back down: first, the same one that made it initially turn up in reverse, and, second, the reflection from surface layer phenomena. (There could be a bottom bounce as well, creating a reflected signal in such an angle that it penetrates the layer structure with ease, but I shall ignore that for now!)
The signal component smoothly turning back down (that is, not quite reaching the area of sharp change in the properties of acoustic medium) would continue traveling on in sinusoidal fashion until reaching... something. But the signal component actually reaching the layer structure of the sea water would both reflect back down in part, but would also scatter above the layer as a perhaps detectable signal, allowing a faint detection potentially at a very long distance.
The physics of the question revolve around the properties of the sea water in terms of how they layer. And as this is, in general, a question dependent on local conditions, it likely makes this fairly difficult to model, unless calculating a fine grid of local temperature profiles, and the other good stuff.
So, to summarize the nonsense, there should be a scattering component model, a reflected component model, and the wave guide travel model, each interacting, for the DSC! :)
< Message edited by AKar -- 6/12/2021 6:22:03 PM >