H. Under this assumption, the eigen-frequencies can be simplified as / \u00E2\u0080\u009E = ( 2 \" + g 1 ) C s , \u00E2\u0080\u0094 0 , 1 , 2 , . . . . (4.10) Equation (4.10) represents the odd mode frequencies of SH waves in a one dimen-sional waveguide. The even modes cannot be excited by a unidirectional stress as described by (4.6). The observed frequency (77SHz) of the pure tone signal (shown in Figure 4.2) indicates that rubbing or dry friction between ice floes has excited a wave similar to the fundamental mode (set n = 0 in Eq. (4.10)). If the above theory has indeed described the rubbing process precisely, one can expect that equation (4.10) can be used to infer the shear wave speed Cs of an ice floe provided its thickness is known. With /\u00E2\u0080\u009E = 77SHz, n = 0 and H = lm, (4.10) yields Cs = 1556m \u00E2\u0080\u00A2 s - l 141 4.5 Concluding Remarks on Chapter 4 I have derived a theoretical model to investigate the generation mechanism of the observed pure tone ice rubbing sound. Based on the observation (Figure 4.2), I believe that the dry friction between adjacent ice floes excites an SH wave in the ice. This SH wave radiates as a pure tone in the water with a frequency equal or close to the fundamental frequency predicted by the theory. To avoid the cumbersome nonlinear boundary value problem indicated by equa-tion (4.4), I have ignored the nonlinear effect by linearizing the boundary condition at the edge of ice floe A (see Eq. (4.5)). As a result, the pure tone frequency due to rubbing has been overestimated since the nonlinear damping coefficient, whether it is positive or negative, will result in a frequency lower than the normal mode frequency given by equation (4.10). Therefore, the inferred shear wave speed (1556m \u00E2\u0080\u00A2 s - 1 ) from (4.10) provides a lower bound for the in situ ice, which seems resonable when compared with 1590m \u00E2\u0080\u00A2 s - 1 , a value inversely solved through the SAFARI model for new ice by Miller [56]. On the other hand, in a marginal ice zone, the primary stress on the ice floe is due to ocean currents and air flow. Therefore, a perfect waveguide does not exist for SH waves \u00E2\u0080\u0094 the stress is always transferred into the ice floe through its boundaries, and the force making the two adjacent ice floes rub against each other is supplied from ocean currents beneath the ice and air flow above. In this case, the frictional force along the ice edge will result in a damping coefficient which is definitely positive, and the problem will no longer be nonlinear. The observations reported here are insufficient to gain a complete picture of the generation mechanism of ice rubbing sound in the arctic ocean. Further experimental study is needed to search for answers to the following questions: 1. Does the pure tone sound result from a nonlinear rubbing process? 2. If a nonlinear effect is indeed involved in generating the sound, then how dominant is this effect in the sound generation? Chapter 5 Conclusions of the Thesis 5.1 Summary of the Study The observations made in the two ambient sound experiments provide me with a unique opportunity to examine the seismic and acoustical behaviour of Arctic sea ice in various environmental conditions. The analysis of the data motivates the devel-opment of 3 theoretical models respectively for ice fracturing due to internal stress (sources not clear), superficial ice cracking due to thermal stress and ice rubbing due to relative motion between ice floes. The surveys of in situ ice by photography and SAR imaging and the acoustical localization of individual cracks (though not very precise), make it possible for the first time to correlate the spatial distribution of cracks with ice conditions. The main research orientation of this thesis focuses on the study of failure mechanisms of the Arctic ice cover through the sound radiated from individual cracks and leads to the potential use of ambient sound to monitor remotely the mechanical properties of sea ice. Both the artificial sound made in the second experiment and some naturally occurring sound, vividly illustrate the filtering effect of an ice cover on a broadband source, and the importance of the water channel in ambient ice sound propagation compared to the contribution of the ice waveguide. The main results from this study are summarized in the following: 142 143 \u00E2\u0080\u00A2 The fracturing of sea ice radiates sound into the ocean. The signals usually consist of a dominant component of low frequency and possibly a small com-ponent of a high frequency. The base band signals correspond to the bulk vibrations of the ice cover. Therefore, the corresponding frequency is a func-tion of mechanical bulk properties of the ice. For most observed events, this frequency is below 700 ~ 1000-ffz, depending on the ice thickness, the Young's modulus, etc. As far as the high frequency signals are concerned, it is believed that they result directly from the actual slip-stick seismic movements over the fault planes at the crack. \u00E2\u0080\u00A2 Rubble ice with tilted snow free surfaces is most sensitive to thermal cooling effects. It responds to thermal stress by surface cracking when the temperature drops to around \u00E2\u0080\u0094 20C. In contrast, ice covered with a snow layer of a few centimetres does not respond to cooling. \u00E2\u0080\u00A2 Thermal cracking tends to radiate sound in directions parallel to ice surfaces. If the ice surface is flat, maximum radiation is in the horizontal direction. \u00E2\u0080\u00A2 Ice failures due to loading or any failure processes that can be represented by a monopole in the ice cover will result in a dipole radiation pattern. \u00E2\u0080\u00A2 For superficial cracking, it is found that the frequency of the cracking sound depends on the radiation angle. There is more high frequency sound radiated downwards than horizontally. \u00E2\u0080\u00A2 Ice rubbing seems to be an efficient mechanism for generating SH waves in the ice. Based on the observed rubbing event, it is felt that the nonlinearity effect is not profound. In concluding this thesis, I would like to point out that the acoustical radiation from an ice cracking event in sea ice is the consequence of geophysical response of ice to various stresses. This response consists of ice fracturing of different scales 144 and vibrations of the ice plate in different modes. The analysis of individual events shows quite often that certain aspects of the ice response are dominant. In order to use the sound of cracking to infer the behaviour and properties of sea ice, realistic theoretical models must be developed to provide a firm basis for interpretation of observations. 145 5.2 Suggestions for Future Research Naturally occurring sound results from failure processes in the ice, and one gains a lot by analyzing this kind of sound. However, it is not an easy task to predict exactly when and where cracks would occur. It is even harder to reconstruct precisely the source mechanisms based on the observed sound. While the monitoring of under-ice ambient sound is an indispensable experimental approach in future ice studies, more effort should be made to use artificial sources to probe the dynamics of the acoustical behaviour of sea ice. Although only one mode of failure can be simulated by artificial sources at a time (say loading failure), the corresponding sound should reveal the ice response to that particular mode of failure. This approach will enable us to represent the geophysical response of sea ice to various environmental stresses in terms of response functions to various modes of failure. Another aspect of the field study of sea ice is to monitor simultaneously the under-ice ambient sound field and strain field of the corresponding ice floe(s) so that the radiated sound can be related to stress release within the ice. The comparison of these two energy budgets will provide us with some information on how efficient the stress releasing process in the ice is in generating under-ice sound (including both water mode and ice mode acoustical radiation). If the acoustically radiated energy is significant compared with the total released stress, then it would provide a motivation for studying the efficiency of water mode acoustical radiation in stress release from sea ice. To fulfill this purpose, a joint experiment is needed in which both acousticians and experts on ice mechanical properties participate. It seems that such an opportunity may occur, when a project to be funded by the Office of Naval Research to study the mechanical properties of the Arctic sea ice gets underway in a couple of years. The following are a few suggestions for future experiments. \u00E2\u0080\u00A2 Using artificial sources to check the angular frequency response of the ice at as many radiation angles as possible. This work can be done both in the 146 laboratory and in the field. \u00E2\u0080\u00A2 If the frequency shift does occur, then the highest frequency that can be ra-diated, should be established. This cutoff frequency should tell us something about the viscous properties of the ice. \u00E2\u0080\u00A2 A laboratory study should be initiated to study acoustical radiation patterns from an ice cover for normal impact forcing and horizontal tensile forcing. The comparison of the two patterns will further confirm that the preferential radi-ation directions of excited sound are closely related to the forcing directions. \u00E2\u0080\u00A2 A laboratory study should also be conducted to study the generation mecha-nisms of the sound of rubbing ice. It will be interesting to determine, for an ideal ice plate subject to rubbing under different magnitudes of normal pres-sure, what is the range of radiated frequencies and how far these frequencies deviate from the first mode SH wave frequency. Answers to these two ques-tions will lead to a better understanding of how important the nonlinear effect is on the sound generated by rubbing ice. Appendix A A n Underwater Sound Field due to a Point Harmonic Source in an Ice Cover In the following, I will show how a solution to the problem given by equations (2.1) to (2.4) is obtained. Using Hankel transform, I can express $i i 2(i?, Z) in terms of their spatial spectra *i, 2(Z,i/): TOO _ $ l i 2 ( i ? , Z ) = / $ l i2(Z,!7)Jo07/2Mi7, (A.l) Jo where n \u00E2\u0080\u0094 ki cos ct\ = fc2 cos a2 is horizontal wavenumber; Jo(rjR) is the zero order Bessel function. The subscripts 1 and 2 again refer to ice and water respectively. It is easy to show that /\u00E2\u0080\u00A2oo 4ir6(R, Z - h) = 2 / S(Z - h)J0{riR)ridq. (A.2) Jo Upon substituting (A.l) and (A.2) into equations (2.1) to (2.4), the original wave problem is represented by two ordinary differential equations in terms of 3>1)2(Z, 77): ^ + = -2S(Z -h), Z< H; (A.3) 147 148 and H; (A.4) where /?i = \Jk2 \u00E2\u0080\u0094 rj2, /?2 = \Jk\ \u00E2\u0080\u0094 t]2 are the vertical wavenumbers in ice and water respectively. The point source is represented by the delta function and this representation causes the source position (0, h) to be a singular point for the sound field. In other words, the wave equation is invalid at the source. To overcome this difficulty, I integrate equation (A.3) with respect to Z around the source region, i. e. from h~ (just below the source) to h+ (just above the source). Thus: \h+ T7F\h- = \u00E2\u0080\u0094 A (A.5) dZ dZ and the sound pressure field is continuous at Z = h+,h~ for a monopole source. Therefore = $iU- (A.6) Equations (A.5) and (A.6) are the famous Pekeris source conditions which state that at the source, sound pressure remains continuous but the pressure gradient is discontinuous. These conditions allow the subdivision of the ice cover into two layers: one is above the source (Z > h); the other below the source (Z < h). Therefore, the source effect is transferred to a boundary condition. The other two boundary conditions at the ice surface and ice-water interface are given in terms of $ 1 ) 2(Z, n) as and $i=0, at Z = 0, d! dZ dH. (A.10) Subtitution of the first expression in equation (A.9) into (A.7) leads to Ax = -Ae**k. Thus, the sound field in the ice cover and water given by (A.9) and (A.10), can be expressed in terms of 4 coefficients, i. e. A,B,C and D. Substituting (A.9) into (A.5), (A.6) and (A.8), I obtain 4 algebraic equations with respect to the 4 coefficients ( A . l l ) 1 _ e2\u00C2\u00AB\"/9ifc -1 -1 0 A 0 2e2iPlh 2 0 0 B -2/ifc 0 eiPi(H- \u00E2\u0080\u00A2h) -to/pi C 0 0 -h) _e-iPi(H-h) -NPx D 0 Solving for coefficient D from (A.ll) yields 2 sin j3\h D m/Jx cos ft\H \u00E2\u0080\u0094 i/32 sin fixH' which is identical to S(j]) in equation (2.7). Thus, equation (A.10) becomes 2 sin fiih pi02(Z-H) Z> H. (A.12) mfti cos ftiH \u00E2\u0080\u0094 i/32 sin faH Substituting (A.12) into (A.l), I obtain the underwater sound field due to a monopole source in the ice cover, i. e. r O O $2 = / Si-nyW-^JoinR^dn. Jo Equation (A.13) is identical to equation (2.6). (A.13) 150 In the following, I will use the stationary phase method to evaluate the inte-gral given by equation(A.13). A similiar problem was solved by Brekhovskikh and Lysanov [57] who used the stationary phase method to evaluate a reflected sound field due to a point source (see pp. 76 to 79 of Ref. [57]). Now let me follow Brekhovskikh's approach to estimate the underice sound field caused by the radia-tion of sound from a monopole. The zero order Bessel function Jo(r}R) can be expressed by a sum of zero order Hankel functions of 1st and 2nd kinds (HQ\T}R) and HQ2\TJR)): MlR) = \(Hl%R) + Hi%R)). Using this relationship and an identity that H^(VRe-n = -Hl%R), I can rewrite equation (A. 13) as follows: $ 2 = f \u00C2\u00B0 S^e^-^H^irjR^drj. (A.14) J\u00E2\u0080\u0094oo For far-field cases, nR >\u00E2\u0080\u00A2 1, and B\u00C2\u00ABW,s\u00C2\u00AB\u00C2\u00AB(,ii)S^-,,(H_L+...). Substitution of this asymptotic expression into (A.14) leads to di, (A.15) where (n) = r,R + px(H -h) + (32(Z - H), which is the phase built up by the propagation of sound (along the direct path) from the source (at R = 0, Z = h) to the observation point (i?, Z) in the water. Since R >\u00E2\u0080\u00A2 A (wavelength), 4>(rj) is a rapidly varying function of rj. Therefore, the principal value of the integral given by equation (A.15) can be estimated by means of the stationary phase method. Note: here I am only considering the contribution of the monpole 151 source to the underwater sound through the direct path sound propagation, i. e. principal radiation. Sound reflected at the ice-air interface will of course contribute to the underwater sound field, but the high attenuation coefficient of sound in the ice makes these higher modes contribute relatively less to the underwater sound than the principal radiation. A full solution of the problem can be obtained by summing the contribution from all the possible modes as illustrated by Ewing et al (pp. 131 to 142 of Ref. [33]). In terms of r), {n) can be expressed as (n) = rjR + y/k\ - rj*(H - h) + Jkj - n\Z - H). (A.16) The stationary phase TJ0 is the root of the following equation: d4\no = R + - ^ = ( H - fc)L + -7==(Z - H)\m. (A.17) From the geometry given in Figure 2.12, it follows that \u00E2\u0080\u009E H-h Z-H R = + . tan orj tan a2 Substitution of this relationship into equation (A.17) leads to rjo = ki cos ai = k2 cos a2. (A.18) Therefore, the sound path giving the stationary phase is the one obeying Snell's law. Expanding {n) around TJ = T / 0 , one has m * * M + \4>\"(m)(v - vo)2. (A.19) Substituting (A.19) into (A.15) yields V 2TR J-OO Bearing in mind that H < X (r}0) w k2 cos a2R + k2 sin a2(Z \u00E2\u0080\u0094 H) \u00C2\u00AB k2r, (A.21) 152 where r is the slant distance between the source and receiver (see Figure 2.12). The second derivative of {ri) at TJ0 is \"(Vo)\u00C2\u00AB-, r 2 . (A.22) K 2 sin a 2 Using the value of the integral /OO j e~s ds = y/%, -oo and substituting equations (A.21) and (A.22) into (A.20) one obtains *a(r, t ) \u00C2\u00AB M\u00E2\u0084\u00A2\u00C2\u00A3fel e*r-^ | ( A . 2 3 ) which is identical to equation (2.8) and is a compact form of the principal integral value for equation (A.13). Appendix B Acoustical Response of an Ice Cover to Loading Failure and Tensile Failure In the following, I will provide detailed derivations of the 2 theoretical models used in chapter 3. B.l Ice Cover Subject to a Point Normal Force In this section, I will show the acoustical response of an ice plate driven by a point normal force at its upper surface as illustrated in Figure 3.23. My goal is to derive the corresponding sound field beneath the ice plate. First, let me briefly describe the main results based on the Steepest Descent Method which will be used in the subsequent derivation. According to Brekhovskikh (see pp. 234 to 237 of Ref. [42]), the path integral 1 = J e\"fMF(()d( ( B . l ) can be evaluated by means of the steepest descent method provided the integrand function possesses some properties. In the above integral, f(() and F(() are analyt-153 154 ical functions of the complex variable \u00C2\u00A3; TJ is a parameter; T is the path of integration in the ( plane. / ( \u00C2\u00A3 ) can be separated into its real and imaginary parts and rewritten as /(C) = /i(C) + tfa(C). If t%r,i2^ oscillates rapidly with ( and e^^i^C) in the integrand fall off rapidly with distance from a certain point, say \u00C2\u00A3 0, then the integration in (B.l) is given by / = c\">W>>./?[*(0) + (-^-)$\"(0) + ...], (B.2) U Tj 47/ where C o is the saddle point (the so called certain point) and is found from the equation | = 0. (B.3) At this point, fi takes its maximum and e71*^) falls off rapidly as ( departs from (0 for large n. $(v) is defined by v) = no% (B.4) with v being a real variable (\u00E2\u0080\u0094oo < v < oo) which specifies the most convenient path for the evaluation of the integration. Therefore, the original integration path T can be changed into the so-called saddle path for the integrand function without affecting the final result. The saddle path is given by / ( C ) = / ( C o ) - v\ (B.5) which passes the saddle point (y \u00E2\u0080\u0094 0). Along this path, / 2 ( C ) = / 2 ( C o ) -Now I consider the acoustical radiation from an ice plate which is subject to a time-harmonic normal point force fn (see Figure 3.23). Lyamshev's reciprocity theory [43] states that if the scattering field Pa of a point source, placed at some point of the space surrounding a plate, is known , then the sound radiation Pr of the plate subject to a normal force / \" is uniquely determined at the same point. Mathematically, 155 where n is the outward normal of the elementary area ds; / \" is a force applied normally to ds. The scattering field Pa due to a point source placed at (7*2,22) is given by equation (3.31). In the plate vibration problem (indicated by Figure 3.23), the surface S is in the xy plane where z = 0 and the normal n is in the direction of the \u00E2\u0080\u0094z axis. Thus, application of equation (B.6) to this problem leads to P ^ = J^QJL where the forcing term is given by equation (3.37). Substituting (3.37) into the above equation, one can easily obtain Pr(r2,z2) = p 2 k 2 k o F n f HJ)1)(k2r2sm0)B21(0)eik*z>cos9cos0osin0d0, (B.8) onpo Jr where HQ1^ is the zero order Hankel function of the first kind, and T represents the saddle path obtained from equation (B.5). For large distance r and radiation angle 0 not equal to 0, the Hankel function can be expressed in terms of its asymptotic form: H^(k2r2 sm9) \u00C2\u00AB J 2 . e ' ^ 8 i n g - / 4 > ( l + 1 . +\u00E2\u0080\u00A2\u00E2\u0080\u00A2\u00E2\u0080\u00A2). (B.9) V7rK2r2SJ.n0 8iK 2 r 2 sin0 From the geometry given in Figure 3.23, it is found that z2 \u00E2\u0080\u0094 H = Rcos02, r2 \u00E2\u0080\u0094 Rsind2 and R ^ > (H,a). These relationships lead to z2 cos e + r2 sin0 w Rcos02 cos 0 + Rsin02 sin0 = Rcos(0 \u00E2\u0080\u0094 02). (B.10) Substitution of (B.9) and (B.10) into equation (B.8) yields 8wp0 V \u00E2\u0084\u00A22r2 Jr Comparison of (B.ll) with (B.l) leads to ( = 0, rj = k2R and p2k2k0Fn cos floe-*'*/4 f~2 f(0) = icos(0-02). (B.13) 156 It follows from (B.l3) that the saddle point (0 = 02, and Hence $(0) = V2e-^F(92). (B.15) Substituting the above relationships into equation (B.2), one obtains = -i , 2 fcoF\"co S ^ 0 B 2 1 (g 2 ) e , f c 2 f l 4irpQR With 40 ) = PoCo/ cos 0O and = p2C2/ cos 02, the above equation can be rewrit-ten as pn, \ p n / D f l X -Z^ k2F COS 02B21(02) i k 7 R Pr(r2,z2) = Pr(R,02) = -i-^ \u00E2\u0080\u0094 e , which is equation (3.39). 157 B.2 Ice Cover Subject to a Pair of Horizontal Point Forces In this section, I will apply the above approach to a more complicated case i. e. the acoustical response of an ice plate driven by a pair of point forces tearing a small cylindrical cavity in the upper surface of an ice plate as illustrated in Figure 3.22. The motivation for searching for an analytical solution to this problem, seemingly very artificial, is to establish a Green's function for thermal cracking sound radiation, which may be useful for the study of sound radiation from a real thermal crack at the ice surface. Here, I emphasize that due to the difficulty in obtaining an analytical solution to a 3-D problem I am going to restrict my solution in a 2-D plane. In other words, I ignore the distortion effect of the cavity on the spherical wave Pq (given by (3.30) emitted from the imaginary point source at observation point M(r2,z2), and keep the scattering field Ps in the same form as that shown by equation (3.31). While the vertical radiation pattern of the field can be obtained relatively easily from the final result, the azimuthal radiation pattern cannot be derived from this model due to the lack of azimuthal dependence of Ps in (3.31). Turning to the original problem, I assume that a pair of point forces are applied normally to the inner surface of a shallow cylindrical cavity (see Figure 3.22). The forcing term in equation (B.6) is given by fr(, z) in equation (3.33) and the normal to the inner surface is opposite to the radial direction, i. e. \u00E2\u0080\u0094 er. Taking all these into account, one has from (B.6) that which is given by equation (3.32). With (3.31) and (3.33) inserted into the above iup0Q 1 dPs(r,z,r2,z2) \u00E2\u0080\u0094dr \r=afr(,z)addz, equation, and dH?>(y) dy V\y) 158 leads to equation (3.34). That is, Pr(r2,z2)= ~F[P2k2 f H^\k2{r2-a)s\n9)B2l{9)eik^cos9k2sin29d9. 4z/Oo Jr For large distance r and radiation angle 9 not equal to 0, the first order Hankel function of the first kind B.[1\k2r2) can be expressed in its asymptotic form Hi1){k2(r2-a)sm9) \u00C2\u00AB J , , \ . A^-*)*^-3-?) ]j irk2(r2 \u00E2\u0080\u0094 a)sm9 (1 \u00E2\u0080\u0094 -ZT7-,\u00E2\u0080\u00943 . . \u00E2\u0080\u009E+\u00E2\u0080\u00A2\u00E2\u0080\u00A2\u00E2\u0080\u00A2), (B.17) v 8ik2(r2 -a)sin0 \" v ' and equation(3.34) becomes Pr(r2,z2) \u00C2\u00AB - F [ P ^ [ [ Z 2 (,fc2(r2-a)ging+,fc2,2COs9)-^ 4ipo Jr V 7rk2{r2 \u00E2\u0080\u0094 a)sm9 B21(9)k2 sin2 9d9. (B.18) With the geometrical relationships indicated by Figure 3.22, the above equation can be rewritten as Pr(r2, zi) \u00C2\u00AB I \l , <2 e^Rco<9-9^ B21(9)k2 sin3/2 9d9. (B.19) \ipQ Jr V nk2(r2 - a) Comparison of the above with equation (B.l) leads to results similar to that for the vertical forcing case discussed in the previous section, except that F(9) in (B.2) is given by HO) = Zl^\lnn^-\u00E2\u0080\u0094^ s i n 3 / 2 OBnWe-'*. (B.20) The saddle point (Q in (B.2) remains the same as that obtained in the previous section, i. e. \u00C2\u00A3 0 = 92. Substituting (B.20) into (B.2), one obtains equation (3.35). That is Pott As mentioned in chapter 3, a real crack due to thermal tension is formed on the ice surface down to certain depth in the ice. Acoustically, the source produced by thermal cracking is physically located at the upper surface of the ice but in the ice interior. In order to use the theoretical model to simulate acoustical radiation from 159 such a source, it is necessary not only to close the cavity shown in Figure 3.22 (i. e. a \u00E2\u0080\u0094* 0), but also to let po,Co,0o \u00E2\u0080\u0094* p\,C\,9\ as well. This leads equation (3.35) to piR corresponding transmission coefficient B2i is obtained through equation (3.24) ZQ\u00C2\u00B0^ replaced by ZJf\ Thus, The with $ B ' l i h ) \" z,z. + 4zf J'+Sr + ^ ' X z . + z.)' ( B ' 2 2 ) where Z^ = p\C\j cos#i and C\ is the longitudinal wave speed in the ice plate; cos 0i is related to cos 02 by Snell's law: sin 0i sin 02 Ci C2 Taking C\ ~ 2C2 which is true for many applications, Snell's law yields: sin 02 = i sin 0X. (B.23) Using the thin plate approximation, \ZS\ ^> \Za\, and considering the case in which \Z,\ > |42)|, I can simplify B2l{02) to 2 7 ( i ) Za + Z^ + Z^ Substituting the above into equation (B.21), one obtains P (R OA ~ ^ r f c 2 s i n 0 2 2 ^ i k 2 R rr{K,\u00C2\u00BB2)~ R ZaCOS01 + piC1 + p2C2(cos01/coS02)e \u00E2\u0080\u00A2 { a ' Z b ) By equation (B.23), it follows that cos 0! cos 9t cos 02 ^ / l - i s i n 2 ^ Therefore, equation (B.25) can be further simplified, and with p\ ~ p2, it follows P r ( R , h ) \u00E2\u0080\u009E J \u00C2\u00A3 ^ \u00C2\u00B1 i ,, K Za{u)yJ\ - 4 sin2 02 + 2p2C2 which is equation (3.41), a final form of the Green's function for horizontal forcing. Bibliography [1] N. Untersteiner, The Geophysics of Sea Ice: Overview, N. Untersteiner , ed. , NATO ASI Series, Plenum Press, New York, pp.1-8, 1986. [2] D. M. Farmer and Yunbo Xie, The Sound Generated by Propagating Cracks in Sea Ice, J. Acoust. Soc. Am. , Vol 85 No. 4, pp.1489-1500, 1989. 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