This section describes those basic physical properties of the sea and the air–sea boundary of relevance to the generation, propagation, and scattering of sound at sonar frequencies. The speed of sound and the density of water influence the generation and propagation of sound. These and other related parameters are described in Section 2.1.1, followed by the relevant properties of air in Section 2.1.2. A more comprehensive description of these oceanographic properties is provided in Chapter 4.
Many parameters of relevance to underwater acoustics vary with temperatureT, salinity S, and hydrostatic pressure (often parameterized through the depth from the sea surface z). Where ‘‘representative’’ numerical values are quoted, they are
evaluated for the following conditions:
T ¼10C;
S¼35;
depthz¼0:
By convention the depth co-ordinate z is zero at the sea surface and increases downwards to the seabed.
For simplicity, in Chapters 2 and 3 the ocean is assumed to extend to infinite depth, with uniform properties occupying the entire half-space satisfyingz>0. For example, the speed of sound and density are assumed independent of depth and range. The purpose of the quantitative numerical calculations based on this idealiza- tion (seeWorked Examplesin Chapter 3) is to illustrate the main principles of sonar performance modeling, rather than to provide realistic estimates of detection per- formance. More realistic examples are presented at the end of the book, in Chapter 11.
2.1.1 Acoustical properties of seawater
The two most important acoustical properties of seawater are the speed at which sound waves travel (abbreviated assound speed) and the rate at which they decay with distance traveled (thedecay rate, or absorption coefficient). A third parameter that can influence sound propagation, through its effect on interaction with bound- aries, is density. The density of seawater, and the speed and absorption of sound in seawater, all depend on salinity, temperature, and pressure. At low frequency the absorption also depends on acidity orpH (see Chapter 4).
2.1.1.1 Speed of sound
For the representative conditions described above, the sound speed in seawater, denotedcwater, is 1490 m/s. More generally, the parametersS,T, andPall vary with depth and therefore so too does the sound speed, resulting in significant refraction. A discussion of these gradients and their important acoustic effects is deferred to Chapters 4 and 9. Here and in Chapter 3 they are neglected for simplicity. The wavelengthat frequencyf is
¼cwater=f: ð2:1Þ
2.1.1.2 Density
The density of seawater (water) under the representative conditions introduced above is 1027 kg/m3. Departures of seawater density from this value are small and for most sonar performance applications may be neglected.
2.1.1.3 Attenuation of sound
Attenuationis the name given to the process of decay in amplitude due to a combina- tion ofabsorptionandscatteringof sound. The term ‘‘absorption’’ implies conversion
to some other form of energy, usually heat, whereas ‘‘scattering’’ implies a redis- tribution in angle away from the original propagation direction, with no overall loss of acoustic energy.
The rate of attenuation of sound in water is less than in air and much less than that of electromagnetic waves in water. Low-frequency sound, of order 1 Hz to 10 Hz, can travel for thousands of kilometers, but high-frequency sound is attenuated more rapidly. The attenuation coefficientwaterincreases monotonically with frequency by about four orders of magnitude in the frequency range from 30 Hz to 300 kHz, and quadratically with frequency thereafter. For frequenciesf exceeding 200 Hz, it can be written (see Chapter 4)
water¼1 f2
f2þf21þ2 f2
f2þf22þ3f2: ð2:2Þ For the specified representative conditions, the three coefficientsi are1
1¼1:40102 Np km1;
2¼5:58 Np km1
and
3 ¼3:90105 Np km1kHz2: ð2:3Þ
The frequenciesf1 and f2, explained further in Chapter 4, are known asrelaxation frequenciesand for the representative conditions are equal, respectively, to 1.15 kHz and 75.6 kHz.
Numerical evaluation of Equation (2.2) gives (to the nearest order of magnitude) water103, 101, and 10þ1Np/km at 300 Hz, 10 kHz, and 300 kHz, respectively. A graph ofwatervs. frequency, computed using Equation (2.2), is plotted in Figure 2.1.
The reciprocal of the attenuation coefficient (i.e.,1), plotted on the same graph, provides a rough measure of the distance that sound can travel in water if unimpeded by physical obstacles. This quantity is referred to here as ‘‘audibility’’, the acoustical analogue of optical ‘‘visibility’’, and varies between 102m Np1 at 300 kHz and 106m Np1 at 300 Hz. By comparison, the attenuation coefficient of green light2 is at least 102Np m1 for clear seawater, so that the optical visibility in water never exceeds 102m Np1 and is usually less than 101m Np1. Thus, for acoustic frequencies up to 300 kHz, the audibility of sound exceeds the maximum visibility of light by up to six orders of magnitude. This is the reason why sound waves have 2.1 Essentials of sonar oceanography 29 Sec. 2.1]
1Two sound waves are said to differ in level by 1 Np if their amplitudes are in the ratio 1:e.
The neper (Np) and the related unit the decibel (dB) are defined in Appendix B.
2The sea is opaque to electromagnetic radiation with the exception of visible light, very low frequency radio waves, and gamma rays. The extinction coefficient is a measure of the decay of light intensity with distance, and in the present notation is equal to 2opt,whereopt is the optical attenuation coefficient in units of nepers per unit distance. For example, the value quoted by (Clarke and James, 1939) of 4 % per meter for the extinction coefficient in the Sargasso Sea means that expð2optxÞ ¼0:96 when the distancex¼1 m. Taking logarithms gives 2opt¼0:04/m.
become the most successful means of probing the underwater environment. It is the raison d’eˆtreof sonar.
2.1.2 Acoustical properties of air
Together with those of seawater, the properties of air determine the reflection coefficient at the air–sea boundary. The sound speed and density of air depend on temperature (T) and pressure (P). For the representative conditions, these are cair¼337 m/s andair¼1.25 kg/m3. Thus, bothandcin air are considerably lower than their counterparts in water, which has important implications for the behavior of underwater sound.
2.2 ESSENTIALS OF UNDERWATER ACOUSTICS 2.2.1 What is sound?
Steady-state pressure increases with increasing depthzand is equal to the total weight per unit area of water plus atmosphere supported above that depth. This quantity is called thestatic pressure(or hydrostatic pressure) and can be expressed quantitatively Figure 2.1. Numerical value of attenuation coefficientvs. frequency of sound in seawater (expressed in units of nepers per megameter) and of its reciprocal,1(in kilometers per neper), calculated using Equation (2.2) for the specified representative conditions:S¼35;T¼10C;
z¼0.
in the form
PstatðzÞ ¼PatmþPgaugeðzÞ; ð2:4Þ where Patm is the atmospheric pressure (approximately 101 kPa); and Pgauge is the additional pressure due to the weight of the water above depthz(the gauge pressure)
PgaugeðzÞ ¼ ðz
0
waterðÞgðÞd: ð2:5Þ
Underwater disturbances result in departures P from this value, for an arbitrary position vectorx,
Ptotðx;tÞ ¼PstatðzÞ þ Pðx;tÞ: ð2:6Þ Once created, provided that certain basic conditions are met (Pierce, 1989), a pressure disturbance propagates with the speed of sound, and P is known as the acoustic pressure, henceforth denotedqðx;tÞ and assumed small by comparison with static pressure.3Such an acoustic disturbance is known as underwater sound. The study of this sound is called underwater acoustics. The assumption of small q simplifies the mathematics and is generally justified because atmospheric pressure is large compared with typical acoustic pressure fluctuations.
A brief account is given here of radiation and scattering of underwater sound from simple sources and for a simple geometry. First, radiation is considered from a point source in an infinite uniform medium, with and without a perfectly reflecting plane boundary (Section 2.2.2). Then the scattering of plane waves is considered, first from a point object and then from a rough surface (Section 2.2.3). The sea surface is considered as an example of a reflecting surface, a radiating surface, and a scattering boundary. A more complete treatment of these phenomena is presented in Chapters 5 and 8.
2.2.2 Radiation of sound
2.2.2.1 Radiation from a point monopole source 2.2.2.1.1 Spherical spreading
Consider a point monopole4 source of power W. To generate sound at a given frequency, the source must expand and contract at that frequency. During expansion the source motion causes an increase in density of the surrounding fluid, with a corresponding increase in its pressure. The resulting high-pressure disturbance propagates outwards in the form of a spherical wave, traveling at the speed of soundcwater. The same sequence follows a contraction, except with a low-pressure disturbance replacing the high-pressure one.
At any fixed moment in time the radiated field comprises a series of concentric
‘‘rings’’ (actually spherical shells in three dimensions) of alternating high and low 2.2 Essentials of underwater acoustics 31 Sec. 2.2]
3The symbolp, introduced in Section 2.2.2, is reserved for a complex variable representing the acoustic pressure. See footnote 5.
4A monopole source is one with a fluctuating volume, such as a pulsating bubble.
pressure. The potential of these rings to do work on the surrounding medium can be expressed in terms of their potential energy density (Pierce, 1989)
EðPÞV ¼ q2
2Bwater; ð2:7Þ
whereEVdenotes energy per unit volume; andBwateris the bulk modulus of water, a measure of its opposition to compression or rarefaction, analagous to the stiffness of a spring, and equal to
Bwater¼waterc2water: ð2:8Þ
The rings also contain kinetic energy, due to the particle velocityu, given by EðKÞV ¼waterjuj2
2 : ð2:9Þ
The superscriptsðPÞandðKÞin Equations (2.7) and (2.9) denote potential and kinetic energy, respectively. If the pressure and particle velocity are in phase, it can be shown that the kinetic and potential densities are equal (Pierce, 1989), so that the average rate of energy flux (i.e., intensity) is
I¼cwaterEðKÞV þEðPÞV
¼2cwaterEðPÞV ¼ q2
watercwater; ð2:10Þ where the overbar indicates an average in time. Conservation of energy demands that the total radiated power at distance sfrom the source, 4s2I, be constant, which means that the RMS pressure (i.e.,
ffiffiffiffiffi q2 q
) must vary as 1=swith distance.
A point monopole source radiatesomni-directionally(i.e., with equal power in all directions). At a distance s from the source, in the assumed uniform medium the energy is distributed uniformly on a sphere of surface area 4s2(Figure 2.2), so the component ofacoustic intensitynormal to the surface of the sphere at a distancesis
IðsÞ ¼ W
4s2: ð2:11Þ
Also of interest is the acoustic pressure resulting from the point source. Using standard complex variable notation for a diverging harmonic spherical wave of angular frequency!, the complex pressure fieldp varies with timetand distances according to Pierce (1989)5
pðs;tÞ ¼ ffiffiffi p2
p0s0eiðks!tÞ
s ; ð2:12Þ
wherep0is the RMS pressure at a distances0 from the source; andkis the acoustic wave number, so that
k¼!=cwater: ð2:13Þ
5The real part of the complex variablepðs;tÞis the acoustic pressureqðs;tÞ. Unless otherwise stated, an expði!tÞtime convention is used for traveling waves throughout.
The true acoustic pressure is obtained by taking the real part of Equation (2.12), so that
qðs;tÞ ¼ ffiffiffi p2
p0s0cosðks!tÞ
s : ð2:14Þ
From Equation (2.10) it follows that
I¼ jpj2
2wcw; ð2:15Þ
where the ‘‘water’’ subscript is abbreviated henceforth as ‘‘w’’. From Equations (2.11), (2.12), and (2.15) it then follows that
p0s0¼ wcwW 4
1=2
; ð2:16Þ
or more generally (for a directional source)
p0s0¼ ðwcwWOÞ1=2; ð2:17Þ whereWO indicates the radiated power per unit solid angle (theradiant intensity).
It is convenient to define a steady-statepropagation factor FðsÞin terms of the ratio of the mean square pressure at the receiver to that at a small distanceðs0Þfrom the source, such that
FðsÞ ¼ q2
p20s20¼ jpj2
2p20s20: ð2:18Þ Defined in this way, the propagation factor has dimensions [distance]2. For a spherical wave in a medium of uniform impedance it is equal to the ratio of received intensityI to the radiant intensityWO.
2.2 Essentials of underwater acoustics 33 Sec. 2.2]
Figure 2.2. Radiation from a point source of powerW in free space.
The intensity at a distancesis I0¼W=ð4s2Þ. At distance 2sthe same power has spread into four times the area, reducing the intensity by a factor of 4.
It follows by substituting forpðs;tÞfrom Equation (2.12), scaled by expðsÞto account for absorption, that the propagation factor for a point source in a uniform medium is
FðsÞ ¼e2s
s2 ; ð2:19Þ
whereis the sound attenuation coefficient introduced in Section 2.1.1.3.
The above arguments apply to a steady-state field due to a source of constant radiant intensity. If the power is transmitted for a short time only, it is useful to think in terms of the transient field resulting from the total radiated energy per unit solid angleEO. The appropriate propagation factor under these conditions is obtained by integrating the numerator and denominator of Equation (2.18) over time instead of averaging them:
FðsÞ ð
q2dt
wcwEO: ð2:20Þ
To summarize, the steady-state mean square pressure for a source of radiant intensity WO, from Equation (2.18), is
q2¼wcwWOFðsÞ ð2:21Þ and for a transient field, the time-integrated pressure squared, from Equation (2.20),
is ð
q2dt¼wwEOFðsÞ: ð2:22Þ Either way,FðsÞis given by Equation (2.19) for an omni-directional point source in an infinite uniform medium. The same equation applies also for a directional source, provided thatWO (orEO) is measured in the direction of the receiver.
The behavior described by Equation (2.19), characterized by itss2dependence due to the spherical nature of the expanding wave front, is known as spherical spreading.
2.2.2.1.2 Reflection from the sea surface
Now consider the effect of placing a reflecting boundary, such as the sea surface, close to the point source of Section 2.2.2.1.1. There are two straight-line ray paths con- necting the source to any given receiver position: the direct path and a surface reflected one. If the source is at depth z0 below the surface (see Figure 2.3), the contribution to the pressure field at the receiver due to the direct path is given by Equation (2.12) with a source–receiver separation equal to
s¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi r2þ ðzz0Þ2 q
: ð2:23Þ
The reflected path can be thought of as originating from an image source at heightz0 above the boundary, with image–receiver separation of
sþ¼
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi r2þ ðzþz0Þ2 q
: ð2:24Þ
Using the method of images, the two contributions from source and image are added coherently to obtain the total pressure at the receiver, scaling the reflected path by the surface reflection coefficientR
p¼ ffiffiffi p2
s0p0 eðikÞs s
þReðikÞsþ sþ
" #
ei!t: ð2:25Þ IfRis real (implying a phase change on reflection of 0 or ) it follows that
Fðsþ;sÞ ¼e2s
s2 þR2e2sþ
s2þ þ2ReðsþsþÞ
ssþ cosð2kxÞ; ð2:26Þ where
xsþs
2 ¼ 2z0z
sþsþ: ð2:27Þ
Equation (2.26) can be interpreted as follows. The first and second terms on the right-hand side are associated with the energy from the direct and surface-reflected ray paths, respectively. The third term is due to interference between these two paths, resulting from the coherent addition of complex pressures. The expression is useful for broadband applications because the third term vanishes after averaging over frequency.
An (equivalent) alternative version, convenient for narrowband applications, is Fðsþ;sÞ ¼eðsþsþÞ ex
s þRex sþ
2
4R
ssþsin2ðkxÞ
: ð2:28Þ 2.2 Essentials of underwater acoustics 35 Sec. 2.2]
Figure 2.3.
Radiation from a point source in the presence of a reflecting boundary.
It is often the case that the distancessþandsare approximately equal, such that the product x is sufficiently small to neglect the x terms, and sþssþþs. Furthermore, for many applications the sea surface can be treated as a perfect reflector with a-phase change (i.e.,R¼ 1; see Box on p. 37). It then follows that
FcohðsÞ 4e2s
s2 sin2 kz0z s
; ð2:29Þ
where
s¼ ðsþsþÞ=2: ð2:30Þ The sequence of sinusoidal peaks and troughs predicted by Equation (2.29) is known as a Lloyd mirror interference pattern. The ‘‘coh’’ subscript stands for ‘‘coherent addition’’, indicating that the two contributions to the total pressure, from the direct and reflected paths, respectively, are added with regard to their phase, before squaring. This means that the phase difference information is used for the purpose of combining the contributions from the two paths. Specifically, if Equation (2.25) is written in the form
p¼ ffiffiffi p2
s0p0ðFþþFÞ; ð2:31Þ where
F¼eðikÞs
s ei!t ð2:32Þ
and
Fþ¼ReðikÞsþ
sþ ei!t; ð2:33Þ
it follows that
Fcoh¼ jFþþFj2: ð2:34Þ The ‘‘incoherent’’ propagation factor is obtained by discarding the phase terms. In other words
Finc¼ jFþj2þ jFj2: ð2:35Þ Alternatively, averaging Fcohover (say) frequency
hFcohi ¼ hjFþj2i þ hjFj2i þ h2jFþjjFjcosð2kxÞi: ð2:36Þ The first two terms are hardly affected by the average and together approximate to Finc. If the average is over several cycles of the cosine function, the third term can be expected to average out to zero, and hence
hFcohi Finc: ð2:37Þ For this reason, there are many situations in which acoherent sum(add and square), followed by an average over frequency, gives the same result as anincoherent sum (square and add).
The reflection coefficient at the air–sea boundary
The reflection coefficient at the sea surface is determined by the impedance of air relative to that of water. The characteristic impedance of air for the assumed representative conditions is given by
Zair¼aircair¼420 kg m2s1;
more than three orders of magnitude smaller than that of water, which is equal to Zwater¼watercwater¼1:53106kg m2s1:
The low impedance of air compared with that of water means that the acoustic pressure required to achieve a given acoustic intensity is much smaller in air than in water. From the continuity of pressure across the boundary it follows that the pressure on the boundary itself must also be small, and to first order this can be approximated by the boundary conditionp¼0 atz¼0. The only way this can be achieved for an incident plane wave of finite amplitude in water is for a reflected wave to be generated at the surface of the same amplitude andopposite phase. Let the horizontal and vertical wave numbers be and , respectively, so that the incident wave can be represented by
pincident¼eiðxzÞei!t and the reflected wave by
preflected¼ReiðxþzÞei!t:
By adding these two terms it can be seen that the only way the total pressure pincidentþpreflectedcan be zero everywhere on thez¼0 boundary is ifR¼ 1. This has two important consequences. First, the unit magnitude corresponds to 100 % reflection of energy, so that sound becomes trapped in the sea, potentially traveling very long distances. Second, the negative sign means a phase change of the reflected wave relative to the incident one, which results in near-perfect cancellation of acoustic pressure close to the sea surface.
In general, the reflection coefficient is a function of frequency, and of the physical properties of the reflecting boundary. For example, the sea surface reflec- tion coefficient depends on the wave height and on the population of near-surface bubbles created by breaking waves (see Chapters 5 and 8.)
2.2.2.2 Radiation from an infinite sheet of uniformly distributed dipoles
One of the factors that limit sonar performance is the presence of background noise in the sea. Much of this background noise originates at the sea surface (e.g., due to breaking waves). Consider an infinitesimal patch of sea surface with surface area A and radiating power per unit surface area and solid angleWAO. The contribution from this patch to the mean square pressure at a receiver situated at a distances(see 2.2 Essentials of underwater acoustics 37 Sec. 2.2]
Figure 2.4) is
q2¼wcwWAO Ae2s
s2 : ð2:38Þ
The sea surface behaves like a sheet of dipoles,6with a radiation pattern proportional to sin2, so that7
WAO¼ 3
2WAsin2; ð2:39Þ
whereis the raygrazing angle(the angle between the ray path and the horizontal), so that
q2¼ 3
2WA Awcwe2s
s2 sin2: ð2:40Þ The solid angle O subtended by the surface element r at the receiver, for an azimuthal increment , is
O¼cos ð2:41Þ
Figure 2.4. Radiation from a sheet source element of width r.
6A dipole source is one made out of two out-of-phase monopole sources, placed an infinitesimal distance apart (in practice, they must be separated by at most a small fraction of a wavelength). An important distinction between a monopole and a dipole is that in the case of the dipole source, there is no net change in volume. See Crocker (1997) for details.
7The constant 3=ð2Þensures that the power radiated per unit area, integrated over all solid angles (into the lower half-space)Ð
2WAOdOisWA. This follows from the use of dO¼cosddand the result
ð2 0
d ð=2
0
dsin2cos¼2=3;
whereis the azimuth angle.