Below are the first 10 and last 10 pages of uncorrected machine-read text (when available) of this chapter, followed by the top 30 algorithmically extracted key phrases from the chapter as a whole.

Intended to provide our own search engines and external engines with highly rich, chapter-representative searchable text on the opening pages of each chapter.
Because it is UNCORRECTED material, please consider the following text as a useful but insufficient proxy for the authoritative book pages.

Do not use for reproduction, copying, pasting, or reading; exclusively for search engines.

OCR for page 74

CAN SEA WAVES CAUSE MICROSEISMS?
By M. S. Longuet-Higgins
Trinity College at Cambridge
Abstract—This 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.
Introduction—It 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 — their
periodic character and the increase of their
amplitude during a "storm" — 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 pressure—Let
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 wave—Consider any peri-
odic, progressive disturbance which moves, un-
changed in form, with velocity c (see figure 1).
Let p (t)• 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

OCR for page 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 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
(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.

OCR for page 74

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
£mz- P | % C dx + constant (2)
J.
where x is a horizontal coordinate, p is the
density, X is the wavelength, and £ (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
• — BOM of gravity
lo,*r
Rigion of compression waves
• Bottom
Figure 4. Waves in a heavy, compressible fluid.

OCR for page 74

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 waves—The 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 motion—The
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-

OCR for page 74

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 fluid—for 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 verification—The 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.

OCR for page 74

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 fluid—
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 •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

OCR for page 74

80
SYMPOSIUM ON MICROSEISMS
—as 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 [6] ). 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/c»
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 waves—So 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; » which, at
any time, has the same value as £ inside S but is
zero outside. Let A1 be the spectrum function
of £', so that
A'(u,v)ei(ukx + vky
(18)
du dv

OCR for page 74

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 «J
A/ *
A(u,,v.)
sn
(U-UI)TI
« co
sin (V-V.)TC -1(0-0,).
—• -i— 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 x— and y—axis 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)—plane 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 time—the waves in S change gradually—
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
•ill
J-ooJ.
A'A'* du dv
(21)
where a star denotes the conjugate complex
function (I equation [189]). a may be called
the equivalent wave amplitude of the motion.
9. General conditions for fluctuations in the
mean pressure—We 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 = £
and z = 1/2 X K a°d the planes x = ± 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 £' vanishes outside the square S. Now
the expression on the right-hand side is closely
related to the potential energy of the motion
£', and can be simply expressed in terms of
the Fourier spectrum—function A1. In fact
(I Section 3.2)
CO OO
/ / «"d,ay =
- CO - OO
(25)
R( 7t
(A'A'* + A'AL e 2l0t) du dv
-oo - oo

OCR for page 74

SYMPOSIUM ON MICROSEISMS
where Al stands for A' (—u, —v), 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)—plane,
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 movement—In
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
6—4
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-
(2°
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*» /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.

OCR for page 74

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/£ 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 7—have 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 • 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 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
i.e. waves spreading out radially in two dimen-
sions (see I Section 5.1). Thus
«. 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

OCR for page 74

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 30°, and
that its mean direction makes an angle of 10°
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 tests—The 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 making•
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.

OCR for page 74

CAN SEA WAVES CAUSE MICROSEISMS
85
13. Historical notes—It 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.

OCR for page 74

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.

OCR for page 74

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

OCR for page 74

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"

OCR for page 74

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

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".

OCR for page 74

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

OCR for page 74

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 = - — 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
OCR for page 74

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
£ = a1 cos (kx - vt) + a2 cos
^2 = - —k 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(kz»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 <•
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 side—perhaps 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.