Underwater acoustics

Underwater acoustics is the study of the propagation of sound in water and the interaction of the mechanical waves that constitute sound with the water and its boundaries. The water may be in the ocean, a lake or a tank. Typical frequencies associated with underwater acoustics are between 10 Hz and 1 MHz. The propagation of sound in the ocean at frequencies lower than 10 Hz is usually not possible without penetrating deep into the seabed, whereas frequencies above 1 MHz are rarely used because they are absorbed very quickly. Underwater acoustics is sometimes known as hydroacoustics.

The field of underwater acoustics is closely related to a number of other fields of acoustic study, including sonar, transduction, acoustic signal processing, acoustical oceanography, bioacoustics, and physical acoustics.

History
Underwater sound has probably been used by marine animals for millions of years. The science of underwater acoustics began in 1490, when Leonardo Da Vinci wrote,
 * "If you cause your ship to stop and place the head of a long tube in the water and place the outer extremity to your ear, you will hear ships at a great distance from you."

In 1687 Isaac Newton wrote his Mathematical Principles of Natural Philosophy which included the first mathematical treatment of sound. The next major step in the development of underwater acoustics was made by Daniel Colladon, a Swiss physicist, and Charles Sturm, a French mathematician. In 1826, on Lake Geneva, they measured the elapsed time between a flash of light and the sound of a submerged ship's bell heard using an underwater listening horn. They measured a sound speed of 1435 meters per second over a 17 kilometer distance, providing the first quantitative measurement of sound speed in water. The result they obtained was within about 2 % of currently accepted values. In 1877 Lord Rayleigh wrote the Theory of Sound and established modern acoustic theory.

The sinking of Titanic in 1912 and the start of World War I provided the impetus for the next wave of progress in underwater acoustics. Anti-submarine listening systems were developed. Between 1912 and 1914, a number of echolocation patents were granted in Europe and the U.S., culminating in Reginald A. Fessenden's echo-ranger in 1914. Pioneering work was carried out during this time in France by Paul Langevin and in Britain by A B Wood and associates. The development of both active ASDIC and passive sonar (SOund Navigation And Ranging) proceeded apace during the war, driven by the first large scale deployments of submarines. Other advances in underwater acoustics included the development of acoustic mines.

In 1919, the first scientific paper on underwater acoustics was published, theoretically describing the refraction of sound rays produced by temperature and salinity gradients in the ocean. The range predictions of the paper were experimentally validated by transmission loss measurements.

The next two decades saw the development of several applications of underwater acoustics. The fathometer, or depth sounder, was developed commercially during the 1920s. Originally natural materials were used for the transducers, but by the 1930s sonar systems incorporating piezoelectric transducers made from synthetic materials were being used for passive listening systems and for active echo-ranging systems. These systems were used to good effect during World War II by both submarines and anti-submarine vessels. Many advances in underwater acoustics were made which were summarised later in the series Physics of Sound in the Sea, published in 1946.

After World War II, the development of sonar systems was driven largely by the Cold War, resulting in advances in the theoretical and practical understanding of underwater acoustics, aided by computer-based techniques.

Sound waves in water
A sound wave propagating underwater consists of alternating compressions and rarefactions of the water. These compressions and rarefactions are detected by a receiver, such as the human ear or a hydrophone, as changes in pressure. These waves may be man-made or naturally generated.

Speed of sound, density and impedance
The speed of sound $$c \,$$ (i.e., the longitudinal motion of wavefronts) is related to frequency $$f \,$$ and wavelength $$\lambda \,$$ of a wave by $$c = f \cdot \lambda$$.

This is different from the particle velocity $$u \,$$, which refers to the motion of molecules in the medium due to the sound, and relates the plane wave the pressure $$p \,$$ to the fluid density $$\rho \,$$ and sound speed $$c \,$$ by $$p = c \cdot u \cdot \rho$$.

The product of $$c$$ and $$\rho \,$$ from the above formula is known as the characteristic acoustic impedance. The acoustic power (energy per second) crossing unit area is known as the intensity of the wave and for a plane wave the average intensity is given by $$I = q^2/(\rho c) \,$$, where $$q \,$$ is the root mean square acoustic pressure.

At 1 kHz, the wavelength is about 1.5 m. Sometimes the term "sound velocity" is used but this is incorrect as the quantity is a scalar.

The large impedance contrast between air and water (the ratio is about 3600) and the scale of surface roughness means that the sea surface behaves as an almost perfect reflector of sound at frequencies below 1 kHz. Sound speed in water exceeds that in air by a factor of 4.4 and the density ratio is about 820.

Absorption of sound
Absorption of low frequency sound is weak. (see Technical Guides - Calculation of absorption of sound in seawater for an on-line calculator). The main cause of sound attenuation in fresh water, and at high frequency in sea water (above 100 kHz) is viscosity. Important additional contributions at lower frequency in seawater are associated with the ionic relaxation of boric acid (up to c. 10 kHz) and magnesium sulfate (up to c. 100 kHz).

Sound may be absorbed by losses at the fluid boundaries. Near the surface of the sea losses can occur in a bubble layer or in ice, while at the bottom sound can penetrate into the sediment and be absorbed.

Boundary interactions
Both the water surface and bottom are reflecting and scattering boundaries.

Surface
For many purposes the sea-air surface can be thought of as a perfect reflector. The impedance contrast is so great that little energy is able to cross this boundary. Acoustic pressure waves reflected from the sea surface experience a reversal in phase, often stated as either a “pi phase change” or a “180 deg phase change”. This is represented mathematically by assigning a reflection coefficient of minus 1 instead of plus one to the sea surface.

At high frequency (above about 1 kHz) or when the sea is rough, some of the incident sound is scattered, and this is taken into account by assigning a reflection coefficient whose magnitude is less than one. For example, close to normal incidence, the reflection coefficient becomes $$R=-e^{-2 k^{2} h^{2} sin^2A}$$, where h is the rms wave height.

A further complication is the presence of wind generated bubbles or fish close to the sea surface. . The bubbles can also form plumes that absorb some of the incident and scattered sound, and scatter some of the sound themselves. .

Seabed
The acoustic impedance mismatch between water and the bottom is generally much less than at the surface and is more complex. It depends on the bottom material types and depth of the layers. Theories have been developed for predicting the sound propagation in the bottom in this case, for example by Biot and by Buckingham.

At Target
The reflection of sound at a target whose dimensions are large compared with the acoustic wavelength depends on its size and shape as well as the impedance of the target relative to that of water. Formulae have been developed for the target strength of various simple shapes as a function of angle of sound incidence. More complex shapes may be approximated by combining these simple ones.

Propagation of sound
Underwater acoustic propagation depends on many factors. The direction of sound propagation is determined by the sound speed gradients in the water. In the sea the vertical gradients are generally much larger than the horizontal ones. These facts, combined with a tendency for increasing sound speed with increasing depth due to the increasing pressure in the deep sea reverses the sound speed gradient in the thermocline creating an efficient waveguide at the depth corresponding to the minimum sound speed. The sound speed profile may cause regions of low sound intensity called "Shadow Zones" and regions of high intensity called "Caustics". These may be found by ray tracing methods.

At equatorial and temperate latitudes in the ocean the surface temperature is high enough to reverse the pressure effect, such that a sound speed minimum occurs at depth of a few hundred metres. The presence of this minimum creates a special channel known as Deep Sound Channel, previously known as the SOFAR (sound fixing and ranging) channel, permitting guided propagation of underwater sound for thousands of kilometres without interaction with the sea surface or the seabed. Another phenomenon in the deep sea is the formation of sound focussing areas known as Convergence Zones. In this case sound is refracted downward from a near-surface source and then back up again. The horizontal distance from the source at which this occurs depends on the positive and negative sound speed gradients. A surface duct can also occur in both deep and moderately shallow water when there is upward refraction, for example due to cold surface temperatures. Propagation is by repeated sound bounces off the surface.

In general, as sound propagates underwater there is a reduction in the sound intensity over increasing ranges, though in some circumstances a gain can be obtained due to focussing. Propagation loss (sometimes referred to as transmission loss) is a quantitative measure of the reduction in sound intensity between two points, normally the sound source and a distant receiver. If $$I_s$$ is the far field intensity of the source referred to a point 1 m from its acoustic centre and $$I_r$$ is the intensity at the receiver, then the propagation loss is given by $$PL=10log (I_s/I_r)$$. In this equation $$I_r$$ is not the true acoustic intensity at the receiver, which is a vector quantity, but a scalar equal to the equivalent plane wave intensity (EPWI) of the sound field. The EPWI is defined as the magnitude of the intensity of a plane wave of the same RMS pressure as the true acoustic field. At short range the propagation loss is dominated by spreading while at long range it is dominated by absorption and/or scattering losses.

An alternative definition is possible in terms of pressure instead of intensity, giving $$PL=10 log (p_s/p_r)$$, where $$p_s$$ is the RMS acoustic pressure in the far-field of the projector, scaled to a standard distance of 1 m, and $$p_r$$  is the RMS pressure at the receiver position.

These two definitions are not exactly equivalent because the characteristic impedance at the receiver may be different from that at the source. Because of this, the use of the intensity definition leads to a different sonar equation to the definition based on a pressure ratio. If the source and receiver are both in water, the difference is small.

Propagation modelling
The propagation of sound through water is described by the wave equation, with appropriate boundary conditions. A number of models have been developed to simplify propagation calculations. These models include ray theory, normal mode solutions, and parabolic equation simplifications of the wave equation. Each set of solutions is generally valid and computationally efficient in a limited frequency and range regime, and may involve other limits as well. Ray theory is more appropriate at short range and high frequency, while the other solutions function better at long range and low frequency. Various empirical and analytical formulae have also been derived from measurements that are useful approximations.

Reverberation
Transient sounds result in a decaying background that can be of much larger duration than the original transient signal. The cause of this background, known as reverberation, is partly due to scattering from rough boundaries and partly due to scattering from fish and other biota. For an acoustic signal to be detected easily, it must exceed the reverberation level as well as the background noise level.

Doppler Shift
If an underwater object is moving relative to an underwater receiver, the frequency of the received sound is different from that of the sound radiated (or reflected) by the object. This change in frequency is known as a Doppler shift. The shift can be observed in active sonar systems, particularly narrowband ones, because the transmitter frequency is known, and the relative motion between sonar and object can be calculated. Sometimes the frequency of the radiated noise (a tonal) may also be known, in which case the same calculation can be done for passive sonar. For active systems the change in frequency is 0.69 Hz per knot per kHz and half this for passive systems as propagation is only one way. The shift corresponds to an increase in frequency for an approaching target.

Sound Fluctuations
Though acoustic propagation modelling generally predicts a constant received sound level, in practice there are both temporal and spatial fluctuations. These may be due to both small and large scale environmental phenomena. These can include sound speed profile fine structure and frontal zones as well as internal waves. Because in general there are multiple propagation paths between a source and receiver, small phase changes in the interference pattern between these paths can lead to large fluctuations in sound intensity.

Non-linearity
In water, especially with air bubbles, the change in density due to a change in pressure is not exactly linearly proportional. As a consequence for a sinusoidal wave input additional harmonic and subharmonic frequencies are generated. When two sinusoids are input sum and difference frequencies are generated. The conversion process is greater at high source levels than small ones. Because of the non-linearity there is a dependence of sound speed on the pressure amplitude so that large changes travel faster than small ones. Thus a sinusoidal waveform gradually becomes a sawtooth one with a steep rise and a gradual tail. Use is made of this phenomenon in parametric sonar and theories have been developed to account for this, eg by Westerfield.

Measurements
Sound in water is measured using a hydrophone, which is the underwater equivalent of a microphone. A hydrophone measures pressure fluctuations, and these are usually converted to sound pressure level (SPL), which is a logarithmic measure of the mean square acoustic pressure.

Measurements are usually reported in one of three forms :-
 * RMS acoustic pressure in micropascals (or dB re 1 μPa)
 * RMS acoustic pressure in a specified bandwidth, usually octaves or thirds of octave (dB re 1 μPa)
 * spectral density (mean square pressure per unit bandwidth) in micropascals per hertz (dB re 1 μPa²/Hz)

Sound speed
Approximate values for fresh water and seawater, respectively, at atmospheric pressure are 1450 and 1500 m/s for the sound speed, and 1000 and 1030 kg/m³ for the density. The speed of sound in water increases with increasing pressure, temperature and salinity. On-line calculators can be found at Technical Guides - Speed of Sound in Sea-Water and Technical Guides - Speed of Sound in Pure Water.

Absorption
Many measurements have been made of sound absorption in lakes and the ocean (see Technical Guides - Calculation of absorption of sound in seawater for an on-line calculator).

Ambient noise
Measurement of acoustic signals are possible if their amplitude exceeds a minimum threshold, determined partly by the signal processing used and partly by the level of background noise. Ambient noise is that part of the received noise that is independent of the source, receiver and platform characteristics. This it excludes reverberation and towing noise for example.

The background noise present in the ocean, or ambient noise, has many different sources and varies with location and frequency. At the lowest frequencies, from about 0.1 Hz to 10 Hz, ocean turbulence and microseisms are the primary contributors to the noise background. Typical noise spectrum levels decrease with increasing frequency from about 140 dB re 1 μPa²/Hz at 1 Hz to about 30 dB re 1 μPa²/Hz at 100 kHz. Distant ship traffic is one of the dominant noise sources in most areas for frequencies of around 100 Hz, while wind-induced surface noise is the main source between 1 kHz and 30 kHz. At very high frequencies, above 100 kHz, thermal noise of water molecules begins to dominate. The thermal noise spectral level at 100 kHz is 25 dB re 1 μPa²/Hz. The spectral density of thermal noise increases by 20 dB per decade (approximately 6 dB per octave).

Transient sound sources also contribute to ambient noise. These can include intermittent geological activity, such as earthquakes and underwater volcanoes, rainfall on the surface, and biological activity. Biological sources include cetaceans (especially blue, fin and sperm whales), certain types of fish, and snapping shrimp.

Reverberation
Many measurements have been made of sea surface, bottom and volume reverberation. Empirical models have sometimes been derived from these. A commonly used expression for the band 0.4 to 6.4 kHz is that by Chapman and Harris. It is found that a sinusoidal waveform is spread in frequency due to the surface motion. For bottom reverberation a Lambert's Law is found often to apply approximately, for example see Mackenzie. Volume reverberation is usually found to occur mainly in layers, which change depth with the time of day, e.g., see Marshall and Chapman. The under-surface of ice can produce strong reverberation when it is rough, see for example Milne.

Bottom Loss
Bottom loss has been measured as a function of grazing angle for many frequencies in various locations, for example those by the US Marine Geophysical Survey. The loss depends on the sound speed in the bottom (which is affected by gradients and layering) and by roughness. Graphs have been produced for the loss to be expected in particular circumstances.

Comparison with airborne sound levels
As with airborne sound, sound pressure level underwater is usually reported in units of decibels, but there are some important differences that make it difficult (and often inappropriate) to compare SPL in water with SPL in air. These differences include:
 * difference in reference pressure: 1 μPa (one micropascal, or one millionth of a pascal) instead of 20 μPa.
 * difference in interpretation: there are two schools of thought, one maintaining that pressures should be compared directly, and that the other that one should first convert to the intensity of an equivalent plane wave.
 * difference in hearing sensitivity: any comparison with (A-weighted) sound in air needs to take into account the differences in hearing sensitivity, either of a human diver or other animal.

Hearing sensitivity
The lowest audible SPL for a human diver with normal hearing is about 67 dB re 1 μPa, with greatest sensitivity occurring at frequencies around 1 kHz. Dolphins and other toothed whales are renowned for their acute hearing sensitivity, especially in the frequency range 5 to 50 kHz. Several species have hearing thresholds between 30 and 50 dB re 1 μPa in this frequency range. For example the hearing threshold of the killer whale occurs at an RMS acoustic pressure of 0.02 mPa (and frequency 15 kHz), corresponding to an SPL threshold of 26 dB re 1 μPa. By comparison the most sensitive fish is the soldier fish, whose threshold is 0.32 mPa (50 dB re 1 μPa) at 1.3 kHz, whereas the lobster has a hearing threshold of 1.3 Pa at 70 Hz (122 dB re 1 μPa).

Safety thresholds
High levels of underwater sound create a potential hazard to marine and amphibious animals as well as to human divers. Guidelines for exposure of human divers and marine mammals to underwater sound are reported by the SOLMAR project of the NATO Undersea Research Centre. Human divers exposed to SPL above 154 dB re 1 μPa in the frequency range 0.6 to 2.5 kHz are reported to experience changes in their heart rate or breathing frequency. Diver aversion to low frequency sound is dependent upon sound pressure level and center frequency.

Marine biology
Due to its excellent propagation properties, underwater sound is used as a tool to aid the study of marine life, from microplankton to the blue whale. Echo sounders are often used to provide data on marine life abundance, distribution, and behavior information. Echo sounders, also referred to as hydroacoustics is also used for fish location, quantity, size, and biomass.