# Waveguides (in the ocean?)

The speed of sound in the ocean is given by

$$c_s(\theta,z) = 1450 + 4.6\theta - 0.055\theta^2 + 0.016z$$

$\theta$ is the temperature in degrees celcius, and $z$ is the depth. In a simplified model, $\theta$ is constant at 10$\,^\circ$C for the part of the ocean above the "themocline". The thermocline is an interface at depth 700 m over which the temperature drops to 4$\,^\circ$C almost instantly.

The question: It claimed that the water below the thermocline can act as a waveguide. Why and what is the extent (in depth) of this waveguide?

My thoughts: Evaluating some relevant speeds: $c_s(10,700) = 1501.7 m\,s^{-1}$ and $c_s(4,700) = 1478.7 m\,s^{-1}$. As the speed changes at the thermocline, there will be refraction and reflection of incident waves from both sides (above and below).

So waves incident from below will be reflected back. However I don't understand what makes the waves reflect on the lower side of this "waveguide". As far as I can see, the speed of sound will increase with depth. If there is no interface with a sudden discontinuity like the thermocline, how does this situation work?

• For internal gravity waves, a region of strong stratification in the water column, i.e. the thermocline, can act as a waveguide for high frequency internal waves. – Isopycnal Oscillation Apr 18 '13 at 20:33

You don't need a sharp discontinuity in the speed of sound to guide the waves. Remember that reflection does not occur right at the interface; rather, the wave always penetrates outside the waveguide to "see" what's going on there. A gradual increase in the speed of sound enforces the wave to reflect as well.

Reflection occurs from above due to the thermocline (700 m), and from below due to the linear increase in the speed of sound, see the red shaded area in the following figure: Guiding from below is possible down to a depth where the speed of sound equals $c_s(10,700)$ which yields 2136.25 m. The reason for this is explained in the following.

A wave which is guided has different properties than a freely propagating wave. In particular, one can show that a guided wave has a larger speed (please note that a guided wave propagates with one single speed throughout the waveguide, it is incorrect to imagine it as a particle bouncing off, e.g., two mirrors) than the minimum speed in the waveguide (in this case $c_s(4,700)$) yet smaller than the maximum speed above or below the waveguide, whichever is smallest (in this case $c_s(10,700)$). The reason for this is because a guided wave in a slab waveguide can be conceptually decomposed into two WAVES (not particles!) bouncing off the "walls" of the waveguide thereby preventing them from leaking out. This interference between the two waves creates a standing wave pattern inside the waveguide perpendicular to the propagation direction. Hence, a wave with a speed larger than $c_s(10,700)$ would immediately leak out of the waveguide and propagate to the upper infinity.

• Ah ok, I understand better now. I will keep this in mind and review waveguide theory. Thanks! – Comp_Warrior Apr 21 '13 at 23:23
• @Petru Welcome to Physics Stackexchange. Since this is not a traditional forum, the standard here (not always adhered to strictly, but still) is to edit new content into your answer rather than have it in the comments or posted as a new answer. The idea is to craft a single really good answer, rather than make a long discussion thread that future readers will have to navigate. I suggest editing the content of one of these into the other. If you need help re-embedding the image (new user restriction, you'll be over that soon enough), you can ping me in a comment. Oh yeah, great answer btw. – user10851 Apr 22 '13 at 8:10
• @Chris, thanks a lot for the information. The idea of creating a single good answer is fantastic. I sense I'll have a great time on the forum. %) I merged the two answers. Cheers! – Petru Apr 22 '13 at 21:46

I wonder if you're mixing up the thermocline and the SOFAR channel. The speed of sound is a minimum at the SOFAR depth, so water at this depth acts as a waveguide.

I imagine sound will reflect off the thermocline, but I don't see how this would act as a waveguide unless the sea bottom acts as the lower reflector.

• I'm not sure, this is actually a part of a comprehensive exam question - and we did not study anything about the ocean. The exact wording of the question is : "Sketch the behaviour of the sound speed as a function of depth assuming the thermocline is located at a depth of about 700 m and has a negligible thickness over which the temperature drops abruptly from 10C to 4C. Explain why the water layer below the thermocline can act as a wave-guide and estimate the depth of the lower boundary of the wave-guide." – Comp_Warrior Apr 18 '13 at 12:41

You don't need a sharp discontinuity in the speed of sound to guide the waves. Remember that reflection does not occur right at the interface; rather, the wave always penetrates outside the waveguide to "see" what's going on there. A gradual increase in the speed of sound enforces the wave to reflect as well.

Reflection occurs from above due to the thermocline (700 m), and from below due to the linear increase in the speed of sound, see the red shaded area in the following figure: Guiding from below is possible down to a depth where the speed of sound equals $c_s(10,700)$ which yields 2136.25 m.

• I thought that as well about the lower edge. But I don't understand completely; why is the lower extent of the waveguide when the speed $c_s(10,700)$ is reached? It seems a bit arbitrary. For instance, why not a bit deeper? – Comp_Warrior Apr 21 '13 at 19:08