55 Geometrical attenuation

An intuitive overview of geometrical spreading can be found in the DOSITS webpage Sound Spreading. We will develop this concept mathematically here, building on concepts that we studied in previous weeks.

Acoustic waves are nondispersive, and thus we can apply what we learned about dilution or concentration of wave energy due the refraction of wave rays by wave speed variations within the fluid. A very important example of sound wave refraction is the creation of the sound channel due to a sound speed minimum at mid-depth in the open ocean. The focusing of sound energy within this “SOFAR channel” carries acoustic signals very long distances through the ocean.

But let’s begin by considering more simple geometrical ray patterns. If there is a localized source of sound within the ocean, then the wave energy will become diluted as wave rays spread out from that source. Recall that the wave energy flux, also called its intensity, is given by

    \[ I = \frac{P^2}{\rho C}, \]

where the sound impedance Z=\rho C. As sound spreads from a localized source, its energy is spread over a larger area, and the intensity (W m-2) will decrease. This kind of reduction in intensity (and associated sound pressure) is called geometrical attenuation. There are two limiting cases.

The first, spherical spreading, corresponds to a point source radiating wave crests that are three-dimensional spheres.  In this case, the area of the spherical shell over which the energy is spread depends on the square of the distance from the source, A=4 \pi r^2.  By convention, a reference intensity, I_0, is located at a 1 m distance from an acoustic source. The total energy transport through the spherical shell at 1 m distance is I_0 4 \pi 1^2=4 \pi I_0. The total energy transport must be the same through the spherical shell located at r meters from the source, 4 \pi I_0 = 4 \pi r^2 I(r). Thus, the intensity at radius, is

    \[ I(r) = \frac{I_0}{r^2} \]

The second, cylindrical spreading, corresponds to the situation where the sound has already reached the top and bottom of the ocean and is repeatedly reflecting from both boundaries. Most of the energy that would have left the system is redirected by these reflections back into the ocean interior (with some loss), i.e., it becomes channelized and subsequently can be thought of as spreading in two- rather than three-dimensions.  This purely horizontal spreading results in distribution of energy over cylindrical shells of area A=2 \pi r H, where H is the depth of the ocean. The intensity at a distance, r, from the source is

    \[ I(r) = \frac{I_0}{r}. \]

Sound traveling in the sound channel behaves the most like cylindrical spreading, because in this case the channelization is caused by continuous refraction rather than reflection off the top and bottom of the ocean; the latter process involves much more loss of sound energy due to interactions with the boundaries. Spherical spreading is usually applied only relatively close to a source, perhaps 1-5 km in distance depending on the depth of the ocean and source. In most modern applications, complex numerical models of sound propagation are used for greater accuracy.

As sound waves travel through the water, the intensity is also decreased due to chemical attenuation, as discussed on the DOSITS webpage Sound Absorption.

Key Takeaways

Two types of common geometrical spreading attenuate the sound intensity (I) with distance (r) from the source (assumed to be at r = 1 m)

Spherical spreading

    \[ I(r) = \frac{I_0}{r^2} \]

Cylindrical spreading

    \[ I(r) = \frac{I_0}{r} \]