Can Sea Waves Cause Microseisms 



83 



The disturbance W (a, r) ei ot at great 

 distances from the oscillating point source ei at 

 consists of one or more waves of surface type, 



W(o,r) e" 



lOt 



p 9 3 9 (2nr) /2 



i.e. waves spreading out radially in two dimen- 

 sions (see I Section 5.1). Thus 



£C m e 



m m 



L[ot- g, 



r + (m+-)Tc"l 

 4 



(30) 



where t 2 is the density of the elastic medium, 

 |3 2 the velocity of secondary waves in the medi- 

 um, 2jt/£ m is the wavelength of the mth wave 

 and C m 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 Ci, C 2 ; — have been computed for some typ- 

 ical values of the constants: p x (the density of 

 the fluid) =1.0 g./cm J ; c' (velocity of com- 

 pression waves in water) =1.4 km./sec. ; (5 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 V3". The results are shown in figure 7, 

 where C 1 , C 2 , C 3 and C 4 are plotted against 

 ah/p 2 . C 1 , for example, increases to a maxi- 

 mum when oh/Pi = 0.85, i.e. when h = 0.27 x 

 2 jic '/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, C3, and C4 have similar 

 resonance peaks when ah/|3 2 = 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 



2\'/ 2 



W = 



P 2 (S 



2 P2 



5/2 



(2 



— ( z c » 2 ) 



nr) y > V m / 



(31) 



11. Practical examples — We 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 



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 x/\\, and 2 Jt/X 2 , where 

 1 1 and X 2 are the least and greatest wavelengths 

 in the spectrum (see figure 8). This region 

 may be divided into two regions Q 1 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 

 a.i = 1.54 x 10 4 cm., 1 2 = 4.00 x 10 4 cm. and 

 hence Qi = Q 2 = Q 12 = 2.15 x 10" 7 cm. -2 . As- 

 suming A = 1000 km 2 (corresponding to a cir- 

 cular storm area of diameter 17 km.), c 12 = 

 2 jt/13 sec." 1 , a, = a 2 = 3m., h = 3 km. and 

 r = 2,000 km. we find from (29) that 1 5 1 = 

 3.2 x 10- * cm., or S.2\i. The peak-to-trough 

 amplitude of the displacement is 6.5^. 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 



