Sound Refraction

Wave rays are imaginary lines that are drawn at right angles to the crests of waves. The term refraction refers to the bending of wave rays due changes in the speed of propagation of the sound wave. Therefore changes of T, p and S can lead to changes in the speed of propagation of sound and consequently refraction. If wave rays are refracted in such a way that wave rays converge then energy is focused in a similar way to a lens might focus light. However, if the wave rays diverge (move apart) then the reverse is true and a reduction in sound intensity results.

The equation governing wave refraction is Snell's Law and is given in two forms below:

Equations 1a & 1b

You can find a simple derivation of Snell's Law in Urick (1975) page 116.

A diagramatic representaion of the parameters in Snell's Law is given in the following figure:

You will notice that the wave ray is refracted upwards towards the velocity minimum. This is an important rule of wave refraction that you should remember.

The diagram shows two different layers in which the sound velocities are c₁ and c₂, respectively where c₁ < c₂. The symbol theta refers to the grazing angle and alpha represents the vertical angles. Snell's Law allows us to compute how much a wave ray will be refracted if a velocity change occurs. Rearranging Snell's Law in terms of the grazing angle gives:

Equation 2 

Now we can compute the refracted wave angle given the incident wave angle and the velocities of the two layers.

Exercise:
A boat is using an echo sounder sounder in an estuary. Here fresh water directly overlies a saline layer at depth. The sound ray emitted from the sounder has a grazing angle of 85 degees at the interface between the fresh and saline water. The sound velocities in the fresh and saline layers are 1425.87m/s and 1518.89m/s respectively. Calculate both the grazing angle of the refracted wave and the amount by which the ray has been refracted.

If you experiment with Snell's law you will soon find that for the example given above if you reduce the grazing angle below 20.16 degrees your calculator will return an error. This is because the value in the brackets in Equation 2 is greater than 1 for these values and you cannot calculate a sine or cosine for values greater than 1. What does this mean in physical terms? This critical angle at which Snell's law fails is termed the CRITICAL ANGLE, it represents the point at which none of the incident wave energy propagates into the lower layer. What happens to the wave energy? The wave energy is reflected at the interface at an angle that is equal to the angle of incidence. The following equation will allow you to compute the critical angle for a given problem.

Try using the equation for the exercise given above and check that your answer agrees with the value of 20.16 degrees.