# 4. FORMING THE ACOUSTIC EQUATIONS

This section of the manual describes the simplest practical situations for the use of parameters linked by the acoustic equations (formerly known as the sonar equations). These parameters relate to the water as a transmission medium for acoustic waves: to fish as acoustic targets in this medium and to the characteristics of the acoustic system.

It is a law of physics that energy can be neither gained, nor lost. This applies to acoustic waves as well as all other forms of energy. The acoustic equations help us to balance the quantities of acoustic energy transmitted and received, rather similar to the way a financial statement should balance. If the separate terms of the equation are accurate we can estimate fish biomass. The effectiveness of acoustic systems can be limited by the sea's own natural background of acoustic energy (noise), which is increased by transfer of energy from the wind, fast currents over certain types of seabed, Harden Jones and Mitson (1982), or from rain acting on the surface. It is necessary to know and understand the units of measurement, their quantities and the relationships which describe the parameters in the acoustic equation to manipulate them correctly.

In the water we are concerned primarily with transmitted acoustic source level measured in dB/1 m Pa/1 m, and in received echo level in dB/m Pa, but familiarity with the concepts of acoustic power (WA, Watts) and intensity I, is useful. To put energy into, and also to take it out of the water there is the energy converter, the transducer, discussed in 3.1.3. It is difficult to make actual acoustic measurements, so the electrical transmitted and received voltages and powers are used.

Echo-sounders and echo-integrators are calibrated in decibels, the use of which is standard for equipment in acoustic survey systems and in the acoustic equation. To see how these are used in calculations with acoustic energy we start by considering a transducer placed in the water. For our purpose the power is concentrated into a beam directed downwards.

## 4.1 Power and Source Level

We have seen how electrical power, when applied to a transducer, becomes acoustic power, or intensity. This section relates the factors which determine the acoustic intensity, or source level (SL) produced by a transducer.

Source Level is defined as

10 log (Intensity of source/reference intensity) (26)

where the reference intensity is that of a wave of rms pressure 1 m Pa.

Power input (Pe), to the transducer is in the form of an electrical pulse and is measured across the cable connections from the transmitter to the transducer. No transducer is 100% efficient in converting electrical to acoustic power, or vice versa. To find the acoustic power output (WA) for a given electrical power input it is necessary to know the transducer efficiency, (h) see 3.1.3. Supposing h is 70%, and the electrical input is measured as 1000 W, then the acoustic power (WA) is

WA = h Pe = (70 ÷ 100) x 1000 = 700 Watts.

What is needed for our acoustic equation calculations, is the SL which results from this acoustic power.

SL is expressed as

SL = 170.8 + 10 log WA + DI dB/1 m Pa/1 m (27)

i.e. reference 1 m Pascal at 1 metre on the axis; a one metre reference distance from the transducer face is normal.

170.8 is a constant for converting acoustic power to source level (see Urick (1975) P67).

DI is the directivity index, section 3.1.3.

From this information and using a figure of 23 dB for DI we can calculate SL at the reference distance of 1 m.

SL = 170.8 + 10 log WA + DI = 170.8 + 28.45 + 23 = 222.2 dB/1 m Pa/1 m

This is the axial transmitted acoustic intensity at the reference distance of one metre from the transducer face. Although source level can be calculated in this way it is not a satisfactory substitute for a direct or indirectly measured value obtained during acoustic calibration, because the electrical waveform is often distorted and Pe cannot always be measured accurately. Having obtained a figure for SL we can apply the-particular laws of propagation for acoustic waves and calculate the intensity for any given distance.

## 4.2 Decrease of Intensity with Distance

If an acoustic beam was infinitely narrow it would suffer no loss except for absorption. But when practical beams are propagated through water they spread so that the power WA covers a continually increasing area as distance from the source increases. We saw in 2.5 that acoustic intensity at any point is equal to power divided by area

I = WA ÷ Area

Knowing how the area increases with distance from the source we can calculate how intensity decreases with increasing distance. If a transducer radiated waves equally in all directions the waves would spread spherically from it. Even though we use transducers which confine the acoustic waves into a beam, the wavefront is still spherical ie it is a small part of the surface of an expanding sphere, see Figure 30.

Figure 30. From geometry we know that the area of the surface of a sphere of radius a is 4p a2. The radius of the sphere from which our beam is taken is the distance from the transducer to the wavefront. Therefore the acoustic intensity on the axis of the beam decreases in proportion to the distance squared.

Power = Intensity x Area so taking the ratio but the distance d1 is the reference distance of 1 m

so in decibel notation d2 represents any distance relative to the reference and is conventionally called R, the range from a source to a given distance.

Thus the normal expression is 20 log R and this is one component of the transmission loss factor TL. Although the term, range R, is more suited to the horizontally directed beam of sonar, it is useful in echo-sounding also. This is because all targets present at the same instant over the spherical surface of the wavefront are at the same range but they are not at the same depth. It will be seen later how these two factors occur in different situations but for the purpose of the acoustic equation, R will now be used throughout this manual.

So TL1 = 20 log R

Another component of transmission loss, is the absorption (a), discussed in 2.6, this follows a linear law with distance so it is added to the expression above in the form a R, and the full transmission loss TL is

TL = 20 logR + a R (28)

Using (28) we can calculate the intensity at a range of R metres from the transducer source. Let R = 50 m and a = 10 dB/km.

SL - TL = 222.2 - (20 log50 + 10 x 50 ÷ 1000) = 222.2 - (34 + 0.5) = 187.7 dB/1 m Pa

At 200 m SL - TL = 222.2 - (46 +2) = 174.2 dB/1 m Pa (see Figure 31(a) below.

Figure 31. (a) Thus the intensity is reduced most significantly by the spherical spreading or 'geometrical' loss, 20 log R, but as the range increases a becomes important. Because a increases greatly with frequency, its effect must be considered at relatively short ranges when high frequencies are used.

The one-way transmission loss is used above, but in order to obtain an 'echo-sounding' the echo must return in approximately the same direction as the transmitted pulse if it is to be received. The one-way transmission loss occurs over the distance travelled by the transmitted pulse and this added to the distance travelled by the echo, gives the, two-way transmission loss = 2TL. For the present purpose we assume that all the acoustic intensity at 200 m is returned towards the transducer.

SL-2TL = 222.2 - 2(20 logR + a R) = 222.2 - (40 logR + 2a R) = 222.2 - 2(48) = 126.2 dB/m Pa

as shown in Figure 31(b).

Figure 31. (b) This is the intensity or echo level (EL) received at the transducer face after the acoustic waves have travelled a total of 400 m on the axis of the beam.

EL = SL - 2TL (29)

Having seen how intensity decreases with range we now consider how it varies across the beam.

## 4.3 Variation of Intensity across the Beam

In 4.2 we examined the decrease of intensity with range from the transducer along the axis of the beam. As the beam gets wider with increasing range, so the wavefront area increases. The decrease in intensity with range can be calculated as shown in 4.2 and compensation applied by varying the amplification of received signals according to their range (by time varied gain i.e. TVG).

This would be quite sufficient if the intensity was constant over the area of the beam at any given depth. But, we saw in 3.1.3 that as the angle from the axis increases, so does the intensity decrease. Thus if we consider the area of the beam at a certain range, the acoustic intensity on the beam axis will be equal to SL - TL. But, at the reference angle from the axis it will be SL - TL - 3 dB. There is a gradual decrease in intensity from the axis to the -3 dB reference level which is illustrated by zones in Figure 32. These zones indicate what is probably the most difficult problem in fisheries acoustics, i.e. we do not know which zone fish are in at any given instant so it is impossible to make direct quantitative measurements with a simple echo-sounder. The effect of the beam pattern can be removed by a number of techniques but there is one in particular which is most widely used, Urick (1975), it forms the basis of section 4.6.

Figure 32. ## 4.4 Single Fish as Acoustic Targets

In 2.8 we saw some of the factors which determine the amount of acoustic energy reflected from fish and its variability. Here we are concerned with the most simplistic way in which single fish targets can enter the acoustic equation. It is shown in 4.2 that the echo level at the transducer when all acoustic energy reaching range R is reflected is

EL = SL - 2TL
or EL = SL - (40 logR + 2a R)

But we are normally concerned with small objects which intercept only a small proportion of the acoustic energy. Figure 33 illustrates this, and that (30)

Figure 33. If all incident energy was reflected and 10 log 1 = 0 db TS

so the echo level would remain unchanged by adding this value of TS.

EL = SL - 2TL + 0

A fish returning a small proportion of the energy has a much lower TS. Supposing the ratio of IR/I0 = 1/1000, i.e. the amount reflected is 1000 times less than the incident intensity, then

TS = 10 log 1/1000 = -30 dB

In a practical situation the EL would be received and we would want to extract TS, given the other factors

TS = EL + 2(20 logR + a R) - SL
TS = EL + 2TL - SL

Using the figures from 4.2, assuming EL = 96.2 dB/1m Pa when R = 200 m and a = 10 dB/km

TS = 96.2 + (92 + 4) - 222.2 = -30 dB.

Of course this applies only to the axis of the beam. Note that 40 logR occurs above, it is always associated with single targets

2TL = 40 logR + 2a R (31)

The target strength of a single fish has been defined, but this cannot be directly applied to large numbers of fish. Instead a figure of mean target strength for a mean length of a particular species, is related to weight so that a figure in dB/kg is obtained for use in the equation to determine biomass. Data are often available from which weight/length relationships can be calculated, these are in the form W = kLx

where

W = weight
k = species related constant
L = length
x is » 3, i.e. weight is proportional to the length cubed, (also species related).

## 4.5 Fish Schools as Simple Acoustic Targets

In 4.4 we assumed that one static, rigid fish reflected a 1/1000 part of the intensity incident upon it. The TS of this fish was found to be -30 dB. With similar reasoning we might say that 1000 fish each of the same TS would reflect all incident acoustic energy if they were filling the beam at precisely the same range. If the school TS is called TSs the relationship is

TSs = 10 logN + TS (32)

where N is the number of fish, each of target strength TS

TSs = 30 + (-30) = 0 dB.

In practise the target strength of schools is only useful when horizontally directed sonar beams are used to measure discrete schools. For the present form of acoustic equation we need a term to describe the amount of energy back-scattered from the school or layer to the echo-sounder transducer. This is known as volume reverberation which depends upon a ratio called scattering strength, in decibel terms it is The term EL has previously been used for the echo received at the transducer but when considering reverberation the analogous term RL, refers to the equivalent plane-wave reverberation level. It is defined as the level of an axially incident plane wave which produces the same transducer output as the reverberation. For general use of RL some assumptions must be made about the scatterers (fish) comprising the scattering layer.

1. The acoustic wave propagation must be in a straight line and the spreading loss taken into account.

2. Fish must be distributed with equal probability throughout the volume contained by half the pulse length at any given range.

3. There must be an absence of multiple scattering, (see Chapter 5).

Point 2 is especially relevant to the acoustic equation because it affects the transmission loss TL. This is because one way TL is 20 log R, i.e. when range increases by 2 times, the area of the wavefront increases by 22. Thus the number of targets intercepted by the beam increases in the same proportion as the TL which effectively cancels out the TL in one direction. This is the second form of the transmission loss equation, used for schools, or layers spread across the beam. Note that a is still a two-way loss.

TL2 = 20 logR + 2a R (33)

Volume reverberation is discussed in section 6.1 based on Urick (1975).

## 4.6 An Equivalent Transducer Beam

When the acoustic beam passes through a school or a layer of fish, fish spread across this beam at a given time and range. The reverberation level (RL) received at the transducer face is proportional to the number of fish and to their distribution across the beam. Even if they were all of the same TS, a small number would be in the 0 to -1 dB to -2 dB, and more still in the -2 dB to -3 dB zone (see Figure 32) the total RL would be much less than if the transmitted intensity were constant across the beam.

The solution is to calculate an equivalent beam. Within this ideal beam there is unity response, but outside, the response is zero. Figure 34 shows the comparison diagrammatically. Preferably measured parameters of the actual transducer are used in the calculation, which takes into account both transmission and reception i.e. two-way pattern, but a different formula is used for different shapes of the transducer face.

Figure 34. In logarithmic terms the formulae are

1. Circular transducers

10 log y = 20 log(l /2p a) +7.7 dB/Steradian (34)

where

l = wavelength (m)
a = radius of the transducer active face
a > 2l

2. Rectangular face transducer

10 log y = 10 log(l 2/4p ab) + 7.4 dB/Steradian (35)

where

l = wavelength (m)
a = length of one side of the active face (m)
b = length of the other face (m)
a, b >> l

These formulae are taken from p.217 of Urick (1975) and assume normal transducer characteristics, e.g. a properly formed main beam and the number, volume and sensitivity of the sidelobes. For accuracy a detailed three-dimensional measurement of the actual beam pattern is needed from which an equivalent solid angle can be calculated. Special facilities are needed for this purpose.

Schools of fish intercept and re-radiate some of the acoustic energy in the echo-sounder pulse. This re-radiation is called scattering and the sum of the scattering in a given volume of water is called the volume reverberation.

Because the scattering of interest goes back in the direction of the echo-sounder transducer it is often specified as 'back-scattering'. Reverberation is due to those fish within the 'ideal' beam contained in the volume of the equivalent beam y, the pulse duration t, and the range R.

## 4.7 Noise Level

The limit to the detection of fish in the sea is noise. In the acoustic equation the noise level NL at the face of the receiving transducer is defined as

NL = 10 log (noise intensity/reference intensity) (36)

where the reference intensity is that of a wave of rms pressure 1 m Pa.

Noise can arise from many sources, see 9.3, the present purpose is to consider it in relation to the acoustic equations.

Figure 35. Figure 35 shows how the ambient noise level varies with wind strength and sea-state, but note that this noise is given as spectrum level (SPL), which refers to the energy of an acoustic wave in a frequency band 1 Hz wide. The actual noise can then be calculated for any bandwidth and it emphasises that, the greater the bandwidth of a system, the more noise is received, regardless of its origin. The noise level affecting an echo-sounder of bandwidth BW is approximated by

BL = SPL + 10 log BW (37)

where BL = band-level of noise (dB/m Pa)

SPL = spectrum level of noise (dB/1m Pa/1Hz)
BW = bandwidth of the receiving system (Hz)

This approximation holds if the bandwidth is not too great, a condition met by present-day fisheries echo-sounders. It is based on adding together the intensities in the adjacent 1 Hz bands across the bandwidth.

To see the extent to which a given wind force or sea-state will affect an echo-sounder we take the spectrum level at the frequency of operation, eg 40 kHz. From Figure 35 the SPL at this frequency for wind force 3 is 30 dB/1 m Pa/1 Hz. An echo-sounder with a bandwidth of 3 kHz would therefore receive a total noise level from this source of

BL = 30 + 10 log 3000 = 64.8 dB/1 m Pa

To appreciate the practical significance of this noise level it must be compared to signals we wish to detect by expressing it in the form of voltage received across the transducer terminals (VRT).

Assuming a transducer with a receiving response (SRT) of -185 dB/1 Volt/1 m Pa,

VRT = BL + SRT = 64.8 + (-185) = -120.2 dB/1 Volt » 0.98 m V

(note the BL is effectively EL or RL).

If the wind increased to Beaufort Scale force 8 the SPL = 42 dB/1 m Pa/1 Hz so that

BL = 76.8 dB/1 m Pa
VRT = 76.8 + (-185) = -108.2 dB/1 Volt = 3.9 m V of noise.

Maximum sensitivity of an echo-sounder is likely to be 1 m V or even less, hence this level of noise could restrict the detection of fish. However for quantitative measurements a signal-to-noise ratio of 10-20 dB is necessary so maximum sensitivity cannot be used always.

Using the equations discussed in section 4, we can calculate the signal voltage from a fish of known TS at a given depth and compare it to the noise. Assuming that an echo-sounder has an SL of 216 dB/1m Pa/1m and it is required to detect a fish of TS = -45 dB at a range of 200 m. Let a » 8.7 dB/km

EL = 216 - 2(20 log 200 + (8.7 x 200)/1000) + (-45) = 216 - 95.5 - 45 = 75.5 dB/1m Pa
VRT = 75.5 + (-185) = -109.5 dB/1 Volt = Antilog - 109.5/20 = 3.35 m V of signal

The fish signal is 0.55 m V less than the average noise and cannot be detected.

Wind-induced noise has been used above to illustrate how noise may be included in the acoustic equations but there are other sources of noise (see section 9.3).

There is rarely a provision for noise to be monitored automatically during a survey. A deterioration in weather conditions often warns the operator before noise appears on the record, although if the system is operating near its maximum range capability the margin between an acceptable and an unacceptable noise level is small. An average level for noise can be taken from Figure 35 for a given wind force, but because of the variability of the wind this may not be adequate for very long. Rain causes a very significant increase in local noise levels and noise transients can occur when a ship slams into a heavy swell.

The speed at which a survey can be run often depends on the noise from the propeller, a factor with a high rate of change with speed, so it may be necessary to choose a survey speed well below that which causes noise to be integrated.

Threshold controls are provided on most integrators to minimise the effects of noise but they tend to bias the results so should never be used unless absolutely essential. Either way there is no direct insertion of a noise figure into the final acoustic equation.

## 4.8 Summary of Acoustic Equation Terms

Individual terms of the acoustic equations used in fishery surveys are briefly explained in Section 4. Further explanation of some of these and their application is given in Section 6.

### 4.8.1 Source Level

SL = 10 log (intensity of source/reference intensity)

where the reference intensity is that of a wave of rms pressure 1 m Pa.

SL is given in dB/1m Pa/1m.

### 4.8.2 Receiving Sensitivity

SRT is given in dB/1 Volt/1 m Pa.

### 4.8.3 SL + SRT

This is a combination of the two parameters which is most conveniently obtained during calibration (see section 7). It avoids difficulties inherent in separate measurements of SL and SRT.

### 4.8.4 Transmission Loss

TL = 20 logR + a R

The one-way loss due to spreading and absorption. Not normally used in fisheries acoustics.

2TL = 40 logR + 2a R

Two-way loss for single targets.

TL2 = 20 logR + 2a R

Two-way loss for schools or layers.

### 4.8.5 Target Strength ### 4.8.6 Volume Back-Scattering Coefficient ### 4.8.7 Reverberation Level where the reference intensity is that of a plane wave of rms pressure 1 m Pa.

### 4.8.8 Beam Factor

10 log y dB

The equivalent beam of solid angle y steradians derived by integration of the actual beam pattern.

### 4.8.9 Biomass Calculation

Biomass is defined as the density of fish (tonnes per square nautical mile) in the area surveyed, derived from the integrated echoes. The integrator output (Vo) is multiplied by a factor which includes

 SL + SRT dB target strength of the species being surveyed dB/kg equivalent beam factor dB and other instrumentation factors such as transmitted pulse duration dB repetition rate of transmitted pulse dB system gain dB integrator constant dB

These factors are considered in section 8.4.1.