Sound generation by ice floe rubbing

Sound generation by ice floe rubbing Zhen Ye a) Institute of Ocean Sciences, PO. Box 6000, Sidney, B.C. V8L 4B2, Canada

(Received 14 April 1994; accepted for publication 13 December 1994)

The acoustic signatures of ice breaking and ice floe interaction have been observed previously by in situ experiments [e.g., Xie and Farmer, J. Acoust. Soc. Am. 91, 1423-1428 (1992)]. It was conjectured that the rubbing of adjacent ice floes in the later phase of an ice breaking event can excite $H waves in the ice that may dominate the response, resulting in the pure tone that is observed in the water. Motivated by these experimental observations, in this paper it is shown that the $H waves are in fact excited by the interaction of rough ice surfaces. In a simple model the characteristics of the sound field in response to the ice floe rubbing are derived. How the roughness of the interacting surfaces affects the observed sound field is discussed. It is also shown that the tonal nature may be due to the resonance of the $H waves within the ice floe. The effects of the rubbing speed are also discussed.

PACS numbers: 43.30.Ma, 43.30.Nb

INTRODUCTION

Acoustic ambient noise in the Arctic Ocean below ice 1-3

floes has been of interest to scientists for many years. Some of the acoustic noise is produced by ice breaking events. In the experiments carried out by Xie and Farmer, 3 three phases of sound generation by ice breaking events have been identified. In the concluding phase, they claimed that a pure tone (around frequency 778 Hz) has been observed for the ice floe that is of approximate 1 m thickness and that the pure tone can last for over 20 s. It was hypothesized in Ref. 3, that in this concluding phase the adjacent ice floes are relatively free to move along the new rupture. Consequently, the ice floes slide longitudinally and rub against each other. Horizontal shear waves can thus be triggered, resulting in the observed pure tone in the water. The reader may consult Ref. 3 for a more detailed description of an ice breaking event. The stimulated $H waves can usually be detected by a geo- phone placed in the ice floe. Moreover, since the $H waves can leak energy into the water through defects at the ice- water interface, the $H waves can also be indirectly detected by underwater hydrophones (Xie and Farmer, personal communication).

Motivated by these experimental observations, we shall further investigate the sound generation process due to ice floe rubbing interaction. We will follow the physical scenario described in Ref. 3, and establish a possible model for the ice rubbing. We shall then calculate the shear horizontal displacement field in the ice floe due to the ice floe rubbing based upon this simple model. We propose that the displacement field can be generated because of the random roughness of the contacting surfaces of the ice floes. This random process of the rough surface can be treated mathematically in analogy to that of the ocean surface. 4'5 Since a$H wave can transform into a compressional wave at the discontinuities or defects on the ice surface, the sound can be radiated into the water at these defects and may be recorded by hydrophones.

a)E-mail: [email protected]

The tonal feature of the sound field observed could be due to

the resonance of the SH waves through the ice floe thickness. At this stage, we are not able to determine exactly how the link takes place between the observed pressure in the water and the stress field in the ice. However, we may generically assume that the sound level recorded by the underwater hydrophones is related to the mean square values of the displacement field inside the ice. The physical significance of the sound generated by the ice-floe rubbing interaction can then be studied generally.

I. FORMULATION OF THE PROBLEM

Consider a two-dimensional rough surface for simplicity and brevity. This surface is pictured in Fig. 1. The random roughness of this surface can be described by a stochastic variable h(z,y) with zero mean such that

(h(z,y))=O.

The brackets denote the ensemble average over the domain of the stochastic variable. When two such random rough surfaces rub against each other, the shear stress thus produced is expected to consist of the stochastic components which have a functional dependence on h(z,y). We denote the stress as F(z,y,t). In general, this stress can be decomposed into two parts, the mean part and the fluctuating part:

F(y,z,t)=(F(t))+ AF(y,z,t). (1)

The mean part may be assumed to be spatially constant while the fluctuating part, AF, may have both spatial and temporal variability. Details concerning the surface deformation and shear force when two surfaces are rubbing against each other are beyond the scope of the present paper.

Now we consider two ice floes A and B as shown in Fig. 2, which have a thickness H. Ice floe A has a finite length L in the x direction and it spans to infinity along the y axis. We assume L >>H. The contracting surfaces are assumed to be rough. Ice floe B is envisaged to rub continuously against floe A. Consequently, shear deformation occurs along the y

2191 J. Acoust. Soc. Am. 97 (4), April 1995 0001-4966/95/97(4)/2191/8/$6.00 ¸ 1995 Acoustical Society of America 2191

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h,•, z) Y

(a)

(b)

FIG. 1. Illustration of a rough surface: (a) view from z direction and (b) view from y direction.

direction, stimulating SH waves in the ice plate. The wave equation for the SH wave has been given in Ref. 6:

•--•- Cs2V 2 u(x,y,z,t) = 0, (2) where u is the ice particle displacement field and Cs=(tx/Pice) m is the shear wave speed. The constants/x and /)ice are the shear modulus and mass density of the ice, re- spectively. The shear stress is assumed to vanish at the ice- water and ice-air interfaces and the corresponding boundary

6 conditions are,

09// =o. (3)

8z z=O,H

We furthermore assume that the end of the ice plate A at x-L is also stress free, namely

8u =0. (4)

8x x=L

Here we have assumed that the ice-water and ic•'--air inter-

faces are flat and cannot support shear stress. At the contacting surface, ice floe A is subject to the frictional coupling with ice floe B and the boundary condition is,

tx =F(y,z,t). (5) x=0

It is noted that stress F may be a function of z,y and time t. [Strictly speaking, the function F(y,z,t) may also be a function of the displacement field u, which will make the prob-

Air

lem more complicated. We will come back to this point later.] Equations (2), (3), (4), and (5) present a well-defined boundary problem that can be solved by standard methods. For brevity, we need only consider the displacement field in ice floe A.

We note that the original boundary conditions presented in Eqs. (3) and (5) of Ref. 3 are incorrectly expressed. Spe- cifically, on the surfaces x=0, and L the stress should be proportional to the derivatives of the displacement field with respect to the variable x instead of z. The reader may refer to Ref. 6 for details. We would also like to point out that a spatially constant stress on the rubbing surface, as proposed in the same paper, cannot yield standing waves in the z direction. To illustrate this point, we follow the procedures described in Ref. 3 and modify the boundary conditions (3) and (5) in that paper according to what has been described above. A general solution to Eqs. (1), (2), and the modified (3) in Ref. 3, can be written as

U = • A n COS[ k 1 n (x -- L ) ] CoS(kn Z), n

where

k n: n rr/H, k•n = x/(tO/Cs) 2- kn 2,

(6)

n=0,1,....

When substituting this solution into the modified Eq. (5) in Ref. 3, we obtain

1, n =0,

gn- •n,0 = 0, /1=/=0, (7) by using f•rdz cos(knz)=HSn, O. This means that a constant stress at x-0 surface can only stimulate the lowest mode n=0. Therefore the observed tonal feature cannot be ex-

plained by the constant stress hypothesis.

II. SOLUTIONS

The above boundary problem could be solved by using Fourier transformations. However, since in the present case we are dealing with a problem with stochastic boundary conditions, the normal Fourier transformation

F(t) = f f(to)e -i•øt dto (8) should be replaced by the so called Stieltjes Fourier integral

F(t) = e -i•øt dr(to) (9)

to avoid the divergence for a stationary random function. The function dr(to) is called the Stieltjes Fourier component. The reader may refer to Appendix A in Ref. 7 for more details concerning the Stieltjes integral. Equation (9) can also be rewritten as

f -itot d•(•) F(t)= e do, (10)

y z water which bears resemblance to the form of the normal notation.

FIG. 2. Coordinates for two rubbing ice floes.

So in the present paper, we will use the normal notation for the Fourier transformation without confusion, i.e., Eq. (8).

2192 d. Acoust. Soc. Am., Vol. 97, No. 4, April 1995 Zhen Ye: Sound generation by ice floe rubbing 2192


The above discussion also holds true for spatial Fourier transformations.

We use the following transformation'

u(x,y,z,t)= f do) dkl eiklYe-iø•t•(x,Z,k1,0)). (11) Substituting this equation into (2) we obtain,

[•72+(k2-k12)]•(x,Z,kl,0))=O (12)

with k=0)/Cs. The general solutions to Eqs. (12), (3), and (4) can be written as

•(X,Z,kl ,0)): E gn(kl ,0))cøS[kln(X-Z )]cøs(knz), n=0

(13)

where

kn=nrc/H, n =0,1,2,3,..., (14) and

kln: •k 2- kl 2- kn 2. (15) At boundary x = 0, we need to solve the following equa-

tion:

tx =F(y,z,t), x=0

which yields,

E An(kl,0))kln sin(klnL)cøs(kn z) n=0

- (2rr)• dt dy F(y,z,t)e -ikiy+iø•t. (16) We solve Eq. (16) as

2 1 1

An(k1 '0))= H[1 + •n,o]kln sin(kin L) tx (2rr) 2

X dt dy dz e -ikly+iwt

X F(y ,z, t) cos(knz). (17)

We note that the coefficients A n are functions of the stochastic stress. The ensemble average of A n is

(gn(kl ,•))= •kln sin(kin L) 8n,08(k1), (18) where we have used the following notation,

(F(w)) = • dt(F(t))e i•t (19) by assuming the averaged stress has no spatial dependence. Due to the delta function 8n,0, only n =0 mode contributes to the averaged displacement field. In the above we have made use of the following relations,

az COS(nZ)COS(mZ) = ( 1 + 8n,O) 8n,m,

o•rdz cos( knz) = H rSn,O , where n,m = 0,1,2... and

1, m=n,

8mn= O, m•n. We define a spatial correlation function

d(x,Z,k1,0)1 ;x',z',k2,0)2)

=(t•(x,z,k 1,0)l)t•*(xt,zt,k2,0)2)) (20)

and a spatial-temporal correlation function

D(x,y,z,t;x',y',z' t')=(u(x,y,z,t)u*(x',y' ' ' , ,z ,t )),

where the average is taken over the surface roughness ensemble and "*" denotes complex conjugate. In order to evaluate this correlation function, we need to calculate the correlation function (An(kl,0)•)A;(k2,0)2)). We note that

(An(k1,0)1)A r•(k2,0)2)):h(An(kl,0)l))(A;(k2,0)2)).

From Eq. (17), direct computation leads to

(A n( k l , 0)l)Ar•(k2,0)2))

4 1 1

=H2(1 + •n,0)( 1 n t- •m,0) kin sin(kin L) k2m sin(k2m L)

X (27r)• /.•2 dtl dyl dt2 dy2 dZl dz2 X e iø•ltl-iklyl-iø•2t2+ik2y2 cos(knZl)COS(kmz2)

X (F(yl ,Zl ,tl)F(y2,z2,t2)), in which

kln- •( 0)l /Cs)2- k12- kn 2, m qT

kin= H ' k,,= H ,

(22)

(23) k2m= •/( 0)2 /Cs) 2- k2 2- k2m .

If the correlation quantity (F(yl,Zl,tl)F(y2,z2,t2)) is known, the correlation function

<mn( kl , 0)1)A ;( k2, 0)2) >

can be calculated from Eq. (22) and the spatial and spatial- temporal correlation functions can be calculated from

d(x,Z,k1,0)1 ;x',z',k2,0)2)

: E <An(k1,0)l)A•(k2,0)2))cøS[kln( X-L)] m,tl

X cos(k n Z ) COS[ k 2m ( X t -- L ) ] cos( k m Zt ),

and

D(x,y,z,t;x',y',z' t')

= f d0)1 dkl d0)2 dk2 d(x,Z,kl,0)l ;x' ,z ,k2,0)2)

X e -iø•lt+ikly+itø2tt-ik2yt.

(24)

(25)

2193 J. Acoust. Soc. Am., Vol. 97, No. 4, April 1995 Zhen Ye: Sound generation by ice floe rubbing 2193


Without specification, the summations over n (or m) take non-negative integers.

Therefore the key to solving the problem is to estimate the quantity

{F(Yl ,Zl ,tl)F(Y2,Z2,t2)}.

Usually this is not easy to achieve. However, if we take some reasonable assumptions, things can be greatly simplified and we may pick up some of the physical essence of this complicated problem. We recall Eq. (1): The stress on the surface can be decomposed into two terms, (a) the mean term can be assumed as a constant, and (b) the fluctuating term has spatial and temporal features. Therefore we have in general that,

{F (y 1 ,z 1 ,t l )F(y 2 ,z2 ,t2))

-(F(tl))(F(t2))+(AF(yl,Zl,tl)AF(Y2,Zl,t2)). (26) Next we assume the fluctuation of the stress function due to

the other surface is spatially uncorrelated on the surface, i.e., the local correlation approximation,

(AF(y 1 ,zl ,tl)AF(y2,z2,t2)}

= (5(Yl-Y2)rS(zl-z2)B(tl ,t2). (27)

This is often a good approximation for a stochastic process, but this approximation could be improved by using the following relation,

(AF(yl ,zl ,tl)AF(y2,z2,t2)}=C(yl ,zl ;y2,z2)B(tl ,t2).

In the present paper, we adopt the local correlation approximation. The reader may refer to, for example, Chap. 6 in Ref. 7 (p. 120) for more details about the local correlation approximation. The temporal function B(tl,t2) may be approximated as B(tl-t2) to describe the stationary random process (steady-state approximation). In actuality, it would be more appropriate to assume that the process is a random process with stationary increments, especially if we want to study the effects of the change of the rubbing speed with time. Nevertheless, the physical significance will not be lost by taking the steady-state approximation, as we shall see in the final section. The discussion of the fundamental concepts of the theory of random process can be found in Chap. 1 in Ref. 8 or Appendix A in Ref. 7. Moreover, we assume the temporal spectral intensity to be tI)(w) for the temporal correlation B(t 1 - t2) , i.e.,

B(tl-t2) = f do (I)(0))e -ita(tl-t2) We require B to be real, which implies •(w)=•*(-w).

After taking the above assumptions, we can easily calculate

(2) (gn(k 1 0)l)g*(k2 0)2)}rag (1) (k 1 0)1'k2 0)2)+gn,m , rn , n,m , , ,

X(kl,0)l ;k2,0)2) (28)

with

A (1) (k Wl'k 2 0)2) n,m 1, , ,

(m(0)l))(m(0)2)) 1 1 11, 2 kin sin(klnL ) k2m sin(k2mL )

X rSn,OrSm,o6(kl) 6(k2), (29)

and

A (2) (k 0)1'k2 0)2) n,m 1, , ,

2•(0)1) 1 1 1

=/.z2H(1 + r$n,O) kin sin(klnL ) k2m sin(k2mL ) 2rr X r$(kl- k2) r$(0) 1 - 0)2)(Sn,m.

In the above we have, as before,

(30)

n?/' m$r

kn H km H kin: 4(0)1/Cs) 2-- kl 2- kn 2,

k2m-- X/( 09 2/Cs) 2-- k2 2- k2m .

The spatial correlation function is thus calculated as

d(x,z,k1,0)1 ;x',z',k2,0)2)

-=d(1)(x,z,k1,0)1 ;x',z',k2,0)2)

+ d(2)(x,Z,k1,0)1 ;x',z',k2,0)2), with

(31)

d(1)(x,Z,k1,0) 1 ;x',z',k2,0) 2)

= Z (F(0)l))(F(0)2)) 1 tl 2 kin sin(klnL) k2m sin(k2m L) n,m

X (Sn,O(Sm,O(5(kl)(5(k2)cos[kln(X-L)]

X cos(knz)Cos [ k2m(X t -L ) ]cos(kmz'), (32)

and

d(2)(x,Z,k1,0) 1 :x',z',k2,w 2)

2•(0)1) 1 1 •(kl-k2) =• 112H( 1+ 6,,,o) [kln sin(klnL)]22rr n

X (5( 0)1-- 0)2)cøS[kln(X-L )]cøs(knz)

X cos[ k2n(X t -L ) ]cos(knzt). (33)

The spatial-temporal correlation D(x,y,z,t;x',y',z',t') is then given by

D(x,y,z,t;x',y ',z',t' ) --D (1)(x,y,z,t;x',y ',z',t' )

+ D(2)(x,y,z,t;x',y ',z',t' ) (34)

with

D(1)(x,y ,z,t;x',y ',z',t' )

f d0) e-i•ø, t (m(0)l)) ( 0)1/½s) sin[ ( 0)1/½s)L ]

COS[ (0)1/½s)( X--L ) ]

X { f d0)2 e-iø"2t' {F(0)2)) ( 0)2 Ic s) sin[ ( 0)2/c s)L ]



x The quantity D(2)(x,y,z,t;x',y',z',t ') is given as,

D (2)(x,y ,z,t;x',y ',z' ,t' )

= dw dk]eit•(Y-Y')-iø•(t-t')•] /z2HTr (1 q- •n,O) n

(35)

1

X [kln sin(k]nL)]2 cøs[k]n(x-L)]cøs(knz) X COS[ k ln(X t - L ) ]cos(knzt). (36)

This equation represents the field correlation function due to the stress fluctuation that is caused by the random roughness of the rubbing surfaces.

We are particularly interested in the sound pressure level in the water. As pointed out in the introduction, the pressure level in the water is assumed to be related to the mean square SH wave displacement field inside the ice, and may be generally expressed as

(p2(t)) = IID(x,y,z,t;x,y,z,t), (37)

where II is assumed to be a transformation operator that converts the SH wave to the sound (pressure) field in the water and x,y,z are the coordinates for the defects on the ice-water interface. The operator II which may contain the spatial derivative operator actually depends on both the po- sition of the recording hydrophone and the coordinates of the defects. However, the coordinates for the hydrophone have not been written out explicitly.

We calculate

(p2(t))__p•2(t) +p22(t) (38) with

p2•(t)=II[ f dw e-iø•t (F(w)} X (W/Cs)Sin[(W/cs)L ] cos •(x-L) (39)

and

f cI) (w) H(k] ,w) p22(t) = dw dk•] /•2H(l+,•n,0)rr [k•n sin(k•nL)] 2 n

X {cos[kln(X-Z )]cos(knz)} 2. (40)

Clearly p• is caused by the mean stress and P2 is caused by the stress fluctuation. The sound field spectrum in the water due to the stress fluctuation is thus obtained as

• cI:, (•) II (/• ,•) p22(m)=•] dk, /z2H(1 + 8n,O)rr [k,n sin(k,nL)] 2 X cos2[k•n(x-L)]cos2(knz). (41)

This equation describes the relation between the possible sound field spectrum in the water and the roughness of the rubbing surface. We mention that Eq. (41) may reduce to a

sum of the residues at poles of the integrand once II(k• ,w) is known. The poles are determined by the roots of the equation

sin(k]n L )=O,

and are

k• =lrr/L /=0 1 2 n • • • • ....

We will not pursue the details of this point here because we are only concerned with the qualitative features of the model. The interested reader may refer to p. 96 in Ref. 5. In the above we have

k = W/Cs, k n = n rr/H,

and

III. DISCUSSIONS AND CONCLUDING REMARKS

Let us first consider the p•2 term in Eq. (38). From the derivation, we know that this term describes the sound generated by the spatially constant rubbing between two ice floes and corresponds to the model study presented in Ref. 3. Since it does not have discrete modes, no tonal features exist in this term. However the temporal variability in the averaged stress field could give different weights to different wave modes (Xie, private communication). We have to note that if the mean stress does have spatial variability, then it could excite the normal modes. We are not interested in p• because it does not incorporate the dynamic parameters such as the rubbing speed.

We now consider the second term in Eq. (38). This term is due to the roughness of two rubbing surfaces. This roughness results in the fluctuation of the surface stress. From Eq. (41), we can calculate the cutoff frequencies from

(W/Cs) 2- k• 2- (n rr/H) 2 >• 0, (42)

where we have assumed that L >>H. Since k•2>•0, the cutoff frequencies are determined by

fn•>ncs/H, n=0,1,2,3 .... (43)

Here we make use of k = 2 rrf/Cs. More detailed discussions on the cutoff frequencies can be found in Ref. 9. Near the cutoff frequencies, resonance may occur because of the fac- tor sin(k•n L) in the denominator. If such is the case, we could observe the discrete bands as implied in Eq. (43). In the experiment carried out in Ref. 3, the second resonance frequency (n= 1) has been observed, i.e., 778 Hz. Xie and Farmer used this frequency to infer the shear wave speed to be 1556 m/s by taking H = 1 m. This result seems agreeable with the measurements described in Ref. 10. In contrast to

the model study in Ref. 3, the present model predicts the existence of the even modes (n = 2,4,...) of the SH wave. A further experimental investigation may be worthwhile. In our model study we also predict the fundamental SH wave with n--0. However, in Ref. 3, this fundamental wave does not seem to appear. One possible reason may be stated as fol- lows. For the n-0 wave, the displacement field in the ice will not depend on the vertical variable z and thus the sound



cannot be radiated into the water if the transformation operator II contains a derivative with respect to z. However we notice that the fundamental mode (corresponding to n =0) seems to appear in a later experiment. • This discrepancy needs further investigation. Here we note that the above features do not depend on the detailed information about how the sound is radiated into the water. It should also be noted

that since Eq. (41) carries an integration, the discrete bands at the cutoff frequencies may become broad bandwidths as in waveguide problems. 12 Further studies are desirable when more information becomes available about how the sound is

exactly radiated into the water. We now study in more detail the sound generation due

to the surface roughness. From Eq. (41), we see that the sound field generated by the rubbing depends on the temporal intensity cI)(w). We now discuss the temporal function B in detail. As said at the beginning of the paper, the distribution function of the roughness (in a form of the fluctuations in the height, referring to Fig. 1) at any point on the rubbing surface can be generally written as P(h) and its Fourier form is

P(h) = f dk •(k)e -ikh, (44) where we assume that the roughness is homogeneous on the surface. We heuristically assume Taylor's frozen-in hypothesis to hold (referring to Fig. 3 and p. 389 in Ref. 7),

B(t)=F[P(Vt)].

"F" denotes the functional of P(. ). In the linear approximation, we may have

B(t)=aP(Vt), (45)

where a is a coupling constant and V is the rubbing speed (cf. Figs. 2 and 3). For simplicity we assume speed V to be constant. In deriving Eq. (45), we have conjectured that the stress temporal correlation function B(t) is related to the height distribution function P(h). This hypothesis implies that the stress temporal correlation at time delay t is proportional to the probability of the height h reached in this time lag, i.e., h = Vt. The validation of this ad hoc hypothesis is subject to future experimental verification. From the qualitative results shown below, the present ad hoc assumption does lead to some insight to the experimental observations.

From Eq. (45), we have

(I) ( to ) = ( 1/ V ) a •( to / V ) . (46)

If we use a Gaussian model for the distribution function

P(h) (refer to p. 336 in Ref. 7 and pp. 19-20 in Ref. 5),

P(h ) = x/•e -ah2 (47) with

a=l/2(h2).

The Fourier components of this model function are

•7)( k ) -- e - k2/( 4a ). (48) So we find

07

•' 06

ø10 10 -8 -6 -4 -2 0 2 4 6 8 h

FIG. 3. Distribution function for the roughness.

(I)( w) = a( 1/V)e -'ø2/4av2 (49) ß

From this relation, we can qualitatively study some interesting features of sound generated by ice rubbing. We specify the rubbing speed V, and plot (I)(w)V vs w2/(4aV 2) in Fig. 4(a). In Fig. 4 we see that at a given speed, the spectral intensity decreases with increasing frequency. Therefore at the low rubbing speed, the low-frequency components dominate the sound field. This is in accordance with the third

phase observation depicted in Fig. 3 in Ref. 3. In this phase the pure tone at 778 Hz is observed and not much higher frequency sound has been observed. When the rubbing (or other movement) speed increases, the contribution from higher frequency components may be important. We plot (I)(oo)V vs V2/(w2/4a) in Fig. 4(b), which clearly shows that the contribution from a given frequency component becomes increasingly important with an increase of speed V. This may explain why broadband sound pulses are observed in the earlier phases of sound generation: 3 Actually, at the initial stages of an ice-breaking event, the movement of the ice fault is relatively large. Therefore high-frequency sounds can be excited and make the contributions, complying with the observations in Fig. 3 in Ref. 3. Furthermore, we plot cI)(w) vs V for three different frequencies in Fig. 4(c). In this plot, we see clearly that at the low rubbing speed, the lower frequency components are dominant. With increasing speed, the different frequency components become equally important which should lead to a broadband sound observation.

From Fig. 4(c), we consider two limits. At the low rubbing speed limit, the spectral intensity is proportional to V. There- fore the sound level approaches zero linearly with decreasing speed in the present Gaussian model. At the high speed limit, the spectral intensity becomes more or less frequency inde- pendent and all the frequencies have the same weight of contribution. Another interesting feature can be inferred from our model: With increasing variance (h 2) of the distribution function P(h), the low-frequency components become more important. This seems reasonable, as a larger variance implies that the distribution function is flatter, which favors the



07

0 6 7 8 9 (a) o •/(4.V •)

O7

-•- 06

05

04

03

02

01

o ø 7

I I I I

1 2 3 4 5 6 8 9

10

9 10

i i i i I i i

i i i i

i 2 3 4, 5 6 7

FIG. 4. Plots of Eq. (49): (a) The spectral intensity •(w) V is plotted against w2/(4a V 2) at the given speed V; (b) the spectral intensity •(w)V is plotted against V2/(w2/4a) at a given frequency; (c) the spectral intensity •(w) is plotted against V/(w2/4a) 1/2 at three different frequencies: w, 1.4w, 3.2w. The x,y axes are in arbitrary units. In these plots we take a= 1.

long wave components. Here we need to note that in actuality, we have to include the attenuation of the sound, which will make the higher frequency components weaker than expected from the present theory.

In our above discussions, we assumed that the stress spatial correlation function is related to the distribution function, as in Eq. (45). Now we show a more general case. Suppose that the stress on a rough surface due to a flat surface has the following spatial correlation,

r(y)= x/ax/•e -ay2 a= 1/2(y 2}

where a is a variance parameter and y is the distaffbe between two points on the rough surface in the y direction. Then the Taylor frozen-in hypothesis states that the temporal correlation on a rough surface could be approximated, as in Eq. (45), by

B(t)=aT(Vt).

Particularly we have assumed that T(y)--P(h) in the earlier discussions.

In summary, this paper investigates the excitation of the horizontally polarized shear wave (SH wave) in an ice plate due to ice floe rubbing interaction. The sound radiated into the water is calculated as a function of the rubbing spectrum ß (w). Discrete cutoff frequencies may be observable if the rubbing spectrum consists of frequencies which are equal or close to one of the cutoff frequencies. We find that the higher the rubbing speed, the more modes can be stimulated, resulting in possible broadband sounds; while at the low rubbing speed, the low-frequency modes may dominate. The essen- tial idea that we communicate in this paper is that the roughness of two rubbing surfaces may play an important role in defining the wave excitation in the ice and the sound field in the water.

We have to admit some limitations to the present study: (1) We have assumed that the rubbing speed is constant in time, (2) we have also assumed that the rubbing stress does not depend on the displacement field, (3) sound attenuation effects have been ignored, and (4) ice-air and ice-water interfaces are assumed to be flat. If we have to consider that

the rubbing speed may change in time and the rubbing stress depends on the displacement field, the problem becomes nonlinear and very difficult to solve. The reader may refer to Ref. 13 for discussions on nonlinear oscillations. In addition, we have not considered other waves such as p waves and vertically polarized shear waves that can also be produced by a breaking event. It may be also worthwhile to pursue a more complicated case in which ice floe has finite length in both y and z direction and the thickness of ice floe is not uniform.

Nevertheless, the present preliminary study sheds some insight on to this complicated physical problem and we hope that our study can stimulate further theoretical and experimental studies.

Finally we point out that the present problem is analo- gous to that in earthquake mechanics. Recently, a theory has been put forward for modeling earthquake rupture as a stochastic process. TM The progress in geophysics may improve our understanding of the ice breaking event and ice-ice interaction and vice versa.



ACKNOWLEDGMENTS

The author thanks Dr. David M. Farmer, Dr. Y. B. Xie, and Dr. Li Ding for discussions and useful references. Craig McNeil at IOS is thanked for English corrections and useful comments. The author would also like to thank two anony- mous reviewers for valuable suggestions and constructive comments. Their comments help clarify the paper greatly. This work received support from the U.S. Office of Naval Research and the Department of Fisheries and Oceans, Canada.

11. Dyer, "Arctic ambient noise: ice source mechanics," Phys. Today 41, 5 and 6 (January 1988).

2 For example, M. J. Buckingham and C. F. Chen, "Acoustic ambient noise in the arctic ocean below the marginal ice zone," in Sea Surface Sound, edited by B. R. Kerman (Kluwer Academic, Boston, 1988), pp. 583-598.

3y. B. Xie and D. M. Farmer, "The sound of ice break-up and floe inter- actioh," J. Acoust. Soc. Am. 91, 1423-1428 (1992); and references therein.

4j. A. Ogilvy, Theory of Wave Scattering from Random Rough Surface (Hilger, Bristol, 1991).

5 L. M. Brekhovskikh and Yu. Lysanov, Fundamentals of Ocean Acoustics (Springer-Verlag, Berlin, 1982).

6L. M. Brekhovskikh, Waves in Layered Media (Academic, New York, 1980).

7A. Ishimaru, Wave Propagation and Scattering in Random Media (Aca- demic, New York, 1978).

8V. I. Tatarskil, The Effects of the Turbulent Atmosphere on Wave Propa- gation (Israel Program for Scientific Translations, Jerusalem, 1971).

9G. V. Frisk, Ocean and Seabed Acoustics (Prentice-Hall, Englewood Cliffs, NJ, 1994).

løB. E. Miller, "Observation and inversion of seismo-acoustic waves in a complex arctic ice environment," M. Sc. thesis, MIT, Cambridge, MA (1990).

n y. Xie and D. Farmer, "Seismic-acoustic sensing of sea ice wave me- chanical properties," J. Geophys. Res. 99, C4, 7771-7786 (1994).

12 Z. Ye, "On acoustic propagation in exponential ocean-surface waveguides," submitted to J. Acoust. Soc. Am. (1994).

13p. Hagedom, Non-Linear Oscillations (Clarendon, Oxford, 1981). 14 Z. M. Yin and G. Ranalli, "Modelling of earthquake rupture as a stochas-

tic process and estimation of its distribution function from earthquake observation," preprint (1994); and references therein.



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Sound generation by ice floe rubbing