# 2. KEY PHYSICAL QUANTITIES

For a proper understanding of the acoustics used in fisheries it is necessary to have some knowledge of simple electricity, electromagnetic waves and acoustic waves.

An important purpose of this chapter is to explain the term power, a matter which is crucial to electrical and acoustic measurements. Power is defined as the amount of energy flow per unit time, or simply as the 'work done' in a given time. It is a well known law of physics that energy can neither be created nor destroyed, but is converted from one form to another. The process of conversion from the form we have, for example electrical energy, to the form we want, perhaps acoustic, cannot be 100% efficient and consequently some may turn into an unwanted form, often heat. It is vital that measurements in fisheries acoustics are based on the true power of the signals and to explain this fully we first consider direct electrical current, then alternating current which is in turn related to acoustic factors in section 2.4.

## 2.1 Direct Electrical Current (DC)

Electrical energy can be stored in many ways but the simplest for the present purpose is the chemical form in a battery of cells from which direct current (DC) is obtained (current means the flow of electrons in a circuit). The laws governing electrical matters are most easily understood by the application of DC to simple electrical circuits.

Figure 1.

A battery presents a steady potential difference (pd), or voltage (V) across its terminals Fig. 1, and if, for example, a lamp is connected to the terminals a direct current will flow through both circuit and battery. Work is being done because electrical energy is being converted to light as the current flows through the lamp filament. The amount of energy changed from electricity to light when measured over a period of time is known as the power (W, watts) dissipated and this is controlled by the voltage (V) of the battery and the amount of current (A) which is flowing. In fact a small proportion of the energy is 'wasted' in the form of heat but this is not important to the present explanation.

Power is the product of V, volts and A, amps

Power, (W, watts) = V x A (1)

Current flow A is directly proportional to V but is controlled by the resistance R (ohms) of the lamp filament.

A = V/R (2)

If we substitute eqn 2 for A in eqn 1

Power = V x V/R = V2/R Watts (3)

The relationship of power to work and energy is given in Appendix I under derived units. It is particularly important to remember that power is proportional to voltage squared.

Sources of direct current are used to provide power for the operation of electronic devices and systems but have no other useful function. Direct current can be produced from rotating machines (generators), but the natural output of any such machine is an alternating current wave, the description of which is given in the next section.

## 2.2 Alternating Electrical Current (AC)

Figure 3.

Electronic and acoustic systems which propagate and receive energy do so by the use of electromagnetic or acoustic waves. The characteristic properties of waves are the same in air, solids, or water. A wave is defined as a progressive disturbance in any of these media. It is formed by the propagation of alternating tensions and pressures, without permanently displacing the medium itself.

An alternating electrical wave can be produced by the rotation of a rectangular loop of wire at a constant speed in a magnetic field. This is shown diagrammatically in Figure 2(a). It can be demonstrated by the laws of electromagnetic induction that the current in the loop will vary according to the angle of the loop relative to the magnetic field. If terminal P is taken as a reference, rotating from 0° to 360°, the direction of the current and its amplitude can be plotted against the angle of rotation as in Figure 2(b). For convenience, intervals of 30° are used.

It is clearly seen that the wave is going in one direction for the first 180°, then it goes through zero and reverses direction (polarity) for the next 180°. The maximum values in each direction occur 90° after passing the zeroes. In its pure (undistorted) state, the amplitude and other characteristics of a sinewave can be easily measured but special care must be taken to avoid confusion. For example it is necessary to be absolutely clear in stating that values are from zero to maximum in one direction (peak) or, if quoting from the extreme peak of one direction to the extreme peak in the other direction (peak-to-peak), see Figure 3.

Although peak, or peak-to-peak values must be recorded, this is simply because they are the recognisable points of a sinewave, it is necessary to convert them before making any calculations of power. The reason is that we are concerned with the actual amount of energy involved, but, whereas with DC this could be obtained by the product of voltage V and current A, in the present case these quantities are continuously alternating so will not yield a correct answer (the average value in fact is zero). We must seek the value of alternating current which does the same amount of 'work' as would a DC current if flowing in the same circuit.

## 2.3 Root-Mean-Square (rms) Values

We have already seen in 2.1 how power is calculated in a DC circuit,

 eqn. 1 W = V x A and transposing eqn. 2 V = AR substituting for V in eqn. 1 W = A2R (4)

which is a more convenient form for our present purpose.

If a DC of 3A flows through a resistor of 1 ohm the power dissipated is

W = 32 x 1 = 9W

and this is represented in Figure 4(a) by the area OX by XX'.

On the same figure there is a sinewave of 3A maximum.

Figure 4. (a)

Figure 4. (b)

Figure 4. (c)

A sinewave can be defined as Amax sin w t

where Amax is the peak amplitude in each direction of the waveform.

w t = 2p or 360° at the end of one cycle, see Figure 3.

The instantaneous value of current is i = Amax sin w t.

If this is flowing in a circuit of one ohm resistance, the instantaneous rate of power dissipation is

i2 x 1 Watts = i2 Watts.

Thus we can square the instantaneous values of the sinewave and, using our example having a maximum of 3A this produces the dotted curve in Figure 4(b). It is easy to see, or to calculate, that the area under the dotted curve is half that of the area OX by XX', that is to say, when the DC has an amplitude equal to the maximum AC the power dissipated by the AC is half of that dissipated by the DC.

Still using the values in Figure 4 it is possible to determine the proportion of the ACmax value which gives the same power dissipation as DC.

The DC to give the same power dissipation as the maximum AC is seen from Figure 4(b) to be 9/2 = 4.5A.

So 4.5 =

Then ie. 3 divided by the square root of 2 but the ACmax value is 3, so we can say 1/21/2 times 3.

(5)

Summing up we can say that the rms value of a voltage, or current, is the square root of the mean (average) of all the instantaneous amounts of energy squared, rms is also known as the effective, or virtual value of AC.

To find the rms value, multiply the peak value by 0.707.

To find the rms value from a peak-to-peak figure, first divide by two, then multiply by 0.707. Electrical powers and voltages can be cumbersome to manipulate and calculate so it is normal practice to convert these to a more convenient form. This is the decibel (dB) which is described in section 2.5.

## 2.4 Acoustic Pressure and Intensity

### 2.4.1 Pressure

Pressure is most easily thought of as a mechanical force. It can be generated by the movement of a piston as shown in Figure 5. If the constant speed rotation of the shaft is transmitted by the crankpin to the piston in the manner illustrated, the result is a sinusoidal motion. This sets up an acoustic pressure wave of sinusoidal oscillations in the water with regions of compression and rarefaction relative to the mean pressure, see Figure 5. In acoustics we are concerned with sinewaves in a similar way to the electrical sinewaves discussed in section 2.2, i.e. the 'work done' or the power used is the rms value. Any practical measurement of acoustic waves is made by observing the electrical output signal (sinewaves) of a pressure sensitive device (transducer). In other words the sinusoidal pressure wave is converted into an equivalent electrical sinewave from which the acoustic properties can be determined.

Figure 5.

Acoustic waves are a form of energy which can be propagated through a medium having distributed mass and elasticity, such as water. At any point where these waves are found in the water, energy is present as a change in the normal state of stress and strain. Energy causes force to be exerted and the oscillations of the particles comprising the medium are therefore transmitted from point to point in the water at a speed depending on the factors given in section 2.6.1.

The relationship between pressure, particle velocity and the quantity r c is similar to that which exists in electricity between voltage, current and resistance (impedance), sections 2.1 and 2.2. Acoustic impedance r c is defined in terms of the constants of the medium, r being the density in kg/m3 and c the speed of acoustic waves in m/s, therefore r c = kg/m3 x m/s = kg/m2.s. The unit of acoustic impedance is known as the Rayl (after Lord Rayleigh) and for seawater it is approximately 1.54 x 106 Rayls.

Pressure is measured in micropascals (m Pa). One m Pa is equal to 10-6 Newtons/m2. Micropascals are used as the standard reference unit because the Pascal (1 Newton/m2) is only a factor of 10 different to the previous standard, the m bar which can be seen in old textbooks. To convert any figures given in dB/1m b it is necessary to add 100, e.g. 120 dB/1m b = 220 dB/1m Pa and -100 dB/1V/1m b = -200 dB/1V/1m Pa. See Appendix I and section 2.5.

### 2.4.2 Intensity

Figure 6.

Acoustic intensity (I) is defined as the amount of energy per second (power), passing through unit area normal to the direction in which the acoustic wave is propagated. This concept is illustrated in Figure 6.

In section 2.4.1 it was stated that pressure, particle velocity and r c are analogous to the electrical quantities of voltage, current and resistance, respectively. Acoustic intensity is analogous to electrical power on the basis of the amount of energy and the time it flows or is used. So equation 3 can be presented in acoustic terms as

Intensity (I) = p2/r c (6)

i.e. the intensity is proportional to the pressure squared divided by the acoustic impedance.

The intensity of an acoustic wave is referred to a plane wave of rms pressure equal to 1 m Pa at a distance of 1 m from the source. For the purpose of fisheries acoustics a plane wave can be described as one which exhibits no significant curvature of its wavefront over the length or extent of a target.

## 2.5 Decibels

A decibel (dB) is one tenth of a Bel, the unit named after Alexander Graham Bell, a famous inventor. It was first used in connection with telephone transmission lines (Martin, 1929) but is now common to all branches of electronics and acoustics.

The decibel is NOT a unit of measured quantity such as metres, kilogrammes or seconds. It is the logarithm to base 10 of a ratio, giving the relationship between quantities. In electronics there are often very large differences, or changes, in the power or voltage between one part of a system and another. In underwater acoustics big differences occur in intensity and pressure due to propagation losses, for example. By converting these changes to decibels using logarithms it is possible to simplify figures and calculations of gains or losses. Since the decibel is based on logarithms, multiplication and division are converted into addition and subtraction.

### 2.5.1 Power and Intensity Ratios

Electrical power, (W, watts) is analogous to acoustic intensity (I) section 2.4, and the same factor 10 log10 is used for both, hereafter in the manual 10 log will be used. The decibel notation for a power W is

N = 10 log W/Wo dB

where Wo is the chosen reference power similarly for an intensity I

N = 10 log I/Io dB (7)

where Io is the chosen reference intensity.

The logarithmic ratio between two quantities one of which is twice the size of the other could be log 2/1 or log 1/2 depending on the reference. Taking logs, the ratios become +0.3010 and -0.3010 respectively. Changing these to decibels in the case of power or intensity the ratios are multiplied by 10

10 log 2/1 = +3 dB and 10 log 1/2 = -3 dB

In other words +3 dB indicates that the quantity is twice as large as the reference, whilst -3 dB indicates that it is only half the reference. Similarly

10 log 10/1 = +10 dB and 10 log 1/10 = -10 dB
10 log 100/1 = +20 dB and 10 log 1/100 = -20 dB.

Table 1 shows the number of decibels equivalent to power and intensity ratios. A ratio relating to a decibel number not in the table can be found by

ratio = Antilog (ndB/10) or 10n/10

alternatively numbers can be selected from the table of dB's to make up the number to be converted, e.g. 75.5 dB is 70 + 3 + 2 + 0.5 and the corresponding ratios are 107, 2, 1.58 and 1.12, which when multiplied together = 2 x 1.58 x 1.12 x 107 = 3.539 x 107.

Table 1 shows the number of decibels equivalent to power and intensity ratios.

Table 1.

 Number of Decibels Power (W) and Intensity (I) Ratio +dB -dB 0.1 1.02 0.98 0.5 1.12 0.89 1 1.26 0.79 2 1.58 0.63 3 2.00 0.5 6 3.98 0.25 10 10.00 0.10 20 102 0.01 30 103 10-3 40 104 10-4 50 105 10-5 60 106 10-6 70 107 10-7 80 108 10-8 90 109 10-9 100 1010 10-10

### 2.5.2 Voltage and Pressure Ratios

In section 2.1 it was shown that power is proportional to voltage squared; there is the same relationship between acoustic intensity and pressure (section 2.4).

The decibel notation is therefore

10 log(V2/R)/(Vo2/R) = 20 log V/Vo (8)
and (9)

where Vo and p0 are the respective reference quantities and r c is the specific acoustic resistance of the water (analogous to resistance in the electrical circuit, (r is the density in kg/m3 and c = acoustic wave speed in m/s).

From the example in the previous section it was shown that the logarithmic ratio between two quantities of power or intensity, one of which is twice the size of the other, is +3.010 and -3.010 depending on which is taken as the reference quantity.

Changing these to decibels of voltage or pressure, the logarithms of the ratios must be multiplied by 20

20 log 2/1 = +6 dB and 20 log 1/2 = -6 dB.

Table 2 shows the number of decibels equivalent to voltage and pressure ratios.

Table 2.

 Number of decibels Voltage and Pressure Ratio +dB -dB 0.1 1.01 0.99 0.5 1.06 0.98 1 1.122 0.84 2 1.26 0.79 3 1.41 0.71 6 2.0 0.50 10 3.16 0.316 20 10.00 0.1 30 31.62 0.0316 40 102 10-2 50 316.0 0.00316 60 103 10-3 70 3162 3.162 x 10-3 80 104 10-4 90 31622 3.162 x 10-4 100 105 10-5

The dB reference for voltage is often 1 Volt, in which case a measurement of less than 1 Volt must be -n dB/1 Volt.

10-5 Volts = 20 x - 5 = -100 dB relative to 1 Volt
1m V = 10-6 Volts = 20 x - 6 = -120 dB relative to 1 Volt.

Figure 7 shows this relationship over a useful range of voltage.

## 2.6 Speed and Absorption of Acoustic Waves

2.6.1 Absorption of Acoustic Waves

The speed of acoustic waves in the sea is denoted by the symbol c and is measured in metres/second (m/s). It is a variable dependent on the temperature, salinity and depth of the water, although the depth factor is not significant for fisheries acoustics.

Depth can only be measured correctly if the acoustic wave speed is known for the area being surveyed. Figure 8 shows the variation of speed against temperature and for a number of salinities. It is evident from this figure that changes of the order of 21/2% might occur between summer and winter but this is usually considered to be insufficient to justify correction to the timing system of the echo-sounder.

Figure 8.

The speed of an acoustic wave is

c = fl (10)

where

c is in m/s
f is in Hz
l is the wavelength (see section 2.7) in metres.

The frequency f is fixed in an echo-sounder but the wavelength varies with the speed of the waves

l = c/f (11)

assuming a speed of 1500 m/s (the most commonly used figure) and a frequency of 38,000 Hz ie 38 kHz, the wavelength is

l = 1500/38000 = 0.0395 m or 39.5 mm

If the speed increases by 20 m/s to 1520 m/s the wavelength increases, by 0.5 mm, to 40 mm, not a significant change. Although thermoclines (layers of water in which the temperature changes abruptly with depth) occur frequently due to a variety of causes, the consequent change in speed of acoustic waves is unimportant for echo-sounding purposes i.e. the difference in acoustic impedance r c is too small to be detected.

### 2.6.1 Absorption of Acoustic Waves

As acoustic waves travel through water, some of the energy is absorbed by chemical processes. The proportion of energy which is converted must be regarded as an acoustic loss and taken into account when calculating results. The loss is linear with distance, ie for each metre travelled a constant fraction of acoustic energy is lost.

Absorption is denoted by the symbol a and is expressed in decibels per kilometre (dB/km), it increases with frequency as shown in Figure 9.

There is a marked temperature dependence but this exhibits different characteristics; at the low end of the frequency scale below 70 kHz where there is a negative gradient, an increase in temperature gives a reduced a. At 120 kHz the gradient is first positive, then negative as in Figure 10(a). These figures are calculated from a formula derived by Fisher and Simmons (1977) after a series of experiments and it is believed to give the most accurate results currently available for fisheries acoustic work. Further research is needed to confirm the validity of the results for all conditions of temperature and salinity likely to be encountered in practical surveys. Nevertheless it is recommended that the graphs in Figures 9 and 10(a)(b)(c) be used to select the value of a for surveys until further results are forthcoming. Variation of the values of absorption for changes in salinity at commonly used acoustic survey frequencies are plotted in Figure 10(b)(c).

Figure 10. (b)

Figure 10. (c)

## 2.7 Frequency and Wavelength

The frequency of a sinewave is defined as the number of peaks of one polarity which occur in one second. Frequency is measured in terms of cycles, ie the number of complete sinewave cycles in one second, but the unit of frequency is now Hertz (Hz) after the physicist of that name. Thus in Figure 11 the sinewave shown has a frequency of 5 Hz.

The term w was used without explanation in section 2.3. It denotes the angular velocity of a wave i.e. the rate at which a particle moves through the cycle, and is equal to 2p f. For the frequency of 5 Hz the angular velocity is 31.416 radians/second.

Periodic time (t) is the time taken for completion of a cycle, or for adjacent peaks to occur, see also Figure 11. The sinewave in section 2.3 was defined as Amax sin w t and the completion of a cycle when w t = 2p, therefore t = 2p /2p f = 1/f so the periodic time of a 5 Hz wave is 1/5 second.

Figure 11.

In fisheries acoustics we are concerned with a limited number of spot frequencies and much of the work has been concentrated at, or close to, one of these ie 38 kHz. Although it is convenient to talk about one frequency, an echo-sounder cannot work unless it operates within a band of frequencies and the one always quoted is nominally the centre frequency of the band. Thus the power is transmitted simultaneously at all frequencies within the bandwidth and this is discussed in greater detail later in 3.1.2.

The frequency of a sinewave is independent of the medium into which it is introduced i.e. the number of cycles to pass a fixed point in one second is constant. But another characteristic of a sinewave, the wavelength (l), is completely dependent on the physical properties of the medium through which the wave is propagated. This is because the wave speed is governed by the density and bulk modulus of elasticity of the medium.

Wavelength is defined as the distance between adjacent peaks or troughs of a sinewave, Figure 12, i.e. the physical measurement in metres (m) relating to this distance, the faster the wave travels the greater is l. When we choose a frequency of operation for an echo-sounder we know that the medium into which it will operate is water, so it is possible to calculate the wavelength within fairly close limits if the speed of acoustic waves in the particular volume of water is known. This is because the speed (c) of wave travel is related to frequency (f) and wavelength (l) as shown in eqn. 11 i.e. l = c/f.

The importance of wavelength becomes apparent in the next section where we discuss some of the effects of acoustic wave interaction with fish.

Figure 12.

## 2.8 Acoustic Waves and Fish

When an acoustic wave strikes a fish a proportion of the energy is reflected and this is called the echo. The factors on which this quantity depends are not entirely understood, but in this section we consider some of the basic theory.

When a plane wave meets a boundary of two different media e.g. the water and the fish body, it may be partially reflected. If the fish has dimensions of the same order as the acoustic wavelength, the intensity reflected in any particular direction depends on the impedance ratio of the water and fish, also the shape and orientation of the fish.

To begin, we assume that the fish remains the same shape whilst its interaction with the acoustic wave occurs. Thus we can look at the directional pattern when the orientation of the fish changes with time. If an arbitrary reference axis passes through a fish whose orientation is being varied and the amplitude of the echo is measured as a function of the direction of this axis, the result can be plotted as a polar diagram. A hypothetical plot is shown in Figure 13(a), but of course this is for one plane only, the whole diagram is three-dimensional and would look roughly like a ball with projecting lumps. The next figure 13(b) shows the same result as before but plotted on a linear angular scale.

Figure 13. (a)

Figure 13. (b)

If the fish were steadily rotated in the plane for which Figure 13 was drawn, the echo amplitude would fluctuate and the effect would be to modulate the echo wave. To modulate means to 'impress information upon' and this is seen in Figure 14 where the fluctuations of the high frequency acoustic wave amplitude are due to the information' about the number and relative size of the lumps, or lobes of the directional pattern.

Figure 14.

A fundamental property of such patterns is the closeness of the lobes and hence the rate at which successive maxima and minima of the echo occur as the fish rotates, ie the angular frequency of the lobes. Patterns with different angular frequencies are shown in Figure 15a and b, and it is clear that (b) has a much higher angular frequency than (a) which will lead to more rapid fluctuations of the echo. A fish could possess both of these patterns because it has a large dimension in one direction and a much smaller dimension at right angles to this, but in practice we are usually concerned only with small variations about the dorsal aspect.

Figure 15.

The presence of a fish in the 'acoustic field' (the surrounding acoustic waves) has the effect of changing the field around the surface of the fish by adding what is called the perturbation field (perturbation means greatly disturbed). It is as if the perturbation field arises from virtual sources of acoustic energy inside the body of the fish. In other words, the fish appears to generate an outward-radiating wave system by extracting a flow of energy from the passing incident waves and re-radiating it in all directions. Around the surface of the fish the perturbation field has a 'waviness' which matches the waves of the incident field so of course there are a finite number of wavelengths occurring in relation to the length of a fish. The number of lobes in the directional pattern is dependent on the size of the fish in wavelengths, not its absolute size e.g. at 38 kHz a 40 cm fish is 10 wavelengths but at 120 kHz it is 32 l. The size of the lobes, on a relative amplitude scale, is determined by the fish size.

Although there is a maximum possible amplitude of echo from any particular fish, the actual size received depends on the angle of the lobe, or lobes of the directional pattern 'pointing' in the direction of the echo-sounder acoustic axis, see Figure 16.

Figure 16.

Discussion so far has assumed that the fish remains the same shape, but this is not true, the shape changes with time as the fish swims. The effect is that the virtual sources within its volume change position slightly which causes another kind of modulation to the echo amplitude. Thus it is clear that trying to predict, or model the response from fish is difficult so we have to make measurements, either of individuals, or known quantities and this is discussed later in section 4.5 and 4.6. Section 2.8 is based on notes written by the late Dr V. G. Welsby. The next chapter deals with the instruments needed for fisheries acoustics.