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CAN SEA WAVES CAUSE MICROSEISMS? By M. S. Longuet-Higgins Trinity College at Cambridge AbstractâThis paper is an exposition of the "wave interference" theory of microseisms. Simple proofs are given of the existence, in water waves, of second-order pressure fluctua- tions which are not attenuated with depth. Such pressure fluctuations in sea waves may be sufficiently large to cause microseisms. The necessary conditions are the interference of opposite groups of waves, such as may occur in cyclones or by the reflection of waves from a coast. IntroductionâIt has long been known that there is some connection between certain types of microseisms and deep atmospheric depres- sions over the ocean; and the similarity be- tween microseisms and sea waves â their periodic character and the increase of their amplitude during a "storm" â naturally sug- gests some causal relation between them. But until recently there have seemed to be many difficulties, both theoretical and observational, to supposing that sea waves could, by direct action on the sea bed, be the cause of all these microseisms; for the latter have been recorded while the corresponding sea waves were still in deep water, whereas theory seemed to show that the pressure fluctuations associated with water waves were quite insufficient, at such depths, to produce any appreciable movement of the ground. However, recent theoretical work in hydro- dynamics has altered this situation: Miche (1944), in quite another connection, discovered the existence, in a standing wave, of second order pressure variations which are not attenu- ated with the depth; a much shorter demon- stration of this result was given by Longuet-Higgins and Ursell (1948), and the result was extended by the present author (1960) to more general systems of waves. In the latter paper it was shown that such pres- sure variations may be quite sufficient, under certain circumstances, to produce the observed ground movement, the chief conditions re- quired being the interference of waves of the same wavelength, but not necessarily of the same amplitude, travelling in opposite direc- tions. This, then, may be called the "wave in- terference theory." In the latter paper (which will be referred to as I) the results on which the theory depends were derived in a general and concise form, with detailed proofs. In view of the interest of the subject it seems desirable to clarify the main ideas behind the theory and to discuss further some of the more unexpected results. This will be attempted in the present paper, in which we shall rely as far as possible on physi- cal reasoning, and refer where necessary to the former paper for rigorous proofs of the results quoted. We shall conclude with a brief histori- cal review of the theory. 1. The importance of the mean pressureâLet us suppose that seismic waves are to be genera- ted by some kind of oscillating pressure distri- bution acting on the surface of the earth or of the sea bed. If the period of the oscillation is T, and the corresponding wavelength of seismic waves is L, then the pressure distribution over an area whose diameter is small compared with L may be regarded as being applied at the same point, so far as the resulting disturbance is concerned; for the time-difference involved in applying any pressure at another point of the area would be small compared with T. Hence the resulting disturbance is of the same order of magnitude as if the mean pressure over the area were applied at the point. Now the wave- length of a seismic wave is many times that of a gravity-wave (sea wave) of the same period. It is therefore appropriate to consider the pro- perties of the mean pressure, over a large num- ber of wavelengths, in different kinds of gravity-wave. We shall first consider some very special but physically interesting cases, when the waves are perfectly periodic and the wave-train is infinite in length. It will be as- sumed for the moment that the water is incom- pressible. 2. The progressive waveâConsider any peri- odic, progressive disturbance which moves, un- changed in form, with velocity c (see figure 1). Let p (t)â¢ denote the mean pressure on a fixed horizontal plane (say the bottom) between two fixed points, A, B, separated by a wave length We may show that p (t) is a constant. Let A and B denote the points, separated from A and B respectively by a distance ct. Then since the motion progresses with velocity c the mean pressure over A'B1 at time t equals the mean pressure over A B at time O, i.e. p (0) ; 74
CAN SEA WAVES CAUSE MICROSEISMS 75 B Figure 1. Positions of the profile ofapro- gressive wave at two different times. the total force on A'BV is X p (O). But since the motion is periodic the force on A A' equals the force on B B'. Hence, by subtraction, the force on A B equals X p- (O) ; and the mean pressure on A B equals p (0) which is inde- pendent of the time. Thus there is no fluctua- tion in the mean pressure on the bottom over one wave-length, or over a- whole number of wavelengths; in any interval containing more than N wavelengths the fluctuation in the mean pressure is less than N"1 pmax where Pmax is the maximum pressure in the interval. In other words, in a progressive wave the contri- butions to the disturbance from different parts of the sea bed tend to cancel one another out. There is a second reason why progressive water waves may be expected to be relativly ineffective in producing seismic oscillations of the sea bed: not only the mean pressure fluctua- tion p, but also the pressure fluctuation p at each point decreases very rapidly with, depth and is very small below about one wavelength from the surface. This fact is closely con- nected with the vanishing of the mean pres- sure fluctuation; the motion below a certain horizontal plane can be regarded as being gen- erated by the pressure fluctuations in that plane; and hence we should expect that the contributions to the motion from the pressure in different parts of the plane would tend to cancel one another out. 3. The standing waveâConsider now a stand- ing wave, and let A and B be the points where two antinodal lines, a wavelength apart, meet the bottom (see figure 2). To a first approxi- mation, a standing wave can be regarded as the sum of two progressive waves of equal wavelength and amplitude travelling in oppo- site directions. Therefore the mean pressure on the bottom between A B vanishes to a first approximation. However, the summation of the waves is not exact; if two progressive mo- tions, each satisfying the boundary condition of constant pressure at the free surface, are added, (i.e. if the velocities at each point in space are added) there is no "free surface" in the resulting motion along which the pressure is always exactly constant; although if the ele- vations of the free surface are added in the usual way, the pressure is constant along this surface, to a first approximation. We should not expect the motions to be exactly super- posable, on account of the non-linearity of the equations of motion. It can be seen from the following simple argument that the mean pressure on the bot- tom, in a standing wave, must fluctuate. Con- sider the mass of water contained between the bottom, the free surface, and the two nodal planes shown in figure 2. Since there is no flow across the nodal planes, this mass consists always of the same particles; therefore the mo- tion of the center of gravity of this mass is that due to the external forces alone which act up- on it. Figure 2 shows the mass of water in four phases of the motion, separated by inter- vals of one quarter of a complete period. In the first and third phases the wave crests are fully formed, and in the second and fourth phases the surface is relatively flat (though never exactly flat; see Martin et al., 1952). When the crests are formed the centre of grav- ity of the mass is higher than when the sur- face is flat, since fluid has, on the whole, been transferred from below the mean surface level to above it. Thus the centre of gravity is raised and lowered twice in a complete cycle. But (a) (b) f f fc) Figure 2. Comparison of a standing wave with a swinging pendulum, at four different phases of the motion separated by a quarter of a p eriod.
76 SYMPOSIUM ON MICROSEISMS (0) (b) Figure 3. Two phases of the interference between two waves of equal length but dif- ferent amplitudes BJ and a9 travelling in opposite directions. The profile of the first wave (dashed line) is reduced to rest by superposing on the system a velocity -c; the second wave appears to travel over the first with velocity-2c. The full line shows the final wave form. the external forces acting on the mass are, first, that due to gravity, which is constant, (the total mass being constant) ; secondly the force from the atmosphere, which is also con- stant, since the pressure p0 at the free sur- face, if constant, will produce a constant down- wards force Xp0 ; thirdly the forces across the vertical planes, which must have zero vertical component, the motion being symmetrical about these planes; and, lastly, the force on the bottom, which equals X p. Since all the other external forces besides I p are constant it follows that p must fluctuate with the time. In figures 2 (a) and 2(c) the mass of water above the mean level is proportional to the wave amplitude a; since it is raised through a distance of the order of a, the displacement of the centre of gravity, and hence the mean pressure fluctuation, is proportional to a2. An explicit expression for p can easily be derived. Let z denote the vertical coordi- nate of a particular element of fluid of mass m, so that z is a function of the time t and of, say, the position of the fluid element when t = o. If F denotes the vertical component of the external forces acting on the mass of wa- ter, we have, on summing the equations of mo- tion for each element of fluid, and cancelling the internal forces: F = D(r z) (1) the summation being over all the particles. The expression in brackets on the right-hand side will be recognized as g "! times the potential energy of the waves; in an incompressible fluid X Â£mz- P | % C dx + constant (2) J. where x is a horizontal coordinate, p is the density, X is the wavelength, and Â£ (x, t) is the vertical displacement of the free surface. But by our previous remarks F = X (p -Po - pg h ), where h is the mean depth of water, equating (1) and (3) we find a5 - g H = Now for a standing wave H, (3) On dx. (4) (5) C = a cos kx cos a t where k = 2 jt/X and a = 2^/t (T being the wave period), and higher-order terms have been omitted. On substituting in (4) we find, after simplification, - g h = -J cos 2 ot (6) This shows that, to the second order, the mean pressure p fluctuates sinusoidally, with twice the frequency of the original wave, and with an amplitude proportional to the square of the wave amplitude. The pressure fluctuation is independent of the depth, for a given wave period, though of course the depth enters into the relation of the wave period to the wave- length, given by Gravity layer Fret lurfoec â¢ â BOM of gravity lo,*r Rigion of compression waves â¢ Bottom Figure 4. Waves in a heavy, compressible fluid.
CAN SEA WAVES CAUSE MICROSEISMS 77 a2 = g k tanh k h (7) There is a close analogy with the motion of a pendulum (see figure 2). In a complete cycle the bob of the pendulum is raised and lowered twice, through a distance proportional to the square of the amplitude of swing, when this is small. The only forces acting on the pendulum are gravity, which is constant, and the reaction at the support. Hence there must be a second-order fluctuation in the vertical component of the reaction at the support. Furthermore the reaction will be least when the pendulum is at the top of the swing (the potential energy is greatest) and will be great- est when the pendulum is at the bottom of its swing (the potential energy is least). It will be noticed that the above analytical proof does not necessarily involve the idea of the centre of gravity, whose vertical coordinate z is defined by m) z = (m z) (8) The theorem on the centre of gravity that was used previously is in fact usually derived from equation (1) : but in the present proof we have appealed directly to the original equations of motion for-the individual particles, without in- troducing z. 4. Two progressive wavesâThe above proof can easily be extended to the more general case of two waves of equal period but unequal amplitude travelling in opposite directions. For, such a disturbance is exactly periodic in space. Thus we may consider a region one wavelength in extent, as for the standing wave. This will not always contain the same mass of 2ir/X (-uK,-vk) / Direction of proportion Figure 5. The spectrum representation of a wave group. water; but, owing to the periodicity, the ver- tical reaction on the bottom due to the flow of water across one vertical boundary will be ex- actly cancelled by that due to the flow across the opposite boundary (see I Section 2.2) ; thus equation (4) is still exactly valid. The wave profile in this case is represented by cos (kx -at) + and so cos (kx + at) (9) 1 r X Jc (a/ + giving - g 2a, ar = - 2a cos 2at) (10) a2 a2 cos 2at (11) The mean pressure fluctuation on the bottom is therefore proportional to the product of the two wave amplitudes ai and a2. When these two are equal (ai = a2 = -j- a) we have the case of the standing wave, and when one is zero (a, = a; a2 = 0) we have the case of the single progressive wave. A physical explanation of this result may be given as follows. Suppose that one of the waves, say the wave of amplitude ai, is re- duced to rest by superposing on the whole sys- tem a velocity - c in the direction of x decreas- ing (this will not affect the pressure distribu- tion on the bottom). The second wave will now travel over the first with a velocity -2c. The crests of the second wave will pass alter- nately the troughs and the crests of the first wave - each twice in a complete period. Fig- ure 3 shows the two phases. One may pass from figure 3 (a) to figure 3 (b) by transferring a mass of fluid, proportional to a2 , from a trough to a crest of the original wave, i.e. through a vertical distance proportional to a i (the transferred mass does not of course con- sist of identically the same particles of water). The vertical displacement of the centre of grav- ity of the whole mass is therefor shifted by an amount proportional to a, a 2 ; and hence the fluctuation in p is also proportional to a, a 2. 5. Attenuation of the particle motionâThe fact that there is a pressure fluctuation on the bottom even in deep water does not, however, mean that there is movement at those depths. In fact it ma~y be shown (Longuet- Higgins 1953) that in exactly space-periodic motion, whether in a simple progressive wave or a combination of such waves, the particle motion decreases exponentially with the depth, apart from a possible steady current. Now if the velocities at great depths are zero, or steady, it follows from the equations of mo- tion that the pressure-gradient must be inde- pendent of the time. Thus if there is a pres-
78 SYMPOSIUM ON MICROSEISMS sure fluctuation it must be uniform in space, i.e. it must be applied equally at all points of the fluid. This indicates that below a certain depth, in a strictly space-periodic motion, the pressure fluctuations are uniform and equal to the fluctuation p(t) in the mean pressure on the bottom, which has been evaluated. The effect of the waves, at great depths, is then the same as would be produced by an oscillating pressure applied uniformly at the upper sur- face of the fluidâfor example an oscillation of the atmospheric pressure. Alternately one may imagine a rigid plane or raft to be floating on the surface of the water and completely cover- ing it, and the pressure to be applied to this plane by means of a weight attached to a spring and oscillating in a vertical direction. 6. An experimental verificationâThe above results were verified experimentally (Cooper and Longuet-Higgins 1951) in the fol- lowing way. Waves were generated at one end of a wave tank and allowed to travel to- wards the far end, where they were dissipated on a sloping beach. The pressure beneath the waves was detected by means of a hydrophone and was recorded continuously. On starting the motion from rest, no appreciable pressure fluctuations were recorded until the wave-front, travelling with approximately the group-veloc- ity of the waves, passed over the hydrophone. The pressure fluctuations then built up quickly to a constant amplitude, and had a period equal to that of the waves. The amplitude agreed well with the first-order theory; it diminished exponentially with depth, and was negligible below about half a wavelength. A vertical barrier was then placed in the wave tank, between the hydrophone and the beach, which reflected the waves back over the hydrophone. As soon as the reflected wave Figure 7. Graph of Cj, C2 , Cj and C4 as function of oh/|32, showing the relative am- plitude of the vertical displacement of the "sea bed" in the first four modes. Figure 8. The form of the wave spectrum in a circular storm. r K Figure 6. The regions of interference of groups of waves in the spectrum. two Figure 9. Wave interference caused by mov- ing cyclonic depression.
CAN SEA WAVES CAUSE MICROSEISMS 79 front arrived over the hydrophone the appear- ance of the pressure record was changed. At moderate depths there were not only first-order pressure fluctuations from the incident and the reflected wave, but also considerable second- order pressure fluctuations, of twice the funda- mental frequency. At greater depths the first- order pressure fluctuations become negligible and only the pressure fluctuations of double the frequency remained. The amplitude of these was in good agreement with equation (6). When the barrier was removed, and the rear end of the reflected wave train had passed the hydrophone, the second-order pressure fluctua- tions rapidly died out. Interference between waves of unequal amplitude was obtained by placing in the tank a vertical barrier extending only to a certain depth below the free surface, which allowed the waves to be partly reflected and partly transmitted. The coefficient of reflection from such a barrier is known theoretically for dif- ferent ratios of the depth of the barrier to the wavelength of the waves, and it was verified that the amplitude of the second-order pressure fluctuations was proportional to the amplitude of the reflected wave. Indeed this property seems to provide a convenient method of ac- tually measuring the coefficient of reflection from different types of obstacles or from plane beaches. Since standing waves produce only second- order pressure fluctuations below moderate depths one would expect that, if pressure fluc- tuations were induced deep in the water, stand- ing waves of half the frequency would be pro- duced at the surface. An experiment of this Figure 10. The spectrum representation of incident and reflected wave-groups. kind was in fact performed by Faraday (1831) ; (see Section 13 of the present paper) who produced standing waves, of half the fun- damental frequency, by means of a vibrating lath inserted in a basin of water. Faraday re- marked that the general result was little in- fluenced by the depth of water: "I have seen the water in a barrow, and that on the head of an upright cask in a brewer's van passing over stones, exhibit these elevations." (1831, footnote to p. 334). The present author has observed a similar phenomenon on board ship: a pool of water on deck, when excited by the vibration of the ship's engines, sometimes shows a standing-wave pattern whose ampli- tude gradually builds up to a maximum, and then collapses; the process is repeated indefi- nitely. 7. Standing waves in a compressible fluidâ The water has so far been assumed to be in- compressible, and we have seen that in this case the pressure fluctuations below about half a wavelength from the surface occur simul- taneously at all points of the fluid. But this can only be true if the least time taken for a disturbance to be propagated to the bottom and back is small compared with the period of the waves. In the deep oceans, where the speed of sound is about 1.4 km/sec and the depth may be of the order of several kilometers, this time may be several seconds. Thus the compressibility of the water must be con- sidered. The first-order theory of waves in a heavy, compressible fluid (in which all squares and products of the displacements are neglected) indicates that water waves of a few seconds' period fall into two classes (Whipple and Lee 1935). On the one hand there are waves approximating very nearly to ordinary surface waves in an incompressible fluid, in which the particle displacement decreases exponentially downwards, to first order; these may be called gravity-waves. On the other hand there are long waves controlled chiefly by the compres- sibility of the medium and hardly attenuated at all with depth; these may be called compres- sion-waves ; their velocity is nearly the velocity of sound in water. The wavelengths of a grav- ity-wave and a compression wave will be de- noted by X g and Xc respectively. For waves of period 10 sec. Xg/Xc is of the order of 10 â¢z . However, the pressure variations which are of interest to us at present are of second order. To investigate the effect of the com- pressibility, therefore, a complete - example, namely a motion which in the first approxima- tion is a standing gravity-wave, has been worked out in full to a second approximation (I Section 4). The result is a*s follows. Near the free surface, that is within a dis- tance small compared with Xc, the waves are unaffected by the compressibility of the water
80 SYMPOSIUM ON MICROSEISMS âas one might expect, since a disturbance could be propagated almost instantaneously through this layer. At a distance of about l/% Kg from the free surface the first-order pres- sure variations are much attenuated, and the second-order pressure variations are practical- ly those given by the incompressible theory (equation  ). Below this level the displace- ments are comparatively small, but, instead of the uniform, unattenuated pressure fluctua- tions in the incompressible fluid, there is now a compression wave, whose planes of equal phase are horizontal: the pressure field in this wave is given by a2 a2 (12) cos 2ah/cÂ» cos 2at very nearly, where z is the vertical coordinate measured downwards from the mean surface level, and c' is the velocity of sound in water. This wave can be regarded as being generated by the unattenuated pressure variation (6). There is a resonance, or "organ-pipe," effect: when cos 2 a h/c' vanishes, the pressure on the bottom (z = h) becomes infinite. This happens when 2oh/c' = (13) (14) du dv where (x, y) are horizontal coordinates, k is a constant and a is a function of (u, v) : 2 = g k (u2 + v 2)* tanh (u2 + v 2)* k h (15) A (u, v) is in general complex, and R denotes the real part. The expression under the inte- gral sign represents a long-crested wave with crests parallel to the line u x + v y - 0 and of wavelength X given by (16) X = 2jt (u v2)*k (17) that is, when the depth is (l/% n + ^4) times the length of the compression wave. In gen- eral, however, the displacements in the com- pression wave are small, being only of the order of a2/Xc ; the displacement of the centre of gravity of the layer at the surface of thickness Vz ^ g is of the order of a */l g. This explains why the compressibility of the fluid below has little effect on the pressure fluctuations at the base of the surface layer. We have then the following picture (see figure 4) : there is a surface-layer, of depth about 1/2 ^g. in which the compressibility of the water is, in general, unimportant: this may be called the "gravity-layer." Below this lay- er there exist only second-order compression waves, generated by the gravity-waves in the surface layer, and of twice their frequency. 8. Application to sea wavesâSo far we have considered only the very special cases of per- fectly periodic and two-dimensional waves. Such waves cannot be expected to occur in the ocean, although the sea surface usually shows a certain degree of periodicity. We shall now consider how the sea surface is to be described in this more general case. It can be shown (See I Section 3.2) that any free motion of the sea surface can be ex- pressed as a Fourier integral: If the point P, = ( - uk, - vk) is plotted in the (x, y) plane (see figure 5) the direction of the vector O P is the direction of propagation of the wave-component and the length of O P equals 2ir divided by the wavelength. Points on a circle centre 0 correspond to wave com- ponents of the same wavelength ; diametrically opposite points correspond to waves of the same length but travelling in opposite directions. When the energy is mainly grouped about one wavelength and direction, the complex ampli- tude A(u, v) will be appreciably large only in a certain range of values of (u, v), say Q, as in figure 5. The narrower this region, the more regular will be the appearance of the waves. The spectrum A(u, v) of the waves is de- termined uniquely by the motion of the free surface, at a particular instant, over the whole plane (see I, Section 3.2) . Since we shall want to consider the wave motion in only a certain part of the plane, say a square S of side 2R, it is convenient to define a motion i; Â» which, at any time, has the same value as Â£ inside S but is zero outside. Let A1 be the spectrum function of Â£', so that A'(u,v)ei(ukx + vky (18) du dv
CAN SEA WAVES CAUSE MICROSEISMS 81 A' is very closely related to A; if k is chosen so that k= n/R (19) and if R is large compared with the wave- lengths associated with most energy in the spectrum then (see I Section 3.3) A'(u,v) = r^r = J Â«J A/ * A(u,,v.) sn (U-UI)TI Â« co sin (V-V.)TC -1(0-0,). ââ¢ -iâ e l du.dv. - ' ' (20) where Oi = o(u,, v,). In other words A' is the weighted average of values of A over neighboring wavelengths and directions. Since u and v are proportional to the number of wave- lengths intercepted by the xâ and yâaxis in S, a "neighboring" wave component is one which has nearly the same number of wave- lengths, in each direction, in S. A' gives a "blurred" picture of A; but the larger the side of the square, the less is the blurring. The region Q ' in the (u, v)âplane which corre- sponds to the blurred spectrum will be almost the same as the region Q corresponding to the original spectrum. A' also varies slowly with the timeâthe waves in S change graduallyâ but this rate of change is slow compared with the rate of change of the wave profile, or com- pared with a A'. The energy of the waves is given very simply in terms of the spectrum function A'; in fact, if a denotes the amplitude of the single long-crested wave which has the same mean energy inside S, oo f co â¢ill J-ooJ. A'A'* du dv (21) where a star denotes the conjugate complex function (I equation ). a may be called the equivalent wave amplitude of the motion. 9. General conditions for fluctuations in the mean pressureâWe shall evaluate the mean pressure p at the base of the gravity-layer, i.e. at a distance of about l/z X g below the free sur- face, over a square of side 2R. (Here Xg re- fers to the mean wavelength of the predomi- nant components in the spectrum.) Consider first the two-dimensional case. The mass of water contained between the surfaces z = Â£ and z = 1/2 X K aÂ°d the planes x = Â± R no long- er consists of the same particles of water; but it is possible to extend the analysis of Section 3 so as to take account of the motion across the boundaries (see I Section 2.2). Provided that the horizontal extent 2R of the interval is large compared with X gthe effect of the flow across the vertical boundaries can be neglected (I Section 3.1). Further, since the motion de- creases rapidly with depth the effect of flow across the horizontal plane z = 1/2 Xg is small. The expression for the mean pressure variation is therefore the same as if the free surface were the only moving boundary: Similarly in the three-dimensional case P - PD 61 6t2 that is P-P0 (23) -R -R 6t2 OW OO J J V> C'2 dx dy, (24) since Â£' vanishes outside the square S. Now the expression on the right-hand side is closely related to the potential energy of the motion Â£', and can be simply expressed in terms of the Fourier spectrumâfunction A1. In fact (I Section 3.2) CO OO / / Â«"d,ay = - CO - OO (25) R( 7t (A'A'* + A'AL e 2l0t) du dv -oo - oo
SYMPOSIUM ON MICROSEISMS where Al stands for A' (âu, âv), and is the amplitude of the wave component opposite to A(u, v). On substituting in (24) we have oo co (26) a2 A'Al e I0t du dv -co - CO This shows that fluctuations in the mean pres- sure p arise only from opposite pairs of wave components in the spectrum; that the contribu- tion to p from any opposite pair of wave com- ponents is of twice their frequency and pro- portional to the product of their amplitudes; and that the total pressure fluctuation is the integrated sum of the contributions from all opposite pairs of wave components separately. The necessary condition for the occurrence of second-order pressure fluctuations of this type is, therefore, that the sea disturbance should contain some wave-groups of appre- ciable amplitude which are "opposite," i.e. such that part at least of the corresponding region in the Fourier spectrum is opposite to some other part. For example, if Q lies entirely on one side of a diameter of the (u, v)âplane, the mean pressure fluctuation, to the present order, vanishes. An important case is when the disturb- ance consists of just two wave groups, cor- responding to regions Q \ and Qz, and of equiv- alent amplitudes a, and a2 (see figure 6). Q i- and Q2. , denote the regions opposite to Qi and Q2 and QU and Qi2 denote the re- gions common to Q i and Q 2. and to Q i- and i}2 respectively. Effectively, then, the inte- gration in (26) is carried out over the two regions Q u and Q u- . When the spectrum is narrow an order of magnitude for the integral on the right-hand side of (26) can be obtained. It may be shown (see I Section 5.2) that P - P ~ 2a ( Q ke 2io (27) where a12 is the mean value of o in QJ2. Thus the mean pressure on S increases proportion- ately to the square root of the region Qi2 of overlap of the wave groups, and inversely as the square root of Q i and Q 2 separately, for fixed values of ai and a2. 10. Calculation of the ground movementâIn order to estimate the movement of the ground, at great distances, due to waves in a storm area A, we suppose the storm area to be divided up into a number of squares S of side 2R such that S contains many wavelengths Xg of the sea waves, but is only a fraction, say less than half, of the length of a seismic wave A,s in the ocean and sea bed. This we may do, since the wavelengths of seismic waves are of the same order as the wavelengths of compression waves in water; therefore Xg/X s is of the order of 10 "2 . The mean pressure or total force on the base the gravity-layer can be calculated as in Section 9; the vertical movement of the ground 5' due to the waves in this square is of the same order as if the force were concen- trated to a point at the center of the square, i.e. 2a, a9 o 1 2 12 where r is the distance from the center of the storm and W (a, r)elot is the movement of the ground at distance r due to a unit pressure oscillation e^ot applied at a point in the mean free surface. The pressure can be considered to be applied in the mean free surface rather than at the base of the gravity-layer, since the latter is relatively thin compared with the length of the seismic waves. To find the total 6â4 12 To calculate W (a, r) we may consider the disturbance due to a force applied at the sur- face of a compressible fluid of depth h (rep- resenting the ocean) overlying a semi-infinite elastic medium (representing the sea bed). Al- though this model takes no account of varia- (2Â° 12,r) (28) displacement 8 from the storm we may add the energies from the different squares S, on the assumption that the contributions from the different squares are independent. Since there are A/4R such squares in the whole storm area, this means that the disturbance 5 ' from each individual square is to be multiplied by A*Â» /2R. Hence we have 2ia 12' (29) tions in the depth of water, or of the propaga- tion of the waves from the sea bed to the land or across geological discontinuities, it can nev- ertheless be expected to give a reasonable esti- mate of the order of magnitude of the ground movement.
CAN SEA WAVES CAUSE MICROSEISMS 83 The disturbance W (o, r) e^ot at great distances from the oscillating point source eiQt consists of one or more waves of surface type, W(o.r) P2 where (. 2 is the density of the elastic medium, P2 the velocity of secondary waves in the medi- um, 2jt/Â£ m is the wavelength of the mth wave and Cm is a constant amplitude depending on the depth of water and on the elastic properties of the fluid and the underlying medium. The first wave has no nodal plane between the free surface and the "sea bed," the second has one nodal plane, the third two, and so on. When the depth h of the water is small, only the first type of wave can exist; the others appear successively as the depth is increased. Graphs of C i, C 7âhave been computed for some typ- ical values of the constants: pi (the density of the fluid) =1.0 g./cmj; cf (velocity of com- pression waves in water) =1.4 km./sec.; p 2 = 2.8 km./sec., and with Poisson's hypothesis, that the ratio of the velocities of com- pressional and distortional waves in the medi- um is Y \/*3'. The results are shown in figure 7, where C1 , C2 , C3 and C 4 are plotted against oh/p 2 â¢ C i, for example, increases to a maxi- mum when oh/pa = 0.85, i.e. when h = 0.27 x 2 jtc '/o, or h is about one-quarter of the wave- length of a compression wave in water. This maximum may therefore be interpreted as a resonance peak. The amplitude, however, does not become infinite as in the case of the infinite wave-train discussed in Section 7, since now energy is being propagated outwards from the generating area. C2, Ci, and C4 have similar resonance peaks when oh/P2 = 2.7, 4.1 and 6.3, respectively, i.e. when the depth is 0.86, 1.31 and 2.0 times the length of a compression wave in water. A measure W of the total disturbance can be obtained by summing the energies from each wave. Thus (31) P2 /325/2(2nr) 11. Practical examplesâWe have seen that a necessary condition for the occurrence of the type of pressure fluctuations studied in this paper is that the motion of the sea surface should contain at least some wave groups of the same wavelength traveling in opposite di- rections. We shall briefly consider some situ- ations in which this may occur. (a) A circular depression. The "eye" or center of a circular depression is a region of comparatively low winds; yet there are often observed to be high and chaotic seas in this region (which indicates the interference of more than one group of swell). Thus, the i.e. waves spreading out radially in two dimen- sions (see I Section 5.1). Thus Â«. r (30) waves in the "eye" must have originated in other parts of the storm. Now the winds in a circular depression are mainly along the iso- bars, but in some parts of the storm they usu- ally possess a radial component inwards. In addition, some wave energy may well be propa- gated inwards at an angle to the wind. This then may account for the high waves at the center of the storm. If wave energy is being received equally from all directions, the energy in the spectrum will be in an annular region between two circles of radii 2 Ji/Xi, and 2 jt/X2, where X i and X 2 are the least and greatest wavelengths in the spectrum (see figure 8). This region may be divided into two regions Q i and Q 2 by any diameter through the origin. Let us take numerical values appropriate to a depression in the Atlantic Ocean. Suppose that the wave- periods lie between 10 and 16 seconds, so that Jli = 1.54 x 104 cm., Jl2 = 4.00 x 10* cm. and x 10 "7 cm- "2 As- suming A = 1000 km2 (corresponding to a cir- cular storm area of diameter 17 km.), o12 = 2 jt/13 sec.' ' , a! = a2 = 3m., h = 3 km. and r = 2,000 km. we find from (29) that|8| = 3.2 x 10-* cm., or 3.2n. The peak-to-trough amplitude of the displacement is 6.5|i. This is of the same order of magnitude as the observed ground movement. (b) A moving cyclone. Consider a cy- clone which is in motion with a speed cojn- parable to that of the waves. Figure 9 repre- sents the position of the cyclone at two dif- ferent times. When the center of the storm is at A, say, winds on one side of the storm (marked with an arrow) will generate waves travelling in the direction of motion of the storm; these will be propagated with the ap- propriate group velocity. When the storm has reached B, winds on the opposite side will gen- erate waves travelling in the opposite direc- tion; and if the storm is moving faster than the group-velocity of the waves, there will be a region C where the two groups of waves will meet. Thus, in the trail of a fast-moving cy- clone we may expect a considerable region of wave interference. (c) Reflection from a coast. The extent of wave reflection from a coast is hard to judge, since the reflected waves are usually hidden by the incoming waves; but when the waves strike a coast or headland obliquely the reflected waves can sometimes be clearly seen. Effec- tive wave interference will take place only on the parts of the coast where the shoreline is
84 SYMPOSIUM ON MICROSEISMS parallel to the crests of some wave components of the incoming waves, but refraction of the waves by the shoaling water will tend to bring the crests parallel to the shore. If the incoming waves are represented by a region Q i in the spectrum, then we may as- sume that the reflected waves are represented by a region Q 2 which is the reflection of Qi in the line through 0 parallel to the shoreline (see figure 10, in which the x-axis is taken parallel to the shoreline). Qi- is then the reflection of Qi in the line through O perpendicular to the shoreline (the y-axis). Suppose that the period of the incoming swell lies between 12 and 16 seconds, that its direction is spread over an angle of 30Â°, and that its mean direction makes an angle of 10Â° with the perpendicular to the shoreline. Then we find Q, = Q2 = 1.4 x 10'8 cm.-2 , Q,2 = 1/3 a, = 0.47 x 10-' cm. If the ef- fective shoreline is 600 km. in length and the region of interference extends, on the average, 10 km. from the shore, then A = 6,000 km2. If also a i = 2m, a 2 = 0. lm(a reflection coeffi- cient of 5%) and if r = 2,000 km., then we find from (26) (assuming h = 0) that 2|8| = 0.3n. Since this amplitude is somewhat smaller than in case (a), we may conclude that coastal re- flection does not give rise to the largest disturb- ances at inland stations, though it may be a more common cause of microseisms near to the coast. Besides the examples given above there is another possible class of cases, namely when a swell meets an opposing wind. For example, coastal swell may be subject to an offshore wind, or there may be a sudden reversal of the direction of the wind at the passage of a cold front.* The wind will doubtless tend to dim- inish the amplitude of the original swell, but it may also tend to generate waves travelling in the opposite direction, the amplitude of which may increase rapidly on account of the roughness of the sea surface. However, in none of the first three cases discussed above is it necessary to assume that such action takes place. 12. Observational testsâThe present theory suggests several possible kinds of experimental investigation. The first is a comparison of the periods of microseisms and of the sea waves possibly associated with them, (which should be about twice the microseism periods). There is a general agreement between the periods, in that the range of microseism periods is from about 3 to 10 seconds while the periods of high sea waves vary from about 6 to 20 seconds. Further, the periods of both microseisms and sea waves both increase, in general, during a time of increased disturbance. The close two- to-one ratio between the periods of sea waves and of the corresponding microseisms which was found by Bernard (1937 and 1941) and re- * See also the author's comment on the paper by Frank Press. lated by Deacon (1947) and Darbyshire (1948) is highly suggestive, though not conclu- sive. A similar, though less detailed study by Kishinouye (1951) during the passage of a tropical cyclone, has not confirmed the relation- ship. Comparisons of this kind are, however, inconclusive, unless it can be shown that the microseisms can be associated uniquely with the recorded sea waves. The meteorological conditions are rarely so simple, and the record- ing stations so well placed, that it is possible to be certain of the connection; the examples se- lected by Darbyshire (1950) were, how- ever, chosen with this requirement in mind. Figure 7 shows that the displacement of the "sea bed" may vary by a factor of the order of 5, depending on the depth of the "ocean." Although the model chosen is extremely simpli- fied, we can nevertheless infer that the ampli- tude of microseisms should, on the present the- ory, depend considerably on the depth of water in the path of the microseisms; the depth in the generating area itself, where the energy-den- sity is greatest, should be of the most critical importance. Comparisons between the micro- seisms due to storms in different localities would therefore be of considerable interest. It should be noticed that the unequal response of the ocean to different frequencies may result in a displacement of the spectrum towards those frequencies for which the response is a maximum. The nature of the frequency spectrum of sea waves under various conditions is of fun- damental importance, and further studies should be undertaken. The wavelengths and directions of the components of the spectrum, both for swell and for waves in the generating area, could be studied by means of aerial photo- graphs or altimeter records taken from an air- plane. An estimate of the amount of wave re- flection from a coast might be obtained by tech- niques similar to those which were used in the model experiments described in Section 7, that is, by comparing the frequency spectra of pres- sure records taken at different depths in the water, or off different parts of the same coast where the bottom gradient varied. The effect of an opposing wind on a swell might be in- vestigated on a model scale, by generating pro- gressive waves in the usual manner and then exposing them to an artificial wind; the growth of the opposing waves would be measured by means of the second-order pressure fluctua- tions deep in the water. It would be of great interest to record the pressure fluctuations on the ocean floor directly, if the practical difficulties of makingâ¢ measurements at such depths can be overcome. A pressure recorder has been designed for this purpose by F. E. Pierce, of the National Insti- tute of Oceanography.
CAN SEA WAVES CAUSE MICROSEISMS 85 13. Historical notesâIt was known to FARA- DAY (1831), who refers to earlier work by Oersted, Wheatstone and Weber, that fluid rest- ing on a vibrating elastic plate will form itself into short-crested standing waves. Faraday was the first to show, by an ingenious optical method, that the period of the standing waves is twice that of the vibrations of the plate. The waves that he used were mostly "ripples," con- trolled predominantly by surface tension, since their wavelength lay between 14 and % inch. In the same paper (1831) Faraday describes many other interesting experimental studies of waves in water, mercury and air. About fifty years later Rayleigh (1883 b) repeated Faraday's experiments and veri- fied, by a slightly different method, the doub- ling of the period. In a theoretical paper (1883 a) Rayleigh gives general consideration to the problem of how a system can be main- tained in vibration with a period which is a multiple of the period of the driving force. He refers in particular to Melde's experiment, in which a stretched string is made to vibrate by the longitudinal oscillation of a tuning fork attached to one end; such a phenomenon is sometimes called "subharmonic resonance." Neither Faraday (1831) nor Ray- leigh (1883) evaluated the second-order pressure fluctuations associated with standing waves. This, however, was done by MICHE (1944) in a different connection, using a La- grangian system of coordinates. Miche noticed the unattenuated terms, and, though he does not mention microseisms, he remarks, "on peut aussi se demander si ces pulsations de pression, malgre leur faible intensite relative, n'exercent pas une action non negligeable sur la tenue des fonds soumis au clapotis." (1944, p. 74.) The wave interference theory seems to have arisen as follows. In 1946 Deacon, fol- lowing similar studies by Bernard (1937, 1941 a) compared the period and amplitude of swell off the coast of Cornwall, England, with the corresponding microseisms at Kew, and found a two-to-one ratio between the periods (Deacon 1947). F. Biesel, then visiting England, pointed out to Deacon Miche's theo- retical work on standing waves. Miche's re- sults, however, cannot be applied directly to sea waves, since exact standing waves do not oc- cur in the ocean. Moreover, his method is not easily generalized, since it involves a complete evaluation of the second approximation to the wave motion. A very simple proof of Miche's result, however, which depended essentially on the idea of the vertical motion of the center of gravity of the whole wave train, was found by Longuet-Higgins and Ursell (1948) ; the advantage of this method was that the second-order pressure fluctuations on the bottom could then be obtained immediately from the first approximation to the surface ele- vation. It then became possible to extend the results to much more general and realistic types of wave motion. A complete theory, giving the necessary conditions for the occurrence of this type of pressure fluctuation, taking into account the compressibility of the ocean, and determining the order of magnitude of the ground movement, was given by Longuet- Higgins (1950). It is interesting that Bernard (1941 a, b) had suggested, with intuitive reasoning, that microseisms might be caused by the standing- type waves observed to occur at the center of cyclonic depressions: "J'ai cru qu'on pourrait trouver la raison de cette particularite dans le charactere que presentent les mouvements de la mer au centre des depressions cycloniques: la houle s'y dresse aux vagues pyramidales constituant un clapotis gigantesque dont les points de plus ample os- cillation peuvent etre autant des sources de pression periodique sur le fond de la mer, pression qui donnera naissance a un mouve- ment oscillatoire de meme periode du sol ... " "Un clapotis analogue, avec oscillations sur place du niveau de 1'eau, se produit lorsque la houle, se reflechissant sur un obstacle, vient interferer avec les ondes incidentes . . . "Au contraire, dans le cas d'un train d'ondes de front continu et de deplacement constant, les points ou les mouvements sont de phase opposee donneront sur le fond de la mer des pressions de sens contraire, et la longeur d'onde des oscillations microseismiques etant beaucoup plus grande que celle de la houle, les mouvements transmis par le sol a une certaine distance seront pratiquement simultanes, mais opposes, et ils interfereront, de sorte que 1'efFet total du train de vagues a 1'exterieur sera nul." (BERNARD, 1941 a, p. 7.) However, Bernard did not apparently see that the corresponding pressure fluctuations must have a frequency twice that of the waves; for he suggests other causes for the observed doubling of the frequencies in the case of coast- al waves." (Bernard, 1941a, p. 10.) REFERENCES BERNARD, P., Relations entre la houle sur la cote du Maroc et I'agitation microseismique en Europe oc- cidental*, C. R. Acad. Sci., Paris, v. 205, pp. 163- 165, 1937. BERNARD, P., Sur certaines proprietes de la houle etu- diees a I'aidc dcs enregistrentents seismograph- iques. Bull. Inst. Oceanogr. Monaco, v. 38, No. 800. pp. 1-19, 1941. BERNARD, P., Etude sur I'agitation microseismique et ses variations. Ann. Inst. Phys. Globe, v. 19, pp. 1-77, 1941.
86 SYMPOSIUM ON MICROSEISMS COOPER, R. I. B., and LONGUET-HIGGINS, M. S., An ex- perimental study of the pressure variations in standing water waves. Proc. Roy, Soc. A, v. 206, pp. 424-435, 1951. DARBYSHIRE, J., Identification of microseismic activity with sea waves, Proc. Roy. Soc. A., v. 160, pp. 439- 448, 1950. DEACON, G. E. R., Relations between sea waves and microseisms. Nature, v. 160, pp. 419-421, 1947. FARADAY, M., On a periodic class of acoustical figures, and on certain forms assumed by groups of parti- cles upon vibrating elastic surfaces. Appendix: On the forms and states assumed by fluids in con- tact with vibrating elastic surfaces. Phil. Trans. Roy. Soc., pp. 319-340,, 1831. KISHINOUYE, P., Microseisms and sea waves. Bull. Earthqu. Res. Inst., v. 29, pp. 577-582, 1951. LONGUET-HIGGINS, M. S., and URSELL, F., Sea waves and microseisms. Nature, v. 162, p. 700, 1948. LONGUET-HIGGINS, M. S., A theory of the origin of mic- roseisms. Phil. Trans. Roy. Soc. A., v. 243, pp. 1- 35, 1950. LONGUET-HIGGINS, M. S., On the decrease of velocity with depth in an irrotational surface wave. (In press) 1953. MARTIN, J. C., MOYCE, W. J., PENNEY, W. G., PRICE, A. T., and THORNHILL, C. K., Some gravity-wave problems in the motion of perfect liquids. Phil Trans. Roy. Soc. A., v. 244, pp. 231-281, 1952. MICHE, M., Mouvements ondulatories de la mer en pro- fondeur constante ou decroissante. Ann. Fonts et Chawisees, v. 114, pp. 25-87, 131-164, 270-292, 396-406, 1944. RAYLEIGH, LORD, On maintained vibrations. Phil. Mao vol. 15, pp. 229-235, 1883. RAYLEIGH, LORD, On the crispations of fluid resting upon a vibrating support. Phil. Mag., v. 16, pp. 50-58, 1883. WHIPPLE, F. J. W., and LEE, A. W., Notes on the theory of microseisms. Mon. Not. Roy. Astr. Soc., Geophys. Suppl., v. 3, pp. 287-297, 1935. Discussion G. E. R. Deacon (National Institute of Ocean- ography at Teddington) The wave-interference theory explains, for the first time, how energy sufficient to gen- erate long, regular, microseisms is communi- cated to the ground. It has been clear for a long time that the occurrence of microseisms is associated with the presence of sea waves, but it could not be proved that the waves played an essential part in the energy transfer. Although each breaker, as it crashes on the coast, must cause a local disturbance, and has been shown to do so, the variations in the moment of impact along a stretch of coast, and the shortness of the wavelength compared with that of 3 to 10 second microseisms, make it most unlikely that the actual beating of surf on a coast could produce the long microseismic waves that can be detected far from the coast. The exponential decrease in wave move- ment with depth was sufficient reason why a train of progressive waves should not disturb the sea bottom at great depths, and at lesser depths the contributions from different parts of the sea bed would tend to cancel each other out. Taking account of the compressibility of the water made no significant difference to this conclusion. If the conviction held by many who had studied microseisms, that sea waves are di- rectly concerned in the generation of micro- seisms were to be confirmed, we had to find a theory which showed that sea waves were modi- fied in such a way that they were able to cause regular changes in pressure, acting simultane- ously over large areas of the sea bed. During the past few years it has, in addition, become necessary to explain why the periods of the microseismic waves are half those of the sea waves, and how the effect of wind and wave- height could vary with the depth of water, being sometimes greater in deep water than in shallow. The new wave-interference theory seems to fill these requirements, and to be capable of withstanding the test of more precise and well- directed observations. It is not easy for the non-mathematician to understand the precise demonstration that two trains of waves of the same wavelengths, meet- ing each other in opposite directions, will cause variations in pressure on the sea bed with twice the frequency of the surface waves, but Dr. Longuet-Higgins has done his best to explain it in non-technical terms. The deduction is simplified by considering the vertical move- ments of the centre of gravity of a water mass bounded by two vertical nodal planes, and by a comparison with the changing tension in the string of a pendulum. It is perhaps not very difficult to accept the result intuitively, as Bernard (1941) did, particularly if we re- member the convincing agreement between theory and observation obtained by measure- ments in a tank. There is also confirmation of the mean pressure changes and their ability to produce microseisms that can be detected far from the coast, in the work of Darbyshire (1950). As Dr. Longuet-Higgins says in his paper, confirmation of the two to one relationship between wave and microseism periods does not completely verify the theory, but when, as Darbyshire showed, the trend of a band of swell from long to short periods was exactly par- alleled by proportionate changes in the micro- seism periods there is little room to doubt that the waves caused the microseisms.
CAN SEA WAVES CAUSE MICROSEISMS 87 If the previous literature is re-examined, bearing the wave-interference theory, and what we already know about waves, in mind, some of the apparent contradictions to which emphasis has been given appear explainable. The example given by Whipple and Lee (1935) of almost identical isobaric charts of two depressions south-east of Greenland, one associated with intense microseismic activity and the other with practically none, is not such an obstacle when the previous histories of the two depressions are studied. One had moved rapidly northwards over the ocean, with plenty of opportunity for wave interference, whereas the other had developed over the land. Similar attempts to estimate wave interference might explain why less microseismic activity was found with a depression over the mouth of the St. Lawrence river and an anticyclone over the Great Lakes than when the positions of the depression and anticyclone were reversed; or why, with a shallow depression off the east coast of Japan, the microseisms were larger on the coast of China while the wind was stronger off the coast of Japan. There is, however, not much to be gained by studying cases which are not fully docu- mented. We must, as Dr. Longuet-Higgins emphasizes, learn more about the conditions which give rise to wave interference; we must select examples in which the metorological con- ditions are sufficiently simple for us to be cer- tain of the connection between the storm and the microseisms, and we must measure the waves and the microseisms as precisely as modern techniques will allow. It is possible that some of the present misunderstanding is due to faulty interpretation of records from seismometers that are highly tuned to the short-period end of the microseism range, and faulty estimation of the sea surface or wave and microseism recordings, in which the size of a long period oscillation can be underestimated owing to the interruption of its swing by minor, shorter, waves. The wave-interference theory is, to say the least, an excellent working hypothesis, and if it is subjected to further question and experi- ment, of the standard set by Dr. Longuet-Hig- gins and his co-workers, we must move rapidly towards a full solution. It seems to me that the subject has now been put on a systematic basis, and that its progress must be more rapid. In spite of some setbacks we shall soon be in a better position to take full advantage of the practical possibili- ties. I think that Dr. Longuet-Higgins's histori- cal note gives a proper account of the develop- ment of the new theory. REFERENCES BERNARD, P., Etude stir I'agitation microseismique et ses variations. Ann. Inst. Phys. Globe, v. 19, pp. 1-77, 1941. DARBYSHIRE, J., Identification of microseism activity with sea waves. Proc. Roy Soc. A., v. 160, pp. 439- 448, 1950. WHIPPLE, F. J. W., and LEE, A. W., Notes on the theory of microseisms, Mon. Not. Roy. Ast. Soc. Geophys. Suppl., v. 3 pp. 287-297, 1935. v Discussion JACOB E. DINGEB Naval Research Laboratory As a discussion of the theoretical paper "Can Sea Waves Cause Microseisms," I should like to present some of the data and interpreta- tions obtained by the Naval Research Labora- tory on various field trips during the hurricane seasons of the past several years. The data considered here is concerned with hurricanes which have followed paths in the Western Atlantic and Caribbean. It has been a primary objective of this work to obtain evi- dence which might help to determine where the area of microseism generation is with respect to the hurricane center and to determine under what condition a hurricane can generate micro- seisms. In furthering this objective it has be- come of interest to study the data in the light of various theories to see if the data lends sup- port to any of these theories. During the hurricane seasons of 1948-1951 records of microseisms have been obtained at points in the Bahamas, Florida, North Carolina and Washington D. C. as various hurricanes have followed varying paths in the Western Atlantic. The following observations have in general been true for all these hurricanes: (1) Storms which generate in the Middle Atlantic and approach the seismo- graph locations do not produce ap- preciable microseismic activity until the storm moves over the continental shelf or, over the shallower waters surrounding the Islands of the Carib- bean Sea. This same observation is pointed out by Donn (1952). (2) As the storm recedes, the microseisms continue at a much higher level of amplitude as compared to the same distance from the seismograph loca- tion during the approach of the storm. (3) The point of nearest approach is not necessarily the time of maximum amplitude. The above observations can be interpreted as giving evidence that the storm must move
88 SYMPOSIUM ON MICROSEISMS over the shallower waters of the continental shelf before microseisms are recorded and that the wake of the storm continues to be important in the generation of microseisms. This and similar observations in the light of the Longuet - Higgins (1950) theory, together with the word of Deacon (1947) and Darbyshire (1950), prompted the Naval Research Laboratory group to conduct field experiments during the 1951 hurricane season designed to obtain data which could assist in determining whether any correlation appears to exist between microseisms and hurricane- generated ocean waves. The installations of the field experiments included the following: (1) A tripartite station on the West End of Grand Bahama. (2) The installation of two wave gages at Cocoa Beach, Florida, through the cooperation of the Beach Erosion Board and the University of Cali- fornia. These gages were of the pressure-sensitive type; the one was similar to the type developed by Woods Hole, and used quite extensi- vely by the Beach Erosion Board, and the other was developed by the Uni- versity of California. These gages were in water depths of about 29 and 46 feet respectively. WASHINGTON I/ CHAPEL HILL / f OCT. 4-1700 OCT. 4-0800 OCT. 9-1700 OCT. 3-0600 HOW SEPT. 9-0900 SEPT. 8-1700 ORLANDO COCOA BEACH SEPT. 8-0900 SEPT. 7- 1700 Figure 1. Map Showing Paths of Hurricanes "Easy" and "How"
CAN SEA WAVES CAUSE MICROSEISMS 89 (3) A single horizontal-component seismo- graph was placed on the grounds of the U. S. Navy Underwater Sound Reference Laboratory at Orlando, Florida. This location is approxi- mately 50 miles inland from Cocoa Beach, and therefore can be con- sidered isolated from local surf vibra- tions, which can cause high seismic noise near the shore. The simultaneous data of microseisms and water waves obtained by these installations during the two hurricanes of the 1951 season is of special interest in that the paths of the storms were radically different. Figure 1 shows the paths of the two storms "Easy" and "How." "Easy" followed a path which was well out over deep water during its entire course (except near its end when it moved over the Banks of Newfoundland). Its nearest approach to Filorida was about 650 miles. Hurricane "How" generated in the Gulf of Mexico, rapidly moved across Florida, and entered the Atlantic with the center passing slightly to the south of the wave-recorded loca- tion. Both of these storms produced high waves on Florida but the character of the waves was considerably different and the mi- croseismic activity was greatly different. The two storms therefore provide an interesting comparison. Figure 2 gives results of the simultaneous recordings of microseisms and water waves throughout the period hurricane "Easy" was in existence. The wave-gage data was an- alyzed by the Beach Erosion Board to give the significant wave height and period plotted as curves C and D respectively. A measure of the amplitude of the microseisms was obtained by measuring the area enclosed by the envelope of the microseisms during a 15 minute interval, an interval being used every two hours and in HURRICANE EASY - SEPTEMBER, 1951 PERIOD OF WATER WAVE AT COCOA BEACH, FLA. 2 - t <E HEIGHT OF WATER WAVE COCOA BEACH, FLA. AMPLITUDE OF MICROSEISMS J)RLANDO. FLA. WASHINGTON. P.O. I T.S-8 SÂ«cond PÂ«rio 1200 8 SEPT. 1200 9 SEPT. 1200 10 SEPT. 1200. II SEPT? 1200 12 SEPT. Figure 2. Microseismic and Water Wave Activity During Hurricane "Easy".
90 SYMPOSIUM ON MICROSEISMS some parts of the record every hour. The rela- tive position of curves A and B has no signifi- cance since the two curves have been shifted with respect to each other. However, the value of the arbitrary units for A and B is the same. The sharp increase in both wave height and period as shown in curves C and D on the morning of September 8 accompanied the arri- val of the swell from "Easy." Data from a Beach Erosion Board gage at Cape Henry and a report from Weather Ship H, several hundred miles east of Charleston, N. C., also gives added evidence that the wave activity shown by curves C and D on Sept. 8 is associated with the arrival of swell from "Easy." The micro- seisms as recorded at Orlando on 8 Sept. show some increase in amplitude at approximately the same time as the maximum wave activity at Cocoa Beach. This increase in amplitude was not at all pronounced; in fact this particular period of microseisms normally would not have received any attention as being an indication of anything unusual. The record was too erratic to permit an analysis of the most pro- HURRICANE HOW - OCTOBER. 1951 12 10 u 111 (O \ u UJ CO tr . AMPLITUDE OF A KROSEIMS ORLANDO. FL j PERIOD OF WATER TWO TIMES THE PER AVES AT COCOA BEA ID OF MICROSEISMS HEIGHT OF WATER WJVE! COCOA BEACH. FLA CHAPEL HILL, N. C. V 1200 I OCT. IZOO 2 OCT. 1200 3 OCT. 1200 4 OCT. 1200 5 OCT. Figure 3. Microseismic and Water Wave Activity During Hurricane "How".
CAN SEA WAVES CAUSE MICROSEISMS 91 nounced period. The slight increase in micro- seisms during the wave activity can be inter- preted as being associated with the swell rather than being generated directly under the storm for these reasons: (1) No simultaneous increase in micro- seisms occurred in Washington. (2) Microseisms generated under the storm should also have shown in- creased activity before the arrival of swell. According to the Longuet-Higgins theory, a standing-wave pattern is required to transfer the water wave energy to microseisms. A standing wave pattern can conceivably be es- tablished upon reflection of the incoming swell by a sufficiently steep coast. The low level of microseismic activity during the swell from "Easy" would indicate, if the Longuet-Higgins theory is of importance, that the reflected wave energy along the Florida coast is very small. Because of the very gradual slope of the shore along Florida one would indeed expect low reflections. The fact that no microseisms of any con- sequence were recorded during the period this intense storm remained over deep water indi- cates either one of two things: (1) microseisms were not generated by any method or (2) the generated microseisms were almost completely attenuated before reaching the continent. The data obtained by NRL is unable to resolve which of these two factors is the important one. Carder (1951) has presented evidence to indicate that the attenuation of microseisms propagated through the floor of the Western Atlantic is much greater than the attenuation over continental land masses. If attenuation is the important factor, then the attenuation may vary with the nature of the ocean floor and thus the results could be different in vari- ous parts of the world. Darbyshire (1950), Banerji (1935) and others have pre- sented evidence that microseisms are generated in deep water and have been recorded at distant points in the case of storms over the Eastern Atlantic, the mid-Bay of Bengal, and the Paci- fic. In view of these observations which con- trasts with the observations in the Western Atlantic it may be inferred that attenuation is a much greater factor in the Western Atlantic than in certain other parts of the world. It is of interest to point out the fact that longer period (7.5 to 8.0 second) mocroseisms are evident on curves A and B, Figure 4, as occurring at Washington and Orlando on the morning of 12 September. The records of these microseisms were nicely formed and of a regular nature. The simultaneity in time and period of these microseisms at Washington and Orlando would indicate a common area of generation. The fact that the storm at this particular time was dissipating itself over the shallow areas off the coast of Newfoundland is further evidence that a storm moving from deep water to shallow water begins to generate micrcseisms. Intense winter microseisms are frequently observed when low-pressure areas move over this portion of the North Atlantic. Let us consider Figure 3 which shows sim- ultaneous data on wave and microseismic acti- vity obtained during hurricane "How." The wave gage was fortuitously placed in a strate- gic location slightly to the north of the area where the storm entered the Atlantic. We may therefore assume that, if waves are re- sponsible for the generation of microseisms, the waves as measured at this time should yield the best possible correlation inasmuch as the waves were confined to the water areas near the gages. Let us therefore compare the water wave amplitude and the position of the storm. We note an abrupt increase in wave amplitude during the early morning of 2 October, reach- ing a maximum about 1200 and dropping off abruptly about 2000. Referring again to Figure 1 we see that the forward part of the storm entered the Atlantic in the morning of 2 October with strong winds blowing from south-southeast and bringing waves toward Cocoa Beach. At about 1200 the center of the storm moved into the Atlantic and by 2000 the winds in the trailing part of the hurricane were from the north, thus effecting a reversal of wind as it existed 20 hours previously over this area. This reversal of wind is evident on the wave records by a rather abrupt decrease in wave amplitude. On Figure 3 we see from curve C that the maximum microseisms oc- curred just after the wind reversal. From curves D and E we observe that during the period when the water wave activity was con- fined to an area near the wave recorder the period of the water waves was closely two times the period of the microseisms. It should also be pointed out that the magnitude of the arbitrary units used as a measure of microseis- mic amplitude on the Orlando records during "Easy" and "How" are the same. It is appar- ent that, although the height of water waves recorded during the two storms is about the same, the amplitude of the microseisms during "How" was five or six times as large as the amplitude during "Easy" and in the case of "How" the amplitude was very outstanding above the normal background. From the above facts one may make the following interpretations: (1) The correlation between one half the period of the waterwave and the period of the microseisms during "How" lends support to the Longuet- Higgins theory- (2) The reversal of wind and the setting up of waves in a direction more or
92 SYMPOSIUM ON MICROSEISMS less in opposition to the waves gen- erated a few hours previously may be a very effective method of producing the necessary standing wave system. One may also refer here to the association of microseisms with cold fronts to support the thought that a relatively sudden reversal of wind over shallow water areas provides a con- dition for microseism generation. Typical weather conditions off the eastern North American coast, prior to the arrival of a cold front, include moderately strong southernly winds. These winds would develop waves travelling in a northerly direction of relatively small amplitude and short period. Following the passage of the cold front the wind direction normally changes abruptly to the northwest. It is reasonable that at some time, shortly after the passage of the front, waves developed by the northwest winds will have periods and wavelengths nearly equal to that of the dying swell from the south. Thus, a standing wave component could exist which would have the potential for excitation of microseisms in ac- cordance with the Longuet-Higgins theory. REFERENCES BANERJI, S. K., Theory of Microseisms. Proc. Indian Acad. Sci., A 1; 727-753, 1935. CARDER, D. C., Earthquake Notes, Vol. 22, Sept. 1951. DARBYSHIRE, J., Identification of Microseismic Activity with Sea Waves. Proc. Roy. Soc., 202A; 439-448, Aug. 7, 1950. DEACON, G. E. R., Relations between Sea Waves and Microseisms. Nature 160; 419-421, 1947. DONN, W. L., Cyclonic Microseisms Generated in the Western North Atlantic Ocean, J. of Meteor. 9; 61-71. Feb. 1952. LONGUET-HlGGINS, M. S., A Theory of the Origin of Microseisms. Phil. Iran. Roy. Soc. A., 243; 1-35, 1950. ocean a second order pressure variation exists which is not essentially influenced by the depth. Moreover, as the frequency of this variation is twice that of the ocean waves and as Bernard had observed that the period of microscisms is roughly half that of sea waves, Longuet- Higgins and Ursell (1948) supposed that this second order effect is the primary cause of some microscisms. The formula obtained by Miche can be derived by a small extension of the theory of gravity waves. Consider the irrotational mo- tion in an incompressible ocean of infinite depth; for simplicity's sake we suppose the movement to be two-dimensional. The horizontal (u) and vertical (w) com- ponents of the velocity are determined by a velocity-potential: u = - Q$/ 9x and w =-di/dz. From the equations of motion 0p Dw dp Du/Dt = - â and = + gp 9x Dt dz where D/Dt = a differentiation following the motion of the fluid, and p = the pressure, we obtain d+ i * - 2 " . + gz = + w 2 and p0 = the constant with q2 = u2 pressure at the free surface. Placing the origin in the undisturbed sur- face the equation of this surface is z = The potential <? has to satisfy the equation of continuity A ^ â 0 and the boundary con- dition Discussion J. G. SCHOLTE Royal Netherlands Meteorological Institute The existence of an unattenuated pressure variation in the ocean was already suspected by Whipple and Lee (1935) and some years later Bernard (1941) also suggested that a standing wave-system produced in some way microscisms, but the well-known expo- nential decrease of gravity waves precluded any understanding of the process. However, in 1942 Miche proved that in the case of standing gravity waves in an incompressible <t> i = â-Sa,sin (kx - J/t) - 'a , or at 9<f> g â + at + 1A q f or z = (D A wave system consisting of two plane progressive waves travelling in opposite di- rections : -kz sin (kx + iÂ»t) ^ e with aj sz a2
CAN SEA WAVES CAUSE MICROSEISMS 98 fulfils A 4> - 0 and satisfies the boundary condition equation 1, to a first approxima- tion if ka i and v 2 = gk. For a second ap- proximation we put # = 0i+i/a2f where d ss d i; neglecting terms of third and higher order in Â£ = a1 cos (kx - vt) + a2 cos ^2 = - âk J al2 cos 2 (kx- vt) + and the pressure = P0 ka we obtain a2 f = a, a2 sin 2 v t . The corresponding surface elevation C - L i + C 2 ,with kx + vt) 2 cos 2 (kx + vt) + cos 2 kx -kz - 2 cos 2 v t. Obviously at large depths(kzÂ»l) the vary- ing part of the pressure is p = 2 Pa,a2j/2 cos 2 j/ t (2) which is the result obtained by Miche for a standing wave system (ax = a2 ). Considering a rather general irrotational movement LONGUET-HIGGINS (1950) was able to generalize equation (2) and to calculate the amplitude of microscisms caused by an arbitrary wave-like motion of the ocean. His final formula (his equation 198) may be in- terpreted in the following (inexact) way. The Miche force of the square X 2, where X = the mean wavelength of the interfering progressive waves, is according to (2) equal to 2 /oaj a2 v 2 X2 If the microscismic amplitude caused by a con- centrated unit force with frequency 2 v at a distance r is denoted by w (2 v , z) the total amplitude will be 2 P a1 &2 W(2 v, z) Supposing the phases of ocean waves at points separated by a distance of a wavelength to be uncorrelated the amplitude generated by a storm with an area A will be of the order , v2 X2 W (2 v, z) With A= I03km2 and X-0.25km( v =%) the vertical amplitude at a distance of 3000 km. appears to be 9.4^, which is of the order of the observed amplitudes. The detailed investiga- tion of Longuet-Higgins shows that this has to be multiplied by a factor which depends on the frequency spectrum of the wave system. For instance, if the energy of the movement is uni- formly divided in every direction within a range of wave lengths between XL and X7 this factor is 2 v2 (a,2 a2 - 2 a, a, cos 2t)e'2kz o i <â¢ the numerical value of this quantity is about 0.54 if X i = 400 meters and X 2 = 154 meters. The vertical amplitude is then 5|i, and the horizontal 3^. This theory undoubtedly explains the phe- nomenon of microscisms in a straightforward way. The only difficulty which it encounters is the fact that microscisms occur very often, while it is a matter of considerable doubt whether standing waves of rather large ampli- tudes are as common. REFERENCES LoNGUET-HicciNS, M. S., and URSELL, F., Sea waves and microseisms. Nature, v. 162, p. 700, 1948. MICHE, M., Mouvements ondulatories de la mer en pro- fondeur eonstante on deeroissante. Ann. Fonts et Chaussees, v. 114, pp. 25-87, 131-164, 270-292, 396- 406, 1944. WHIFFLE, P. J. W., and LEE, A. W., Notes on the theory of microseisms. Mon. Not. Roy. Astr. Soc., Geophys, Suppl., v. 3, pp. 287-297, 1935. Discussion from the Floor Haskell. (Questioning Longuet-Higgins.) Ocean waves are coherent over more than just one wave length, so shouldn't the area of gen- eration be subdivided into areas that are larger than one wave length on a sideâperhaps the wave lengths? (Longu&t-Higgins answered* perhaps so.) Longuet-Higgins. (In answer to Press's ques- tion, "what if the wave periods on the surface occur off the peak of your resonance curve?") The sea waves must be considered as possessing not a single period, say 12 seconds, but a fre- quency spectrum of a certain width, say 8-16 seconds (the pressure fluctuations would then be from 4 to 8 seconds period.) The spectrum of the microseisms should be a combination of the spectrum of the pressure variations and that of a response curve. If the most promin- ent period of the pressure variations occurs off the peak of the resonance curve, the most prom- inent period of the microseisms would be ex- pected to be displaced towards the peak.