Underwater Acoustics

Many useful issues can be addressed using the concept of a simple source, as shown in the Concepts home page animation. This is a source which is effectively omnidirectional, either due to its size or shape. Clearly, a piezoelectric ceramic sphere, driven in the "breathing" mode will excite spherically spreading waves, but the same effect will be achieved by any small projector which creates the same change in volume, and which is small compared with a wavelength.

The source strength Q is defined as the rate of change of volume, or "volume velocity", so the SI units are m³/s. If the pulsating sphere shown on the entry page is measured, the maximum velocity of the surface in m/s can be multiplied by the surface area in m². This concept is valuable in deriving the theoretical effects of transducer reciprocity (linking receive and transmit sensitivities), and the effects of baffles, amongst other issues.

There is a simple relationship between the source strength Q and the acoustic pressure P at range r, and thus the source output P·r:

where ρ is the fluid density and f is the frequency of oscillation. This is applicable with no reverberation, giving spherically spreading sound energy.

It is beneficial to check that the units in this equation balance. The source output has units of Pa·m, N/m or kg/s². When divided by Q in m³/s, the ratio becomes (kg/m³)·(1/s), as expected.

The equation relates the magnitudes of P and Q, both of which are oscillatory and sinusoidal for a single frequency simple source. The relative phases of P and Q will change with range r. However, when the simple source is close to a rigid boundary, the acoustic pressure is doubled as the sound is reflected:

Note that the total output power in watts, proportional to P², but only integrated over a half space of 2π steradians, has doubled for the same source strength, due to the change in the impedance into which the source is being driven. By halving the solid angle, the compressibility also halves, doubling the impedance and source power requirement for a fixed Q.

Source strengths are not commonly quoted on a decibel scale, unlike the widely used source level, the decibel scale format for the source output, which is more immediately related to practical measurements.

The term 2r/(ρ·f) is known as the free field reciprocity factor for spherical spreading.