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Couple of days ago I discovered a YouTube channel SmarterEveryDay, where I watched the submarine Nuclear Submarine Deep Dive series. I found it very interesting especially the episode about the Sonar.

I am university student and we only briefly mentioned acoustics, so I tried to challenge myself. Currently I find the topic of surface duct very interesting.

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The change of propagation speed with depth will manifest itself through refraction of the sound. A graphical method of illustrating the effects is called ray tracing. The basic idea is to draw lines perpendicular to the wave fronts and follow their paths.

As the rays enter deeper water the propagation speed increases and the rays bend upwards. When the rays reach the surface, the will be reflected back downwards and the same process begins again. Naturally, some of the energy is lost and the reflection, but the overall effect is to trap the sound in a relatively small layer below the surface. This effect is called a surface duct.

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I have a multiple part question about this phenomenon.

I would like to get analytical formula for; how does the distance $d$ beetween the sonar (point-source of sound waves) and the reflection point depend on angle $\theta_0$ at which we emit waves.

I tried something like this $$\nabla^2 {\bf u} - \frac{1}{c^2}\frac{\partial^2 {\bf u}}{\partial t^2} = A \; ,$$ where $\bf u$ is vector field of particle velocity and $A$ is our sonar/point-source which looks like $$A = \alpha e^{-i\omega t} \delta({\bf r}) \delta(\theta - \theta_0) \; .$$ Is this correct???

Our sonar is positioned directly at surface at origin of our coordinate system therefore $\delta({\bf r})$, and I am interestied only at waves emited at angle $\theta_0$ therfore $\delta(\theta - \theta_0)$. Does my differential equation look alright or should I use acoustic pressure $p$ instead of ${\bf u}$? If this is the case, how should I formualte my eqaution because the pressure in water changes with deepth $z$ like $\rho g z$?

Then I tried to use ${\bf u}({\bf r}, t) = {\bf \hat{u}}({\bf r})e^{-i\omega t}$ to get rid of time dependence and got $$\nabla^2 {\bf \hat{u}} - \frac{\omega^2}{c^2} {\bf \hat{u}} = \alpha \delta({\bf r}) \delta(\theta - \theta_0) \; .$$

But new problem arises, the speed of sound $c$ is not a constant, but is some linear function of depth $z$, like $c(z) = k_0 z + c_0$. How to get solutions of such differential equation? I tried using Green function, but on wikipedia where there is table of Green funstions for different operators the one I need isn't listed there. Should I derive it myself? How would that go? And In which coordinates is best to solve given differential equation (cartesic, cylindrical or spherical)?

Yet again another problem! Even if I find Green function for our function, I have to know the boundary codnition at surface. It can be one of two, $\frac{dp}{dt}|_{surface} = 0$ or $p|_{surface} = 0$. Which one is correct and why exactly?

Thanks for every bit of help!!! I will update the questin as I go!

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The acoustic wave equation (time independent Helmholtz-Equation with the assumption of time-harmonic excitation of the form $exp(\pm iwt)$) is typically solved in potential form where $u$ would be the velocity potential in your equation.

$$\nabla u(x,y)+k^2u(x,y)=0$$

where $k=\omega/c$ is the wave number. If theres an isotropic source (= Greens Function $G(x,y)=exp(ikr)/(4\pi r)$) present in the medium you get

$$\nabla G(x,y)+k^2G(x,y)=\delta(x,y)$$

Deriving Greens Functions is tough. Mostly they are guessed through physical intuition of the underlying problem. I dont think theres a general way to derive Greens Functions for PDEs.

The ratio of the wave impedance of water and air is very large that the waves are mostly reflected and the boundary behaves "sound hard". The problem could therefore be solved as a Neumann problem (assuming $\partial u/\partial n =0$, where $n$ is the surface normal vector) in a good approximation.

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