Finite-Amplitude Waves

CHAPTER

Finite-Amplitude Waves

13 L. Bjørnøy

UltraTech Holding, Taastrup, Denmark

Definitions The world we live in is nonlinear. Relations between characteristic parameters like pressure, density, and temperature in fluids and relations between material constants in solids are nonlinear. Since Robert Hooke (1635e1703) in 1660 put forward his linear elastic relation (named Hooke’s law) between force and deformation of solids, attempts have been made to “linearize” the world. The driving force behind the linearization attempts has in particular been lack of fundamental understanding of nonlinearity concepts and lack of tools to handle nonlinear problems. Only the strong development in computer technology, with faster and more powerful computers over the recent 40e50 years, has given access to understand and to exploit nonlinear phenomena. One of these phenomena is nonlinear acoustics. As shown in Chapter 10, the maximum power radiated by a sonar (sound navigation and ranging) system is directly proportional to its piezoceramic volume. Moreover, the stored electric energy in the piezoceramic materials is limited by factors like electric insulation breakdown, ceramic depolarization, and efficiency deterioration caused by increased dissipation at high electric fields. Too large a driving voltage on the piezoceramic materials will make them to respond nonlinearly. This means that the waveform of the pressure signal radiated from the sonar will not be a replica of the driving electrical voltage. The material nonlinearity in the sonar system will most frequently, for high acoustic pressure amplitudes radiated into the water, be accompanied by a series of nonlinear effects along the propagation path for the acoustic signals. Some nonlinear effects may limit the applications of sonar for desired purposes. However, they can also be exploited to enhance the sonar acoustic effects in finite-amplitude waves in water. Section 13.1 examines nonlinear acoustic phenomena and the associated physics. Section 13.2 focuses on nonlinear underwater acoustics including parametric acoustic arrays. This is followed in Section 13.3 with a discussion of underwater explosions including other sources of high-intensity sound. Section 13.4 provides a list of symbols and abbreviations and the page of first usage for each, and the references for this chapter. y

30 March 1937e24 October 2015.

Applied Underwater Acoustics. http://dx.doi.org/10.1016/B978-0-12-811240-3.00013-8 Copyright © 2017 Elsevier Inc. All rights reserved.

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13.1 PHYSICS AND NONLINEAR PHENOMENA The nonlinear acoustic effects include generation of harmonics of an originally sinusoidal signal during its propagation, limitations in the acoustic pressure amplitude to be transmitted over a given distance due to acoustic saturation, and other nonlinear effects that may be found in focused acoustic fields, which, for instance, can be produced by the sound velocity profile in the water column. Creation of cavitation at positions of intense sound, most frequently near the sonar system, also involves nonlinear acoustic effects. The formation of sum and difference sound frequencies when two intense, coexistent sinusoidal signals interact with each other during their propagation in water, forming a so-called parametric acoustic array with some advantageous beam patterns, constitutes an exploitation of nonlinear acoustic effects. The jet formation and circulation in the water near an intense sound source, a phenomenon called acoustic streaming, and the creation of forces on objects in water by the so-called acoustic radiation pressure are also nonlinear acoustic effects.

13.1.1 HARMONIC DISTORTION The fundamental equations of nonlinear acoustics may be derived from the fundamental equations of fluid mechanics. These equations are an equation of continuity, three equations of motions, and an equation of energy. Moreover, a constitutive equation, the equation of state, for the fluid is necessary to describe all types of motion in fluids, see Bjørnø [1]. These six fundamental equations can be reduced to one single, nonlinear fundamental equation, the Burger’s equation [2], which describes the propagation of finite-amplitude sound waves in a viscous and heat conducting fluid. The dimensionless Burger’s equation is given by: vV vV v2 V V ¼ G1 2 vs vy vy

(13.1)

where G describes the ratio of the influence of nonlinearity to the influence of dissipation. G, called the Gol’dberg number, was first introduced by Gol’dberg [3] as a criterion such that shock formation should not be likely to take place if G < 1. G is expressed through the relation:  

B u0 r0 x 0 B þ2 þ 2 Rea ¼ G¼ (13.2) A A b where Rea ¼ uoxc/(b/ro) constitutes an acoustic Reynolds number analogous to the hydrodynamic Reynolds number. b {unit: kg/(ms)} represents the viscosity and heat conduction effects through the relation: b ¼ 4 m/3 þ z þ (1/cv þ 1/cp), where m and z are the shear viscosity and the bulk viscosity {unit: km/(ms)}, respectively, while cv and cp are the specific heats at constant volume and at constant pressure, respectively. In Eq. (13.1) the dimensionless ratio V ¼ u/uo is the ratio between the local particle velocity u {unit: m/s} in the finite-amplitude sound wave and the particle

13.1 Physics and Nonlinear Phenomena

velocity amplitude uo {unit: m/s}. s ¼ (B/2A þ 1)M x/xc is a dimensionless propagation distance parameter with the one-dimensional distance x {unit: m}, x ¼ 0 at the finite-amplitude wave source, and the characteristic distance xc ¼ l/ 2p, where l {unit: m} represents the wavelength in an originally sinusoidal wave of finite amplitude. y ¼ co(t  x/co)/xc is the time relation with t {unit: s} as the time and with co {unit: m/s} as the local sound velocity. M ¼ uo/co represents the acoustic Mach number on a par with the hydrodynamic Mach number. The dimensionless ratio B/A is the second-order nonlinearity ratio, an important material constant in nonlinear acoustics. This ratio is defined through a Taylor series expansion of the equation of state for the fluid for constant fluid density r {unit: kg/m3} at r ¼ ro, and by taking into account only terms up to second order [1]. B/A can be expressed by:

 B vc 2c0 Tb vc ¼ 2r0 c0 þ (13.3) A vp T cp vT p where p {unit: N/m2} is the local pressure in a sound wave in the fluid and T {unit: K} is the absolute temperature in the fluid. b denotes the isobaric compressibility {unit: K1}. Values of B/A for water at different temperatures and pressure are given in Table 13.1, while Table 13.2 gives B/A values for seawater at various salinities and temperatures at p ¼ 105 N/m2. A solution to Eq. (13.1) according to Mendousse [6] can be expressed by: " #!
 N P G n2 a n v Log en ð1Þ In e G cosðnyÞ 2 n¼0 2 (13.4) Vðs; yÞ ¼ vy G where the Neumann factor en ¼ 1 for n ¼ 0, and en ¼ 2 for n  1. In is the Bessel function with an imaginary argument. Table 13.1 Values of B/A for Freshwater Without Gas Bubbles as a Function of Temperature and Pressure (1 bar ¼ 105 N/m2) Temperature Pressure (bar)

08C

308C

408C

508C

608C

808C

1 250 500 1000 2000 5000 10,000

4.08 4.90 5.58 6.35 6.78 6.44 e

5.21 5.43 5.63 5.83 6.08 6.14 e

5.49 5.59 5.69 5.84 6.00 6.02 e

5.55 5.62 5.69 5.80 5.93 5.93 5.36

5.61 5.70 5.75 5.83 5.88 5.77 5.52

5.74 5.79 5.84 5.86 5.82 5.58 5.12

Reproduced from Hagelberg, M.P., Holton, G., and Kao, S., Calculation of B/A for Water from Measurements of Ultrasonic Velocity versus Temperature and Pressure to 10000 kg/cm2, J. Acoust. Soc. Amer., 41, (3), pp. 564e567, 1967, with the permission of the Acoustical Society of America.

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Table 13.2 Values of B/A for Seawater at Various Salinities and Temperatures at 1 atm Temperature Salinity (%)

08C

108C

208C

308C

33 35 37

4.89 4.92 4.96

5.06 5.09 5.12

5.21 5.25 5.27

5.37 5.41 5.42

Reproduced from Coppens, A.B., Beyer, R.T., Seiden, M.B., Donohue, J., Guepin, F., Hodson, R.H., and Townsend, C., Parameter of Nonlinearity in Fluids. II, J. Acoust. Soc. Amer., 38, (5), pp. 797e804, 1965, with the permission of the Acoustical Society of America.

For G [ 1, the distortion of a finite-amplitude, originally sinusoidal, plane acoustic wave during propagation through a dissipative fluid can be found from expression (13.4) and is shown in Fig. 13.1. Fig. 13.1 shows the change of the shape of a single period of the finite-amplitude plane wave during propagation from the source at s ¼ 0 and through different dimensionless distances s. In the region between s ¼ 0 and s ¼ 1 nonlinear effects are stronger than dissipative effects

FIGURE 13.1 The distortion of a single period of an originally plane sinusoidal, finite-amplitude wave during its propagation in a thermoviscous fluid from the source at the dimensionless distance s. Reprinted from Physics Procedia, 3, (2010), Leif Bjørnø, Introduction to Nonlinear Acoustics, page 9, Copyright (2010), with permission from Elsevier. http://www.elsevier.com with permission of Elsevier.

13.1 Physics and Nonlinear Phenomena

and the wave front steepness increases until at s ¼ 1 the wave reaches its maximum steepness at the zero crossing. This distance is frequently called the discontinuity distance. Only the zero crossing will propagate with the infinitesimal velocity of sound, co, while the local phase velocity in the wave will receive contributions from convective (nonlinear) terms of the equation of motion and from effects caused by the nonlinearity of the equation of state of the fluid. The local phase velocity in the compressional part of the wave will be the sum of the local sound velocity, c, and the local particle velocity, u, while in the rarefaction part the local phase velocity will be the difference between these two quantities. For increasing values of s the dissipative effects will grow relative to the nonlinear effects and a “sawtooth” wave shape may appear. Due to energy dissipation the wave profile will gradually lose its steepness, and the wave amplitude, and therefore, the nonlinear influence on the wave shape, will be reduced. For values of s [ s3 w 0.6G, as discussed in Ref. [1], the wave profile returns to its original sinusoidal shape with a strongly reduced amplitude. This forms the so-called “old age region” of the finite-amplitude wave. A Fourier analysis of the wave shape transformation in Fig. 13.1 shows that higher harmonics of the original sinusoidal finite-amplitude wave will be formed during the wave shape propagation. Since absorption in most fluids is proportional to the square of the frequency, the transfer of acoustic energy from the fundamental frequency to higher harmonics increases the dissipation of energy in the finite-amplitude wave. The stronger dissipation at the higher amplitudes may also lead to sound beam broadening when the sound amplitude on the acoustic axis is attenuated more than the side lobes. For sufficiently high values of the original wave pressure amplitude the later amplitude of the distorted, originally sinusoidal wave becomes relatively independent of the initial pressure amplitude. This effect, termed acoustic saturation, forms an upper limit for the sound pressure amplitude transmitted over a given distance. Therefore, even for an increase in the finite-amplitude wave amplitude at the source, the same amplitude will be received after wave propagation over a given distance as more and more energy goes into the higher harmonics with their higher rate of dissipation. Acoustic saturation can form an important factor limiting the propagation distance for sonar signals.

13.1.2 FOCUSED SOUND FIELDS The sound velocity gradients in seawater may lead to refraction of sound beams as discussed in Chapter 2. The refraction of the sound beams may lead to focused sound fields, where increases in the acoustic amplitude can take place. The increased amplitude can lead to finite-amplitude effects with formation of higher harmonics and to excess attenuation, i.e., attenuation above the linear attenuation of sound waves, as discussed in Ref. [1]. Focused sound fields in seawater may comprise influence of nonlinearity, diffraction, and attenuation. An equation which covers all three effects is the parabolic, KhokhloveZabolotskayaeKuznetzov (KZK) equation, which has been used extensively for studies of focused, finite-amplitude, ultrasonic fields in medicine as, for instance, the field in and around the focal point formed by a

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therapeutic body stone disintegrator, the Lithotripter, as discussed by Neighbors et al. [8,9]. The KZK equation in dimensionless form can be written as:   v2 v3 r 0 v 2 P2 2  Vt  4ar0 3 P ¼ 2 (13.5) 4 vsvs vs [d vs2 where P ¼ p/po is the normalized pressure in the finite-amplitude wave with po expressing the pressure at the wave source. s ¼ x/ro, where x is the on-axis distance from the source and ro (¼ua2/2co) is the Rayleigh distance for a monochromatic source with radius a. V2t is the two-dimensional transverse Laplacian. s ¼ u (t  z/co) is a retarded time, where u and t denote the angular frequency {unit: rad/s} and time, respectively. a is the attenuation coefficient, while co and ro are the isentropic speed of sound and the fluid density, respectively. [D (¼ro co 3 /(B/ 2A þ 1)upo) is the discontinuity distance, originally being characterized as the source distance for the first formation of a shock wave in a lossless fluid. Originally, Eq. (13.5) was derived without thermoviscous loss, e.g., a ¼ 0, by Zabolotskaya and Khokhlov [10], while the inclusion of dissipation was developed by Kuznetsov [11]. Eq. (13.5) accounts for nonlinearity, diffraction, and dissipation to an equal order of magnitude, where the 2-D transverse Laplacian term accounts for the diffraction and the term 4arov3/vs3P incorporates thermoviscous losses. In the derivation of Eq. (13.5) a well-collimated beam is assumed, i.e., ka [ 1, with k ¼ u/co. When P is expanded as a Fourier series and inserted into Eq. (13.5), the KZK equation becomes a series of coupled partial differential equations, the numerical solution of which follows a procedure initially developed by Aanonsen et al. [12] and enhanced by Hart and Hamilton [13] for application to focused sound fields. An example of the solution to Eq. (13.5) is given in Fig. 13.2, where the distortion

FIGURE 13.2 The time history for the distortion of a focused, originally sinusoidal, finite-amplitude wave during its propagation in a thermoviscous fluid toward the geometrical focal point situated at s0 ¼ 0. The wave source is situated at s0 ¼ 1. The figure shows the increase in the wave amplitude due to focusing effects and the wave front steepening due to nonlinear effects during the wave propagation. Reproduced from Neighbors, T.H. and Bjørnø, L., Monochromatic Focused Sound Fields in Biological Media. J. Lithotripsy Stone Dis., 2, (1), pp. 4e16, 1990 with permission of Futura Publishing Company.

13.1 Physics and Nonlinear Phenomena

propagation is calculated for the focusing of an originally sinusoidal, finite-amplitude wave propagating in a thermoviscous fluid. The wave source is situated at the dimensionless distance s0 ¼ 1, while the geometrical focal point is situated at s0 ¼ 0. Calculations show that the diffraction effects in the focused wave contribute to an increase in the wave amplitude of the compression part of the wave, while it reduces the amplitude of the rarefaction part. The increasing amplitude, and due to it, the nonlinear effects steepen the waves. Just before the geometrical focal point the wave amplitudes will achieve their maximum value and the increasing dissipative effects will influence the nonlinear process in such a way, that the geometrical focal point will not become the venue for the maximum focused wave amplitudes.

13.1.3 CAVITATION As mentioned in Chapter 10 on sonar systems, the maximum acoustic output power from a projector is limited by a number of factors introducing electrical, mechanical, and thermal limitations. However, an additional limiting factor is introduced by cavitation in the water around and near the projector. At shallow depths this factor may impose more serious limitations on the output power from a projector than other factors. The alternation acoustic pressure in an acoustic signal produced at the projector face is superimposed on the ambient pressure of the water. In the rarefaction half cycle of the emitted pressure signal, the absolute pressure is reduced below the ambient pressure. With increasing acoustic intensity in the emitted signal, the absolute pressure in the rarefaction half cycle may be reduced to zero or become negative. Dissolved gases in the water, frequently connected with micro particles or existing as free microbubbles, will form nuclei for water rupturing into a large number of frequently visible bubbles, which vibrate and collapse. Cavitation is the overall name for the bubble formation and the vibration and collapse of the bubbles. The collapse phase of the bubbles, where tiny water jets with speeds exceeding some hundreds of meters per second are formed, may lead to erosion effects on the projector face. The same erosion effects from cavitation are frequently found on part of ship propellers which may lead to propulsion system imbalances. Apart from projector mechanical erosion effects cavitation may also affect acoustic transmission. The formation of a large number of gas bubbles close to the projector face acts as a pressure release surface between the projector and the water. This pressure release surface will strongly reduce the radiation impedance seen from the projector, limit the radiated acoustic power from the projector, and distort the radiated acoustic beam pattern. Increased power transfer to the projector after cavitation formation will only lead to increased internal losses, such as heat formation in the projector. Moreover, due to strongly increased nonlinear acoustic qualities of bubble clouds, see Bjørnø [14], the formation of higher harmonics in the acoustic signal transmitted by the projector may take place and cavitation will also produce a substantial amount of noise which will interfere with the signal transmitted by the projector.

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The cavitation threshold, i.e., the acoustic pressure in the signal at cavitation inception, is a function of several factors, such as the amount of gas dissolved in the water, the number of microparticles, the number of microbubbles and their sizes, pollution, temperature, viscosity, and the ambient pressure and thus the depth where the projector is operating. Also, the signal frequency and the duty cycle have a strong influence on the cavitation threshold. At the sea level, the absolute pressure is reduced to zero if the peak pressure, po, in the acoustic signal is 105 Pa. If this pressure is used for definition of the acoustic intensity leading to cavitation inception at the sea level, the theoretical cavitation threshold, T, can be defined as:  5 2 10 p2 W T¼ 0 ¼ ¼ 0:33$104 2 m 2rc 2$1:5$106 which means that the maximum power transmitted by a plane, circular projector face with a diameter of 0.1 m at the sea level to avoid cavitation is below 26 W. Since T is proportional to the ambient pressure squared, the maximum power is transmitted by the projector before cavitation inception increases rapidly with projector depth. For a 30 m depth, this projector will be able to transmit up to 415 W before cavitation inception, unless other factors, such as the number and size of microbubbles influence the cavitation threshold. The theoretical cavitation threshold, TdB {unit dB rel 1 mPa} at the sea level is:
 T TdB ¼ 10 log ¼ 217 dB rel 1 mPa I0 when Io ¼ 6.5$1019 W/m2 for po ¼ 1 mPa. If the totally radiated acoustic power is, P {unit: W}, at the start of the cavitation, which is assumed to be uniformly distributed over a projector’s radiating surface area, A {unit: m2}, the relation between P and projector depth, d {unit: m} for the cavitation threshold T is: 
d 2 (13.6) P ¼ AT 1 þ 10 Due to other factors besides the ambient pressure, such as dissolved gases and the microbubbles number and size, accurate cavitation threshold prediction in seawater is difficult and the scattering of the measured threshold values is substantial. Also, Eq. (13.6) is only valid for low frequencies and long pulses. This is due to the fact that the duration of the rarefaction phase of the signal cycles, controlled by the signal frequency, influences the cavitation threshold since the bubble formation process during cavitation requires a finite time, which also is influenced by the amount of dissolved gas and other factors. Higher frequencies, with a shorter rarefaction phase duration, increase the cavitation threshold. Fig. 13.3 shows the cavitation threshold as a function of the acoustic frequency. As discussed by Urick [15], the curves in Fig. 13.3 are based on measurements published by Esche [16], Strasberg [17], and Flynn [18]. Fig. 13.3 shows that below 10 kHz the cavitation threshold is low, while

13.1 Physics and Nonlinear Phenomena

FIGURE 13.3 Cavitation threshold pressure in Pa, as a function of signal frequency (kHz). The hatched region around the average curve represents the scattering of the measured results found by various authors. The hatched region separates the region, where cavitation always produced, from the cavitation-free region. Adapted from Urick, R.J., Principles of Underwater Sound, (3rd. Ed.), McGrawHill Book Comp., 1983, Peninsula Publishing, Figure 4.6 with permission of Peninsula Publishing.

above 10 kHz it increases substantially. Also, the pulse length has a strong influence on the cavitation threshold. Longer pulses lead to lower cavitation thresholds, and for pulses of more than 10 ms duration, the cavitation threshold is nearly constant. Pulse lengths less than 10 ms show an increasing cavitation threshold.

13.1.4 ACOUSTIC RADIATION PRESSURE AND ACOUSTIC STREAMING A propagating sound wave carries acoustic momentum and acoustic energy. The average acoustic momentum carried through a unit area in a fluid per unit of time produces an acoustic radiation pressure, ps {unit: Pa} which, according to Stephens and Bate [19], can be written as: umax p2max ¼ 2 (13.7) ps ¼ ru2max ¼ pmax c rc where pmax {unit: Pa} and umax {unit: m} denote the maximum pressure and particle velocity, respectively, in a plane acoustic wave, while c {unit: m/s} is the sound

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velocity. Since the acoustic radiation pressure depends on the acoustic pressure squared, the radiation pressure is more influential at higher than at lower acoustic pressures. Acoustic streaming is the steady flow which may be associated with sound propagation in a fluid and was first recognized by Faraday. An in-depth exposition of acoustic streaming is given by Nyborg [20]. The streaming velocity is directly dependent on the coefficient of absorption in the fluid, and since the radiation pressure will fall off with distance from a projector due to the absorption, a net force, F {unit: N/m3} will be exerted on the fluid in front of the projector. This force will accelerate the fluid in a motion away from the projector. However, the motion will be resisted by the fluid viscosity, and in a final steady state a uniform fluid motion will result, provided that a suitable return path for the fluid can be established. The net force F for a plane, circular projector is provided by: vps F¼ ¼ 2aEo ðrÞ (13.8) vx where x {unit: m} is the direction normal to the projector surface while r {unit: m} is the radial coordinate on the projector surface. a {unit: Np/m} is the absorption coefficient and Eo(r) {unit: J/m3} is the energy density variation across the projector surface. As shown by Stephens and Bate [19] the streaming velocity v(r) {unit: m/s} as a function of the radial distance r from the center of a plane, circular projector with diameter 2a {unit: m} may be written as: aEo a2 vðrÞ ¼ fðrÞ (13.9) m where f(r) represents the variation of the velocity v across the diameter of the projector and m {unit: Pa$s} is the coefficient of shear viscosity in the fluid. Acoustical streaming was used by Liebermann [21] to calculate the magnitude of the bulk viscosity coefficient h0 {unit: Pa$s}, see Chapter 4, which explained a major part of the deviation found between theoretical and experimental values for absorption of sound in seawater.

13.2 NONLINEAR UNDERWATER ACOUSTICS The application of nonlinear acoustic phenomena in underwater acoustics, where nonlinear effects mostly had been considered a nuisance, started in the late 1950s when the exploitation of the sum and difference frequencies produced by the interaction between two sound waves of finite amplitudes for production of sound beams was suggested by Westervelt [22,23]. In particular, the difference-frequency beam, proposed by Westervelt was likened to an end-fire array and named the parametric acoustic array. Due to its advantageous beam qualities it has found a rather widespread use in underwater acoustics. The parametric acoustic transmitting array and the parametric acoustic receiving array have been studied extensively over many years, and sonar systems based on the parametric transmitting principles are now commercially available. An early exposition of the parametric acoustic array can be found in Bjørnø [24].

13.2 Nonlinear Underwater Acoustics

13.2.1 PARAMETRIC ACOUSTIC TRANSMITTING ARRAYS The generation of sum and difference frequencies by the interaction between two finite-amplitude sound waves has been the subject of discussions for more than 200 years. Helmholtz [25] and Lamb [26] credit the original observation of difference-frequency tones to Sorge in 1745 and Tartini in 1754. A considerable step forward in the studies of difference frequencies and their potential applications was produced by Westervelt’s classical article [23] “Parametric Acoustic Array.” In this article the nonlinear interaction between two superimposed, high-frequency, primary waves of finite amplitudes was calculated based on the solution to an inhomogeneous wave equation derived from Lighthill’s [27] theory for aerodynamic sound generation. Westervelt’s inhomogeneous wave equation in a general form may be written as: 1 v2 ps vq ¼ ro c2 vt2 vt

(13.10)


 B 1 vp2i 1þ 2 2A ro c4o vt

(13.11)

V2 p s  where q¼

ps {unit: Pa} in Eq. (13.10) is the pressure of the scattereddi.e., the difference frequencydwave generated by the collimated primary waves which are designated by the subscript i. t denotes time {unit: s}, while ro {unit: kg/m3} and co {unit: m/s} are the fluid density and velocity of sound, respectively. B/A is the dimensionless second-order nonlinearity ratio, and q is the source strength density of the process responsible for the generation of acoustic energy through the nonlinear interaction between the primary waves. Inserting Eq. (13.11) into Eq. (13.10) and solving for ps as a function of the distance R {unit: m} from the observation point to the center of the projector emitting the primary waves and the observation point angular coordinate q {unit: radian} produces Eq. (13.12) for ps. ps ðR; qÞ ¼

u2s p2o S ½1 þ B=2A qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 4 8pro co R a2 þ k2s sin4 ðq=2Þ

(13.12)

where us is the angular frequency of the difference-frequency wave, po denotes the pressure amplitude of each primary wave, S {unit: m2} denotes the cross-sectional area of the collimated wave interaction zone, ks is the difference-frequency wave number, and a is the absorption coefficient of the primary waves, and represents the only influence in Eq. (13.12) from the primary frequencies. The a value used is normally the average of the a values for the two primary frequencies. Westervelt’s solution to Eq. (13.12) is restricted to the far field of the scattered wave by the condition: ksR > (ks/a)2. From Eq. (13.12) the far-field, half-power beam width qh {unit: rad} of the difference-frequency wave can be expressed by: rffiffiffiffi a (13.13) qh w2 ks

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This shows that beam narrowing takes place for a decrease in the primary frequency, opposite to the case for a conventional linear projector. Also, a beam width reduction follows a difference-frequency increase. This directivity has the same form as Rutherford scattering in atomic theory. Eq. (13.12) shows that ps increases for higher primary wave amplitudes po; for increasing difference frequency, i.e., for lower downshift ratio (uo/us); for increasing fluid material nonlinearity of B/A; for decreasing fluid density ro; and in particular for decreasing fluid sound velocity. Variations of these parameters have formed the basis for attempts to increase the parametric acoustic conversion efficiency which is expressed by the ratio of sound energy in the differencefrequency wave to the sound energy in the primary waves, see Bjørnø et al. [28]. Because the effective array length can be made very large by selection of primary frequencies, highly directive difference-frequency beams can be produced by a projector small compared to the difference-frequency wavelength. Due to the exponential shading resulting from the primary wave absorption, very low side-lobe levels are usually found in parametric arrays. Fig. 13.4 shows a reduction in the sidelobe levels of 35e40 dB relative to the main-lobe level. The low side-lobe level will reduce the influence of clutter and thus improve the signal-to-noise level, and reverberation will decay quickly, which permits a high pulse repetition rate when the array operates in a pulse mode. Considerable frequency agility characterizes

FIGURE 13.4 Parametric beam profiles for Kongsberg Defence Systems’ TOPAS PS 40. (A) The primary beam at 40 kHz. (B) The difference-frequency beam with a downshift of 10 to 4 kHz. (C) The difference-frequency beam with a downshift of 4 to 10 kHz. The strongly reduced side-lobe levels at the difference-frequency beams are clearly visible with a reduction of 35 to 40 dB relative to the main-lobe level. Images provided courtesy of Drs. Arne Løvik and Johnny Dybedal, Kongsberg Defence Systems, Norway.

13.2 Nonlinear Underwater Acoustics

the parametric array, where large changes in difference frequency can be generated with only small changes in primary frequency, and the projector can continue to operate near its resonance frequency for a large range of difference frequencies. These superior qualities make high-resolution sonar feasible where only small projectors can be deployed, and the lack of side lobes encourages the use of parametric transmitters in reverberation-limited regions of the sea, for instance, in relation to shallow-water communication as discussed by Kopp et al. [29]. Westervelt’s [23] model is based on the assumption that the primary fields are collimated plane waves and the extent of the interaction region is limited by viscous absorption, while no nonlinear absorption is present. Also, the observation point is situated far from the interaction region. The influence of the absorption magnitude, i.e., small-signal absorption or nonlinear absorption, and the possibility that most absorption takes place in the near field or in the far field of the primary waves, form four distinct operating regimes for parametric sources. These regimes are: 1. Arrays are limited by small-signal absorption in the transducer’s near field. This is the Westervelt case and comprises most experiments based on low-power, high-frequency primary beams [30e32]. For lower primary frequencies the absorption will decrease. 2. Substantial parametric generation may take place in the far field of the primary waves, where the absorption influences the array length. The divergence in the far field of the beams from a piston source, which is most frequently used, leads to reduced primary signal amplitudes. However, far-field absorption is the effect terminating the parametric processes. The far-field absorption-limited array forms the second operating regime for parametric sources. Contributions to the study of this regime are found in Refs. [33e35]. 3. The third parametric array operating regime is formed when the array length is limited by nonlinear absorption in the near field of the primary beams. This array type is normally called saturation limited due to the influence of harmonics and shock formation in the primary beams. Since the near field of a piston source may be approximated by a plane-wave field, models for saturation-limited arrays are normally based on plane-wave assumptions. Theoretical and experimental studies of saturation-limited arrays may be found in Refs. [36e38]. 4. The fourth regime is of minor practical importance since saturation effects in the far field rarely occur, unless the primary beams are spherically spreading. Attempts to produce a theoretical treatment dealing simultaneously with all four regimes have been made by several authors [39e42]. The references for the four parametric ray regimes in the years 1960e85 reflect the most productive time in parametric acoustic array research. Ref. [42] is a valuable review of the development in parametric acoustic arrays since it combines research results obtained in the United States and Europe with results obtained in the former USSR. The physical qualities and the characteristics of the difference-frequency beam in regimes (1) through (3), when the field point is outside or inside the interaction region, show some individual variations. While the field point originally was considered to be outside the interaction region in the parametric array, the limitation of test

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facility dimensions or measurements near the signal source require a consideration of the field point inside the interaction region. When the field point is inside the interaction region, diffraction effects dominate over viscous effects, and in Ref. [43] a full account of diffraction effects was given through a 3-D scattering integral expressed by:
 R Z v2 p2i x; t  1 þ B=2A co ps ðx; tÞ ¼ R dV (13.14) 4pro c4o vt2 V

where R {unit: m} is the distance between the field point for the measurement and the volume element dV of the interaction region. The computational time required for each prediction limits the application of Eq. (13.14). Through a coordinate transformation which reduces Eq. (13.14) to a single integral and by assuming that the primary waves are spherically spreading from their origin, while neglecting viscous absorption and nonlinear attenuation, a close agreement between theory and experimental data was obtained by Rolleigh [44]. The saturation effects by high-amplitude wave interaction in parametric arrays cause an effective shortening of the array length which results in a broadening of the difference-frequency beam, a reduction in the difference-frequency source level, and an increase in side-lobe effects relative to the main beam. A comprehensive study of the parametric array behavior at increasing primary wave amplitudes has been reported by Moffett and Konrad [45]. Fig. 13.4 shows the parametric gain G as a function of the scaled primary source level SL0 expressed by: G ¼ SL  SLo and SLo ¼ SLo þ 20 log fo funit: dBg

(13.15)

where SLo and SL denote the rms source level of one primary-frequency componentdthe two primary-frequency components are assumed to have equal amplitudesdand the rms source level of the difference-frequency signal, respectively. fo {unit: kHz} is the mean primary frequency. The curves in Fig. 13.5 are for a downshift ratio, fo/f ¼ 10, and for various degrees of absorption expressed by aRo {unit: dB}, where Ro {unit: m} is the Rayleigh distance, which expresses collimation length of the primary wave near field. f {unit: kHz} is the difference frequency. The reduction in parametric gain for increasing scaled primary source level at lower absorption due to saturation effects can be seen from Fig. 13.5. The parametric gain will in most cases be nearly 40 dB or more down from the primary source level, see Fig. 13.6. This figure shows calculated and experimental data for the field point inside and outside the primary wave’s interaction region according to Bjørnø et al. [28]. The difference between the sound pressure level of the primary waves and the difference-frequency wave depends on the distance from the projector. Sound pressure level extrapolations show that the two curves for sound pressure levels will intersect at a projector distance of nearly 300 m with a sound pressure level of 110 dB rel 1 mPa. The low conversion efficiency has given rise to several studies focused on improving it. Based on Eq. (13.12) an increase in B/ A and a decrease in density and sound velocity of the fluid will result in increased conversion efficiency. Replacing the liquid in the primary waves’ near field by ethyl or methyl alcoholdboth with higher B/A and lower density and velocity of sound

13.2 Nonlinear Underwater Acoustics

FIGURE 13.5 Parametric gain curves for fo/f ¼ 10 and for various absorption coefficients a {unit: dB/m} and as a function of the scaled primary source level SL0 {unit: dB}. Reproduced from Moffett, M.B. and Mellen, R.H., Model for Parametric Acoustic Sources J. Acoust. Soc. Amer., 81, (2), pp. 325e337, 1977, with the permission of the Acoustical Society of America.

than waterdshowed increasing conversion efficiency. It was found [28] that it was possible to obtain an average gain relative to the use of water of 10.2 dB in ethyl alcohol and 13.6 dB in methyl alcohol. Moreover, using tone bursts with the burst frequency equal to the difference frequency and with the carrier frequency equal to the average frequency of Westervelt’s two primary frequencies increases the conversion efficiency about 2 dB while a 100% modulation of the carrier frequency with the difference frequency will result in more than a 2.5 dB increase in conversion efficiency. Also, Merklinger [37] found conversion efficiency improvements through signal processing procedures. Replacing the fluid in the primary waves’ near field with a solid, silicone rubber cylinder, is reported by Ryder et al. [46], who found that the lower sound velocity and the higher B/A of the silicone rubber together with the cylinder’s slow waveguide antenna effect in water result in a conversion efficiency increase of 2e5 dB. However, these conversion efficiency improvements were not able to substantially improve the nearly 40 dB in parametric gain.

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CHAPTER 13 Finite-Amplitude Waves

FIGURE 13.6 Measured and calculated primary wave pressure amplitudes and the difference-frequency wave exposed to absorption and spherical spreading as a function of distance to the projector. Reproduced from Figure 1 Bjørnø, L., Christoffersen, B. and Schreiber, M.P., Some Experimental Investigations of the Parametric Acoustic Array. Acustica, 35, (2), pp. 99e106, 1976, with permission from Acustica e Acta Acustica.

13.2.2 PARAMETRIC ACOUSTIC RECEIVING ARRAYS The possibility of developing a parametric acoustic receiving array was first suggested by Westervelt [23]. In a parametric receiver, the nonlinear interaction process may take place between a low-frequency signal wave of low intensity and a locally generated high-frequency pump wave of higher intensity. The sum- and differencefrequency signals are then received by a hydrophone on the acoustic axis of the pump wave. Low- and high-amplitude receiving arrays have been studied, and their difference is in the inclusion of finite-amplitude pump wave effects. Far-field reception was considered by Barnard et al. [47]. They studied the first-order sound field consisting of the interaction between a low-amplitude, spherical, harmonic pump wave and a plane, harmonic signal wave. A comprehensive theoretical and experimental study of the parametric receiving array is reported in Refs. [48,49], including interaction effects between the signal wave and the pump wave in the pump wave near

13.2 Nonlinear Underwater Acoustics

field and far field. Also the misalignment of the pump or the receiver transducer was studied showing that for a misalignment the difference-frequency beam pattern is asymmetrical and is a mirror image of the sum-frequency beam pattern. The influence of transducer vibration either due to platform motion or to transducers not firmly mounted, reported in Ref. [50], showed that transducer motion is a significant factor in parametric receiving array performance. The detrimental effects may be lessened by proper system design. However, one of the main reasons why the parametric receiving array has not received the same attention as the transmitting array may be ascribed to the influence on the reception from transducer motion, water noise at the signal frequency or at the sum and difference frequencies, electronic noise in the equipment, and other factors.

13.2.3 APPLICATIONS OF THE PARAMETRIC ACOUSTIC ARRAY Around 1985 most theoretical studies which formed the basis for constructing parametric acoustic transmitting arrays had been carried out and the necessary knowledge base was created. Not all aspects of the parametric array had been investigated; unification of theories and relations to other part of the fundamental expressions in nonlinear acoustics were still missing. However, a useful theoretical/numerical basis had been formed and the operation of experimental test models of parametricdtransmitting and receivingdarrays in small and large scales had been studied to verify the theoretical/numerical basis. Parametric acoustic array experiments had also been performed before 1985 for detecting biomass in the water column, for seabed studies, subbottom imaging, underwater communication, and transmission of television pictures over shorter distances in a lake. Raytheon, in cooperation with the US Naval Underwater Systems Center (NUSC), had in the late 1960s produced a test model of a parametric subbottom profiler, which in 1970 gave the first qualified pictures of fish schools and seabed structures, thus proving the superior beam qualities of the parametric array. During the 1970s the University of Birmingham carried out studies of harbor bottom profiles in the Netherlands and a large high-power towed parametric sonar (TOPS) had been produced and tested by NUSC [51]. This sonar had the following technical data: 0.5  2 m transmitter, 480 mass-loaded elements operating at a mean primary frequency of 24 kHz, and a primary source level of 250 dB rel 1 mPa at 1 m. Its primary 3 dB beam width was 2 degrees  8 degrees. The theoretical/numerical studies of parametric arrays since 1985 have involved substantial contributions from the “Bergen School” in Norway. In a series of papers dealing with sound wave interaction, beam shape influence, near-field contributions, influence of the source field shape, and lack of axis symmetry on the parametrically generated sound field were studied in depth [52,53]. Also, the importance of nonlinear effects on parametric signal generation [54] and the effects of focusing on the nonlinear interaction [55] were studied via solutions to the nonlinear parabolic equation. The interference of the difference-frequency sound after reflection at a surface with the original difference-frequency sound was studied theoretically and experimentally [56]. Also, the influence of misalignment of pump, source, and hydrophone on the performance of the parametric receiving array was quantified

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CHAPTER 13 Finite-Amplitude Waves

in numerical examples which showed that the best performance would be obtained for a good alignment, a high pump frequency, and the hydrophone placed not too far from the source near field [57]. Also the mechanisms of a parametric beam penetration into the seabed, when the nonlinear interaction between the primary waves is terminated at the interface between the water column and the seabed, have been studied theoretically and experimentally for parametric arrays operating in a transient mode [58e60]. Pace et al. [60] used the plane wave spectrum and reflection coefficients at the water columne seabed interface to calculate in 2-D and 3-D the reflected and refracted acoustic fields for conventional (linear) and parametric beams incident on the interface below and above a critical angle of 62 degrees. Figs. 13.7 and 13.8 show 3-D calculations of the contour plot of reflected and refracted fields due to a parametric beam incident on the interface at a 55 degrees and 65 degrees beam axis incidence. The refracted beam strength and direction for incidence beyond the critical angle are caused by the width and position of the plane wave spectrum and their relation with the plane wave reflection coefficients in the vicinity of the critical angle. The plane wave spectrum width around the critical angle leaves a portion of the spectra in the angular range below critical to enable a refracted beam to exist even as the incident beam axis increases beyond critical. Several successful attempts to exploit the parametric transmitting array for bottom and subbottom profiling occurred in projects under the European Union’s former Marine Science and Technology (MAST) program, such as the project “Sediment Identification for Geotechnics by Marine Acoustics” (SIGMA), which provided valuable information about the seabed structure and the relation between seabed material acoustical and geotechnical qualities. The narrow beam qualities –0.20 Depth into seabed in metres

874

–0.10

0.00

0.10

–24 –18

–27 –15

–9 –21

–30

0.20 0.30 0.0

–18 –15

–12 –24

–21

–27 –24

–21 24 – –270 –3

–30

–18 –27

–30

–0.2

–0.4

–0.6

Distance along interface in metres

FIGURE 13.7 3-D contour plot of reflected and refracted field due to a parametric beam incident on the watereseabed interface at a 55 degrees beam axis incidence with 3 dB contours. The 5 degrees dotted lines are centered in the 5 degrees interval incidence point. From Pace, N.G. and Bjørnø, L., The reflection and refraction of acoustic beams at water sediment interfaces. Acust. Acta Acust., 83, pp. 855e862, 1997, with permission from Acustica e Acta Acustica.

13.2 Nonlinear Underwater Acoustics

Depth into seabed in metres

–0.20 –30 –27 –18

–0.10 –21

0.00 0.10

–15 –12

–12 –8 –15 –24 –27 –30

–9 –18

–18 –21 –24 –27 –30

–24 –21

–27 –30

0.20 0.30 0.0

–0.2 –0.4 Distance along interface in metres

–0.6

FIGURE 13.8 3-D contour plot of the reflected and refracted field due to a parametric beam incident on the watereseabed interface at a 65 degrees incidence angle with 3 dB contours. The dotted lines at 5 degrees intervals are centered in the point of beam axis incidence. From Pace, N.G. and Bjørnø, L., The reflection and refraction of acoustic beams at water sediment interfaces. Acust. Acta Acust., 83, pp. 855e862, 1997, with permission from Acustica e Acta Acustica.

of the parametric array have also been exploited in relation to underwater communication in the MAST project Acoustic Communication Using Parametric Array (PARACOM) [61], where the interaction between the parametric beam and the water column boundaries was of interest. An example of the calculation of a parametric array transducer and its sound field is given by Bjørnø [62] in connection with the SIGMA project for seabed studies by a vertically incident parametric beam. The fundamental projector frequency was 60 kHz and the difference frequency was 6 kHz. The projector consisted of four 0.15  0.15 m squared disks with effective side lengths of 0.6 and 0.15 m. The environmental data were: ao ¼ 0.0164 dB/m (at 60 kHz); water temperature t ¼ 10 ; a tow-fish ambient static pressure at 10 m depth; Po ¼ 2$105Pa; PH ¼ 8Pa; and salinity S ¼ 35 ppt, with a sound velocity co ¼ 1490 m/s. From Eq. (13.13), the half-power beam width qh ¼ 1 degrees and the directivity index DI is about 19 dB. The acoustic power transmitted is 400 W for the primary frequency, which gives a transmitted power per primary frequency of 0.89$104 W/m2 or a sound pressure of 1.63$105 Pa. From Section 10.9 in Chapter 10 the primary frequency source level will be 216 dB rel 1 mPa at 1 m, and from Eq. (13.15) the scaled primary frequency source level is SLo ¼ 252 dB relative to 1 mPa at 1 m. Fig. 13.6 gives for the downshift ratio of 10, the gain value of G ¼ 42 dB, and SL will then be 174 dB rel 1 mPa at 1 m. Fig. 13.6 shows that no acoustical saturation is involved. For an absorption-limited parametric array the length of the interaction region will be [a ¼ ½ao, which will be [a ¼ 263 m. With a depth of operation of 80e120 m, a truncation of the parametric array is caused by the seabed. The influence of the rectangular shape of the projector represented by its aspect ratio N ¼ 1:4 can be calculated as follows. The source level coefficient Q, see Berktay [63], may be determined from Fig. 13.9. The scaled Rayleigh distance ro ¼ (ls/lo)Ro, where Ro is the Rayleigh distance for the projector, and ls and lo denote the wavelengths of the difference frequency and the primary frequency, respectively, will

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CHAPTER 13 Finite-Amplitude Waves

FIGURE 13.9 The source level coefficient Q as a function of the product of the absorption parameter aT and scaled Rayleigh distance ro for various projector aspect ratios N. Adapted from Berktay, H.O., Parametric sources e Design considerations in the generation of low-frequency signals. In: Ocean Seismo-Acoustics e Low-frequency underwater acoustics. T. Akal and J.M. Berkson (Eds.), NATO Conference Series, Plenum Press, pp. 785e800, 1986, with permission of Springer.

give ro ¼ 36.2 m, and with the absorption parameter aT ¼ 2ao, the abscissa value in Fig. 13.9 becomes 0.14, which gives Q ¼ 4 dB for the difference-frequency wave. The source level for a fully developed array, i.e., without seabed truncation, but with the projector aspect ratio taken into account, is SL ¼ 165 dB. The normalized parametric source beam width for a projector aspect ratio of 1:4 is 1.35 degrees and the difference-frequency half-power beam widths in the two directions for a fully developed projector array are qh ¼ 1.43 degrees and qh ¼ 3.39 degrees. A few commercially available parametric transmitting systems are produced and sold. The systems are marketed by Atlas Hydrographic GmbH, Germany, by Kongsberg Defence Systems, Norway, and by Innomar Technologie GmbH, Germany. The main application has been bottom and subbottom profiling. Parasound produced by Atlas Hydrographic operates with primary waves in the frequency range 18e33 kHz and difference frequencies from 0.5 to 6 kHz. The maximum bottom penetration is above 200 m for the highest power transmitted. Source levels for the 70 kW transmitted power are SLo ¼ 245 dB rel 1 mPa at 1 m and SL ¼ 206 dB rel 1 mP at 1 m. For a 35 kW transmitted power they are SLo ¼ 242 dB rel 1 mP at 1 m and SL ¼ 200 dB rel 1 mPa at 1 m. Kongsberg Defence Systems offers TOPAS PS Systems: the PS 18 for water depths from 10 m to full ocean depth; the PS 40 for water depths from 5 m to 1000 m, see Fig. 13.9; and the PS 120, a portable, high-resolution system, for depths between 3 and 400 m. The PS 40 primary frequencies are in the range 35e45 kHz with SLo w240 dB rel 1 mPa at 1 m, and parametrically generated frequencies are in the range 1e10 kHz with SLo ¼ 185  206 dB rel 1 mPa at 1 m (depending on the difference frequency). Fig. 13.4 shows the TOPAS PS 40 primary and two differencefrequency beam profiles, and Fig. 13.10 shows the subbottom profiles produced by

13.2 Nonlinear Underwater Acoustics

FIGURE 13.10 Subbottom profiles produced by Kongsberg Defence Systems’ TOPAS PS 18. Primary frequency: 18 kHz. Primary SL is about 243 dB rel 1 mPa at 1 m. Difference-frequency signal: Chirp 1.5e5 kHz, and 20 ms duration. Difference-frequency SL w195e207 dB rel 1 mPa at 1 m (Data from R/V G.O. Sars taken in Trondheim Fjord, Norway.). Water depth about 350 m. Penetration into the seafloor >120 m. High lateral and range resolution due to the narrow beam and the high bandwidth. Figure provided courtesy of Drs. Arne Løvik and Johnny Dybedal, Kongsberg Defence Systems, Norway.

TOPAS PS 18. This figure emphasizes the high lateral and range resolution produced by the narrow difference-frequency beam and the high bandwidth used in the Chirp difference-frequency signal covering 1.5e5 kHz with a 20 ms signal duration. Innomar Technologie’s parametric subbottom profiler, SES-2000, operates with primary frequencies around 100 kHz and difference frequencies in the range

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CHAPTER 13 Finite-Amplitude Waves

5e15 kHz, and for deep-water studies the primary frequencies are around 35 kHz and the difference frequencies are between 2 and 7 kHz, which permit sediment penetration up to 150 m.

13.3 UNDERWATER EXPLOSIONS High-intensity sound in the sea can be generated by using chemical explosives, air guns, or electrical discharges. Chemical explosives have been used as signal sources for single, but reproducible, broadband high-amplitude signals. The characteristic feature of a signal from a chemical explosive is the formation of a high-frequency shock wave followed by the low-frequency pulsation of a gas bubble. More low-frequency air gun signals have found applications in studies of the seabed, where return signals from layers in the seabed provide information about characteristic structures of interest in prospecting for oil or gas. Electrical discharges between two electrodes have been used for to study shock wave propagation in water, since the signals are very reproducible and the amplitudes and signal time histories can be controlled by the capacity of the condenser bank, the spark gap electrode distance, and other factors. This section deals with the characteristic features of signals based on the use of chemical explosives, while other sources of high-intensity sound are also mentioned.

13.3.1 THE SHOCK WAVE A chemical explosive in most cases consists of the elements C, O, H, and N in various combinations. Some of the explosives used are TNT (trinitrotoluene, C7H5O6N3), tetryl (C7H5O8N5), RDX (C3H6N6O6), and PETN (C5H8O12N4). Also the explosive pentolite, consisting of 50% TNT and 50% PETN, has been used extensively to produce underwater, high-intensity sound due to its reproducible signals. The energy available in TNT, for instance, is around 4.5$106 J/kg. The frequently used SUS (Signal, Underwater Sound) charges are mostly TNT in about 0.82 kg amounts. The detonation process in the explosives is most frequently started by a shock wave produced by an especially sensitive explosive which is used as a primary material to start the detonation process. In the detonation process a detonation front will move with high velocity through the explosive transforming into reaction products under very high pressure, about 1010 Pa, and temperatures, above 3000 C. A detonation is characterized by a constant detonation front velocity, which for TNT and tetryl are about 6900 and 7500 m/s, respectively. The calculation of the transition from explosive before the detonation front to reaction products after the front can be done by use of the RankineeHugoniot relations and the Chapmane Jouget condition, D ¼ c þ u, where D is the detonation front propagation velocity {unit: m/s} of propagation and c and u are, respectively, the velocity of sound and the particle velocity in the reaction products after the front; see Cole [64]. When the detonation front reaches the boundary between the explosive and water, a shock wave is transmitted into the water. The time history of the shock wave is characterized by a very fast rise in pressure from the hydrostatic pressure to a peak pressure, Pm {unit: Pa}, with the magnitude depending on the explosive type and the amount in kilograms. The time for the growth in pressure from

13.3 Underwater Explosions

hydrostatic to the peak pressure is only a small fraction of a second which depends on the type and amount of explosive used. The shock front is followed by an exponential decay in pressure with a time constant q {unit: s} for the pressure being reduced to Pm/e. Similarities between peak pressures and time histories of shock waves from chemical explosives, and thus their impulses I {unit: Ns/m2} and energy flux densities E {unit: J/m2}, were found very early and have given rise to power laws of the form: constant $(W1/3/R)a, where W {unit: kg} and R {unit: m} denote the amount of explosive and the distance from the detonation site, respectively, and a is a constant. The influence of the density of the explosive forms an integrated part of the constants. The power laws, which give a compact and reasonably precise method for representing the shock wave data in water, are: !a W 1=3 Pm ¼ K 1 R q ¼ K2 W

1=3

I ¼ K3 W 1=3

E ¼ K4 W

1=3

!b W 1=3 R !g W 1=3 R W 1=3 R

(13.16)

!d

where the constants K1eK4 and the exponents aed are individual for each type of explosive. For TNT and tetryl the constants and exponents are given in Table 13.3, taken from Bjørnø [65].

Table 13.3 Experimentally Obtained Values of the Constants K1eK4 and Exponents aed in Eq. (13.16). Explosive Range r ¼ R/W Pm {unit: Pa}

1/3

q {unit: s} I {unit: Ns/m2} E {unit: J/m2}

K1 a K2 b K3 g K4 d

TNT

TETRYL

0.46e11.1 521.6$105 1.13 96.5$106 0.18 5760 0.89 9.8$104 2.10

4.31e50 506$105 1.10 87$106 0.23 5900 0.87 11$104 2.12

Table extracted from Bjørnø, L., A comparison between measured pressure waves in water arising from electrical discharges and detonation of small amounts of chemical explosives. Trans. ASME J. Eng. Ind., 92, Ser. B, (1), pp. 29e34, 1970, with permission of the American Society of Mechanical Engineers.

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CHAPTER 13 Finite-Amplitude Waves

13.3.2 THE GAS BUBBLE While nearly half the explosive energy goes into the shock wave, for tetryl the energy amount was measured to 46% [65], most of the residual energy is found in the gas bubble. The gas bubble expansion after the shock wave emission is associated with energy emitted in the form of pressure waves which propagates radially from the bubble. Since the pressure in the reaction products of the gas bubble, after the detonation and shock wave emission, is substantially higher than the hydrostatic pressure at the detonation site, the gas bubble expands continuously for a relatively long period. During the bubble expansion its gas pressure decreases gradually to the hydrostatic pressure, but the bubble expansion continues due to the inertia in the outward flowing water. When the pressure in the gas bubble falls below the absolute environmental pressure, the difference between the bubble pressure and the environmental pressure gradually stops the bubble expansion. The bubble boundaries now contract at an increasing rate. This inward motion continues until the gas compressibility and the increasing pressure in the gas bubble stop the inward motion of the bubble boundary and its environment. The bubble now has its first minimum radius and a new expansion and contraction cycle may start. In nearly 80% of the bubble pulsation time, the pressure in the bubble will be below the hydrostatic pressure. Depending on the explosive detonation depth gas bubble oscillations may persist for a number of cycles. In spite of the fact that an oscillating gas bubble, due to Bernoulli effects, will receive a repulsive force from a free surface and will be attracted toward a rigid boundary, the buoyancy effects on the bubble make it migrate toward the water surface. The speed of migration is highest when the bubble has its minimum radius. The period, Tn {unit: s}, for the gas bubble oscillation depends on the detonation depth, d {unit: m}, and on the amount of explosive W {unit: kg}. The number of oscillations, n, increases with the detonation depth, and the oscillation period Tn is approximately given by Ref. [64] Tn ¼ K5

W 1=3 ðd þ 10:33Þ5=6

(13.17)

where K5 is a constant which is about 2.1 for most explosives. Eq. (13.17) assumes that no buoyancy influence is present and that the oscillations take place in the same depth far away from limiting surfaces like the water surface or the seabed. For each cycle of expansion and contraction, bubble energy will be lost in pressure radiation, flow, and turbulence, and the buoyancy will move the bubble to lower water depths, which all reduce the oscillation period. An approximate expression for the peak pressure, Pmp {unit: Pa}, in the first bubble pulsation, achieved at the first bubble minimum radius, can be found from: W 1=3 (13.18) R where W {unit: kg} and R {unit: m} are the explosive weight and distance between the detonation site and measurement point, respectively. Eq. (13.18) assumes spherical pressure signal propagation. Pmp ¼ 7$106

13.3 Underwater Explosions

While the peak pressure in the first bubble pulse is no more than 10e20% of the peak pressure in the shock wave, the bubble pulse duration is much longer. In fact the impulses of the two pressure waves are nearly equal. This is the reason that the most serious damage on a ship caused by an underwater explosion frequently is due to the gas bubble pulsation, in particular when the explosion takes place under the keel and the gas bubble migrates toward the ship bottom which is already damaged by the shock wave. The interaction between the sea surface, shock wave, and subsequent gas bubble gives rise to characteristic visible phenomena. When the shock wave is reflected at the water surface, an expansion wave is produced which moves backward into the water column. The water at the surface layer is thrown up with a particle velocity proportional to the arriving shock wave pressure and a rounded dome of whitish water forms on the surface. The white color is caused by the cavitation produced by the expansion wave. At a greater radial distance along the water surface from position of the first encounter between the shock wave and the surface, the shock wave produces a rapidly advancing ring of darkened water due to the shock wave influence on the index of refraction of the water. When the gas bubble reaches the water surface, characteristic surface phenomena may occur. If the gas bubble is in its high-pressure phase, it will shoot up a narrow plume of spray to considerable heights above the surface. If it is in its expanded phase, only low plume formation will occur since the gas bubble motion is nearly radial, and the plumes are projected outwards in all directions through the spray dome. For detonations close to the water surface the bubble-pulse influence on the high-intensity signal may be avoided due to gas venting out the bubble before its contraction phase, and only the shock wave will be available for the measurements.

13.3.3 OTHER SOURCES OF HIGH-INTENSITY SOUND Other sources of high-intensity signals are, for instance, air guns, in particular for seismic investigations, electrical discharges over a spark gap, and boomers. These signal sources are given a brief exposition in this section. An air gun is a mechanical device which releases a high-pressure air bubble under water. Similar to the gas bubble after a chemical explosive detonation, the released air bubble expands and contracts in cycles. During the cycles it emits pressure waves as sources of seismic waves used in reflection seismology in the layers below the seafloor. The air gun consists of one to several tens of pneumatic chambers, which can form an array that is pressurized with compressed air at pressures normally ranging from 14 to 21 MPa. Air guns are submerged in the seawater and are normally towed behind a vessel at a depth of about 6 m. The air gun is fired by an electrical signal which triggers a solenoid valve, which releases air into a fire chamber causing a piston to move. The piston movement allows air to escape from the main storage chamber into the water, producing a high-pressure air bubble. The release of large bubbles gives low-frequency signals, while small bubbles lead to more highfrequency signals. The pressure variation in the water due to the air bubble pulsation is called the air gun signature. The released air bubble is nearly spherical, and its

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CHAPTER 13 Finite-Amplitude Waves

expansion due to the air pressure exceeding the hydrostatic pressure produces a signal with a shocklike front, but not a shock front like the one found by explosive detonation. Similar to the gas bubble from a chemical explosive detonation, the air bubble is exposed to the same mechanism and may go through many expansion/ contraction cycles, where it emits pressure waves into the environment. The bubble oscillation amplitude is damped with the increasing number of cycles and the oscillation period is not constant from one cycle to the next. This nonharmonic motion is due to energy transmission into the water at each cycle. The first cycle produces the highest pressure amplitude in the primary wave, frequently exceeding 250 dB rel 1 mPa at 1 m. Even the largest single air gun will frequently not produce enough energy to permit the seismic signal to penetrate and produce return signals from deeper layers about 5 km below the seafloor. Single air guns produce pressure pulses which oscillate through several cycles, and each cycle produces pressure signals whose reflected signals oscillate through the same cycles as the source signal. This makes it difficult, if not impossible, to separate the primary from the secondary peaks in the return signal from the layers in the seabed. What is needed is a single, well-defined pressure spike from the air gun. The amplitude deficiency and the problems caused by the bubble oscillations can be solved by use of arrays of air guns. Air gun arrays can contain up to several tens of individual air guns with different size compressed air chambers. The aim is to create the optimum pressure in the initial shocklike pressure wave with minimum bubble reverberation after the initial wave. The multigun array produces increased amplitude due to the increased number of guns. Since different volume air guns produce bubbles with different oscillation periods, an air gun array can be built where the bubble pulses peaks after the primary pulse cancel each other due to being out of phase. Since the air gun array is towed near the sea surface, the reflection of the bubble pulse pressure signals in the surface, the so-called Lloyd’s mirror effect, discussed in Chapter 2, gives rise to a pulse with a negative peak. When the guns and their bubbles have different volumes, the bubble peaks occur at different times. It is possible to tune the array by choosing air gun volumes such that a bubble pulse is canceled by the negative pulse from a smaller air gun. Since the air gun array is towed in the same depth, the negative pulse arrival time is the same in all air gun signatures. The signature of an air gun array depends on the individual air guns and the stability of the air gun array geometry. Also the synchronization of the repeatability of firing time for the individual air guns will influence the array signature. The time synchronization is normally computer controlled within a time window of 100 ms. The stability of the array geometry involves maintaining of the distance between the individual air guns and keeping their tow depth constant. The tow ship turning and the sea state will influence the stability of the geometry. The sea state, i.e., waves on the surface, will also influence the magnitude of the negative pulse peak produced by the reflection at the sea surface. When one air gun in an array is fired, its signature may influence the signature of subsequently fired air guns if the distance between the two air guns is small. The first

13.3 Underwater Explosions

air gun’s pressure pulses will change the hydrostatic pressure around the next air gun to be fired. If the air guns are less than 1 m from each other, their bubbles may coalesce into one single bubble. Air guns forming a cluster without the coalescence between the bubbles may produce a much stronger peak pressure during their interaction than tuned air gun arrays. The high pressure amplitudes and the low-frequency contents of the air gun array signals are so loud that they may disturb, injure, or kill marine life. The impacts of the signals may include temporary or permanent hearing loss, habitat abandonment, disruption of mating and feeding, and even beach stranding and marine mammals deaths. Air gun blasts may also kill eggs and larvae and scare fish from important habitat and thus harm commercial fisheries. The potential damage to life in the sea caused by seismic air gun blasting repeated every 10 s, frequently for 24 h a day, and through weeks at a time, is discussed in Chapter 12 on bio and fishery acoustics. Electrical discharges over a spark gap establish an electrical spark channel in the water. The time for the spark channel formation is dependent on the voltage gradient over the spark gap and may last from a few to about 100 ms, see Bjørnø [65]. When the conducting path between the electrodes in the spark gap has been established, a considerable rise in the electric current through the spark gap takes place and a fast transfer of a substantial amount of energy to the small volume of water will occur, which causes a rapidly rising temperature followed by a fast expansion of the spark channel. This expansion causes a high transient pressure in the water around the spark channel and a shock wave formation. By changing the capacity C {unit: F} of the condenser bank, voltage Vo {unit: V} over the spark gap, electric circuit induction L {unit: H}, or the spark gap length [ {unit: m}, the discharge energy and duration can be controlled. It can be shown empirically [65] that the peak pressure Pm {unit: Pa}, the impulse I {unit: Ns/m2}, and energy flux density E {unit: J/m2} are given by:  2 1=3 5 [Vo Pm ¼ 6$10 R  E ¼ 0:32

[Vo2

2=3

R2

(13.19)



1=3 [Vo2 I ¼ 2:1 R where R {unit: m} is the distance from the spark gap center to the field point. Since the pressure in the spark channel can be assumed to be directly proportional to the discharged energy in the water spark gap, it is possible to establish a scaling law similar to the pressure variation produced by a chemical explosive detonation. [Vo2 represents the energy contained in the spark gap on a par with the amount of

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CHAPTER 13 Finite-Amplitude Waves

chemical explosive in kg in Eq. (13.16). The 1/R dependence of pressure amplitude given in Eq. (13.19) is due to the longer time to establish the spark channel compared to the interaction time between the explosive detonation wave and the surrounding water. High-intensity signals from electrical discharges show a very high degree of reproducibility. The boomer may produce pressure amplitudes above 7$107 Pa. The highintensity pressure waves can be produced by the discharge of a condenser bank through a flat coil with a flat copper membrane situated, insulated from the coil, above the flat coil and in contact with the water. The discharge will produce eddy currents in the copper membrane which cause the membrane to be pushed away from the coil due to self-induction in the membrane. When the membrane is pushed away from the coil, a pressure wave is established in the water above the membrane.

13.4 LIST OF SYMBOLS AND ABBREVIATIONS Table 13.4 provides a list of the symbols and abbreviations used in this chapter and the page each symbol or abbreviation first appears. Table 13.4 List of Symbols and Abbreviations Symbol or Abbreviation SONAR G x Rea b m z t co s M B/A p T en In co

Description SOund NAvigation and Ranging Gol’dberg number One-dimensional distance {unit: m} Acoustic Reynolds number Represents viscosity and heat conduction effects {unit: kg/(ms)} Shear viscosity {unit: km/(ms)} Bulk viscosity {unit: km/(ms)} Time {unit: s} Local sound velocity {unit: m/s} Dimensionless propagation distance parameter Acoustic Mach number Dimensionless second-order nonlinearity ratio Local sound wave pressure {unit: N/m2} Absolute temperature {unit: K} Neumann factor, en ¼ 1 for n ¼ 0, and en ¼ 2 for n  1 Bessel function with imaginary argument Fluid isentropic speed of sound {unit: m/s}

Page of First Usage 857 858 859 858 858 858 858 859 859 859 859 859 859 859 859 859 862

13.4 List of Symbols and Abbreviations

Table 13.4 List of Symbols and Abbreviationsdcont’d Symbol or Abbreviation ro [D T TdB P A d ps pmax umax c ps qh G SL0 R fo Ro f NUSC TOPS MAST SIGMA TNT D SUS W Tn Pmp C Vo L [

Description Fluid density {kg/m3} Discontinuity distance, i.e., the source distance for first formation of a shock wave in a lossless fluid Theoretical cavitation threshold {unit: W/m2} Theoretical cavitation threshold {unit: dB rel 1 mPa} Totally radiated acoustic power {unit: W} Projector’s radiating surface area {unit: m2} Projector depth, d {unit: m} Acoustic radiation pressure {unit: Pa} Maximum pressure {unit: Pa} Maximum particle velocity {unit: m/s} Sound velocity {unit: m/s} Scattered difference-frequency wave pressure {unit: Pa} Half-power beam width {unit: rad} Parametric gain {unit: dB} Scaled primary source level {unit: dB} Field point measurement distance {unit: m} Mean primary frequency {unit: kHz} Rayleigh distance {unit: m} Difference frequency {unit: kHz} US Naval Underwater Systems Center TOwed Parametric Sonar Marine Science and Technology Sediment Identification for Geotechnics by Marine Acoustics Trinitrotoluene (C7H5O6N3) Detonation front propagation velocity {unit: m/s} Signal, underwater sound Explosive weight {unit: kg} The period {unit: s} for gas bubble oscillation First bubble pulsation peak pressure {unit: Pa} Condenser bank capacity {unit: F} Spark gap voltage {unit: V} Electric circuit induction {unit: H} Spark gap length {unit: m}

Page of First Usage 862 862 864 864 864 864 864 865 865 865 865 867 867 870 870 870 870 870 870 873 873 874 874 878 878 878 880 880 880 883 883 883 883

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