Can Sea Waves Cause Microseisms 



75 



Figure 1. Positions of the profile of apro- 

 gressive wave at two different times. 



the total force on A' B Ms I p (0). 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 I p_ (0) ; 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 p ma x 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 wave — Consider 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 



(°) 



(b) 



J L 



(c) 



1 L 



(d) 



Figure 2. Comparison of a standing wave with 

 a swinging pendulum, at four different phases 

 of the motion separated by a quarter of a 

 period. 



