Geophys. J . R. astr. SOC.(1969) 17, 63-92.
A Study of Primary and Secondary Microseisms recorded
in Anglesey
J. Darbysbire and E. 0. Okeke
(Received 1968 May 20)
Summary
The paper describes a theoretical investigation of the generation of
primary microseisms which have the same period as the generating sea
waves and secondary microseisms which have half the period, following
the work of Longuet-Higgins and Hasselmann. In particular, the mechanism in both cases by which a high phase velocity pressure wave can be
formed on the sea bottom to resonate with the ground wave, as suggested
by Hasselmann, is investigated. By introducting a damping term in the
elastic equations for the ground movement, quantitative estimates can be
made of the ratio of the wave to microseisms intensity. In the case of
secondary microseisms, the theory is confined to generation in shallow
water and a value has to be assumed for the reflection coefficient of waves
off the coast. In the case of primary microseisms, the ratio increases with
the fifth power of the period, the steepness of the coast and inversely with
the width of the breaker zone.
These theoretical relations are checked with simultaneous observations
of waves on the coast of Anglesey and on weather ships in the North
Atlantic Ocean and microseisms recorded at Menai Bridge. Three types
of microseisms could be identified: (1) primary microseisms generated in
the Atlantic, probably off the coasts of Ireland and Iceland; (2) secondary
microseisms generated in the Atlantic; (3) primary microseisms generated
in the Irish Sea. In the case of (1) and (2), there was agreement with
theory to about one order of magnitude. In the case of the Irish Sea off
the coast of Anglesey, the coast is very steep down to a depth of 5 to 10
fathoms so that a steep coast is presented to short waves of periods less
than 9 s and these tend to be more effective in generating primary microseisms than longer waves. This is borne out by the observations but the
numerical factor is much greater than the one derived theoretically and
as yet no satisfactory explanation can be found for this.
1. Iotroduction
1 . 1 Origin of microseisms
Microseisms are small Earth tremors of amplitude up to lop. There has been
a good deal of discussion about the origin of those with angular frequencies in the
range 0.5 to 2.0 rad s-l but it is now generally agreed that they are mainly caused
by sea waves. An account of previous work is given by Darbyshire (1962).
63
64
J. Darbyshire and E. 0. Okeke
Two conditions must be fulfilled for the microseisms to be energized by sea
waves. Firstly, the wave pressure must still be appreciable at the sea bottom and
secondly as shown by Hasselmann (1963), the wave pressure variation must take the
form of a progressive wave along the sea bottom with a phase speed equal to that
of the ground wave so that resonance can take place.
1 .2 Primary and secondary microseisms
Primary microseisms are defined as microseisms which have the same frequency
as the sea waves which appear to pioduce them. Although their existence had been
suspected for some time e.g. Banerji (1930), it was only definitely established recently
by the work of Oliver (1962), Haubrich et al. (1963), and Hinde & Hatley (1965).
In all the cases considered by these authors, the primary microseisms were found
towards the low frequency end of the wave spectrum, the frequency being usually
less than 0.5 rad s-'.
These microseisms seem to be generated mainly in shallow water where there
would be appreciable pressure variations at the bottom due to wave action. To
obtain a high velocity pressure wave to resonate with the ground wave, would,
however, require the wave energy to be spread over a wide enough range of wave
numbers to include the wave number of the ground seismic wave. Hasselmann
(1963) suggests that this is obtained by modulation of the sea waves as they approach
shallow water. A similar approach will be made in this paper but the width of the
breaker zone where the waves decrease in height and the effect of the imperfect
elasticity of the sea bottom will also be taken into account.
Secondary or double frequency microseisms have double the frequency of the
generating gravity waves and can be generated in either shallow or deep water.
They were observed by Bernard (1941), Deacon (1947) and Darbyshire (1950).
According to the first-order theory of wave motion, the wave pressure disappears
at depths greater than half a wavelength, but Miche (1944) showed that stationary
waves had a non-linear time-dependent second-order term which did not vanish
with depth. Longuet-Higgins (1950) showed that this second-order term can be
generated by the interference of two waves with equal frequencies, but not necessarily
equal amplitudes, moving in opposite directions. Hasselmann showed that when the
two frequencies are not quite equal, a fast moving pressure wave travels along the
bottom and if the frequency difference is just right, the speed will equal that of the
ground wave.
The generation of both primary and secondary microseisms thus depends on the
response of the ground to a travelling pressure wave of high speed and so this will
now be considered in detail.
2. Response of the ground to long pressure waves
2.1 Introduction
The ground is assumed to be a semi-infinite solid medium and any layering
effects will be ignored to simplify the mathematical problem. In this case, it can be
expected that a state of resonance will be obtained when the phase velocity of the
pressure wave equals the speed of Rayleigh waves along the bottom. If c,,, is this
speed, then c,,, = o / k , and as o the angular frequency is assumed to be constant
and equal to that of the sea wave, it follows that the pressure wave must have some
energy corresponding to the wave number k,.
The analysis for working out the velocity of a Rayleigh wave in a semi-infinite
elastic medium is given in many textbooks of seismology, such as Bullen (1963),
p. 89. In this analysis, the conditions of zero pressure at the surface and zero damping
are assumed. We modify these by superimposing on the semi-infinite medium a thin
65
A siudy of primary and secondary miaoseimns
layer of liquid through which the pressure is applied but the thickness of the liquid
is assumed to be too small to affect the solution in any other way. The wave effect
as shown by Bullen decays exponentially with kz where z is the vertical distance
downwards; with sea water, the velocity of sound is 1.5 km s-l so that for an angular
frequency of I-Orads-', k = 1/1.5km-', so this assumption would be justified for
depths less than 0.75 km or 400 fathoms.
Some damping is also introduced into the equations. There are many difficulties
to surmount in producing an adequate theory for the damping of waves in an elastic
solid. The usual assumption following the Kelvin-Voigt hypothesis is that the
damping force is proportional to the time rate of strain. This assumption leads,
however, to the result that the decrement factor is proportional to the square of the
frequency but this is not supported by the experimental evidence. Our interest in the
problem is merely to obtain a realistic estimate of the damping factor. This can be
done more easily by assuming that the damping factor is proportional to the particle
velocity as this leads to the result, largely supported by the experimental evidence,
that the decrement factor is proportional to the frequency. The damping factor can
then be immediately obtained from the experimental observations of the decay of
seismic waves with distance travelled.
2.2 Equations of motion
If ps is the density of the solid medium, pw the density of water, and the subscripts
1, 2, and 3 refer respectively to the x-direction, the y-direction, and the z-direction
taken vertically downwards, we have by modifying the conventional equations as
given by Bullen
+$)
P33=Pex,ik(x-ct)=l($$
+2p($
-*)ax
aZ
+yps$,
(2.1)
where ps y is the damping factor and where the displacement is indicated by u l ,
u2 and u3.
and from equations (2.2) and (2.3)
p,y
dt = -iyp,kc- a4 +iyp,kc- a*
dU3
az
ps y
ax
du1
a4
a*
dt
= iyp, kc - - iyp, kc - .
ax
az
If it assumed that
= Aexp {ik(-rz+x-ct)}
$ = B exp {ik(-sz+x-ct)}
where
ti
J. Darbyshire and E. 0. Okeke
66
Then at the surface when z = 0, and taking
x = ps 2 - 2 1 1
cc = P A 2
(2* 9)
(2.10)
where B is the velocity of compressional waves, B the velocity of distortional waves,
we have on substituting equations (2.5)-(2.10) in equations (2.1) and (2.4) and
dividing throughout by the exponential term
P
- = A(k2-2B2)(-k2 r 2 - k2)+2B2(-k2 r2 A- Bk2 s)-Ayk2 rc- Byk2 c
(2.11)
PS
0 = B2(2k2rA+k 2 B - k 2 s2 B) +yk2 cA - yk2 cs B.
6 = J3B
Let
and
(Poisson's relation)
P
(2.12)
(2.13)
-=p.
(2.14)
p = A{B2(-k 2 r2-k2)-2B2 k2 r2}- 2B2 k2sB -yk2 rc A -yk2 c B
(2.15)
PI
Then equation (2.1 1) becomes
= -Ak2{B2(1 +3r2)+yrc}-Bk2(2f12 S + Y C )
and equation (2.12) becomes
0 = A{2B2r + yc} +B{B2- B2 s2 - ycs}.
(2.16)
(2.17)
Therefore eliminating between equations (2.15) and (2.13)
B = - A{2/32r+yc}/{B2-fi2s2-ycs}
(2.18)
p = Ak2{-B2(1 +3r2)-yrc}+Ak2{2B2 r + ~ ~ } { 2 B ~ s + y ~ } / {S ~~-~Y C- B
S }'(2.19)
Alp = k-2{B2- fi2 s2 - 7 ~ s )
[{-p2(1 + 3 r 2 ) - y r c } { f 1 2 - ~ 2 s 2 - y ~ s }(+2 ~ 'r+yc)(2B2 s+rc)]-'
= k-2{/32- 8' s2
-ycs} { - B4(1 + 3r2)(1-8')
+4p4 rs-yrc/12(1 - s2)
+ y 2 c2rs+2/12ycs+y2 c2+2B2 ycr+fi2(1+3r2)ycs}-'.
Let
A = -B4(l+3r2)(1-s2)+4/14rs.
(2.20)
(2.21)
Let
f(y) = -yrcB2(I-s2)+y2c2rs+2~2ycs+y2c2+2B2ycr+B2(1+3r2)ycs.
(2.22)
Then equation (2.20) becomes
A/p = k - 2 { B2( 1 -s')
- ~ C S{A
} +f (?)}- '.
(2.23)
The equation A = 0, gives the solution for the Rayleigh wave velocity in the simple
case of zero damping and no applied pressure. Similarly it can be shown that
B/p = -k-2(2B2r+y~){A+f(y)}-1.
(2.24)
Therefore the amplitude of the vertical displacement from equations (2.3), (2.7)
and (2.8) is, if u3 now represents the amplitude and the exponential term is divided
out,
- B2 s- ycs)+ ik(2B2r + y ~ ) } p k{- A~+f(y)}-'
uj = { = ipk -
'(f12 s2 r2+ycsr + B2 r +yc){A +f(y)}-'
= ipk-'{B2r(s2+ l)+yc(sr+ l)}{A+f(y)}-'.
(2.25)
67
A study of primary and secondary microseisms
This gives the amplitude of the displacement u3 in terms of that of the applied pressure p ( = P/p,) but to find numerical values it is necessary to find the value of y.
y can be found from the decay of seismic waves with distance when they are
travelling freely. Then p = 0 and from equation (2.20) a finite value of u3 is only
possible with a value of c that makes A+f(y) = 0. Now A = 0 gives a value of
c = 0.92p (Bullen 1963) and the introduction of the f (y) term which is usually much
smaller than the A term will only give a slightly different value for c and this can
thus be obtained from the numerical solution of A+f(y) = 0 by using Newton's
formula. If there is a time decrement term, however, part of the change in the value
of c must be imaginary and this part will give the value of the decrement term to a
first order of approximation. Thus if
A+f(Y)
(2.26)
= F(4
(2.27)
where
F(c+dc) = 0.
If c = 0*92p, then froiii the definitions of
equations (2.21) and (2.22)
F(c) = 0.95 y i p 3 -0.57 7'
= 0.95iys3
I'
and s,
I'
= 0.85i,s = 0.39i and from
b2
(2.28)
ignoring terms in y2.
(2.29)
To obtain F'(c), we expand A+f(y) and as p 4 y, the terms involving y can be
neglected, so we have
P(c) = /?4[(c2 /9-'-2)(2-c'
/P)-4((3c2
8-2-
l)(CZ
1)}*]
(2.30)
(2.31)
= -4-3/3'
at c = 0-928
(2.32)
and as F(c) = -0-95is3y from equation (2.29),
6c =
therefore
0-95iy
= 0.22iy
4.3
(2.33)
and therefore substituting in equation (2.7)
t$
= A expik{-rz+x-ct-0-22iyt}
= A exp -0-22kytexp {ik(--rz+x-ct)}.
(2.34)
Therefore the time decrement term
6 = 0.22yk
(2.35)
and if A is the distance decrement factor,
c l = 6.
(2.36)
As k = o l c and c is independent of the frequency for Rayleigh waves in an unlayered
medium, this result implies that the decrement is proportional to the frequency.
The only results published about the decay of microseisms over large distances
are those of Savarenski et al. (1959) who find a value of 1 = 7 x lO-'km-' which
68
J. Darbyshire and E. 0. Okeke
did not vary much with frequency. For earthquake surface waves, Bullen (1963,
p. 98) suggests 12 = 2 x 10-4km-1. Attewell & Ramana (1966) have summarized a
good deal of results on the transmission of sound through rocks and for Rayleigh
<f < 1 c/s, find that the attenuation coefficient =
waves in the frequency range
5 x lO-'f dB/cm where f is the frequency in c/s. When converted to our units this
corresponds to a A of 9 x 10-4km-' for an angular frequency of l-orads-' and
is of the same order of magnitude as that found for microseisms by Savarenski
et d. It was thus decided to use the value obtained by Savarenski and apply it to
for o = 1.0 rad s-',
waves of 1.0 angular frequency in rad s-'. Thus A = 7 x
B = 2.8 km s-' (as assumed by Longuet-Higgins (1950)), then c = 0.92 p =
2.6 km s-', and by equation (2.36)
s = 2.6 x 7 x 10-4 s - 1 =
1.83 x 10-3
s-1,
(2.38)
and for o = 1-0, k = 242.6 x 2n = 0.40km-'. Therefore by equation (2.37),
A = 0.22 y x 0*#/2.6 = 0.34 7,
but if A = 7 x
km-', then
7 = 2.05 x
lo-' km s-'.
(2.39)
2.3 The response curve
Once y is known, it is possible to calculate the response for various values of c/p.
p = P/p, by equation (2.14) and if the surface wave amplitude is a, then
for a shallow depth, the wave bottom pressure amplitude
P = gap,
and so p = gapw/p,
(2.40)
then from equations (2.25) and (2.40),
(2.41)
a
and from equations (2.25) and (2.14)
2
= i { / 3 z r ( ~ 2 + 1 ) + ~ ~ ( ~ + 1 ) } / p , k(7)).
{A+f
P
(2.42)
The values shown in Table 1 are based on equation (2.41). The values in the third
~ plotted in Fig. 1. It will be seen that
column which are proportional to ( u ~ / u )are
the value is maximum at c/B = 0-919 and drops to approximately 1/10 of this value
for c/p = 0.915 and 0.925 and so the effective range of c//3 must be within 0.5 per
cent of the undamped value of 0.92.
Table 1
Variation of response with wave velocity
ClB
0.80
0.85
0.88
0.91
0.915
0.917
0919
0.922
0.925
kps USIBP~
0
(kps ualgpw 0)'
(lul-2
s2)
0.259
0.393
0.632
2.330
4.640
7.400
12.670
6.960
3,610
(lul-4
s4)
0.067
0.154
0.399
5.460
21.530
54.900
160.580
48.500
13.040
A siudy of primary and seeoadary microseismS
69
(a) Amplitude
151
(b) Energy
FIG.1. Variation of intensity with c / j .
2.4 Application to a power spectrum of waves
If we have a monochromatic wave train a sin (or-k .x), then the variance or
the power is given by +a2. When the wave energy is by some mechanism (which may
be frequency-dependent), spread over a range of wave numbers, then, if there is no
loss of energy, the total power is still &.z2 but the power per unit wave number is now
taken to be
3a2Sa(k),
at this frequency and
f
+a2Sa(k), dk = +a2
-W
so that
W
.
J Sa(k),dk
-W
= 1.
(2.43)
70
J. Darbyshire and E. 0. Okeke
As w is taken to be constant, the response curve can be plotted against k = w/c
rather than c/s and if the response expressed as a function of k is R(k), then the
total power transmitted
1
00
+a’ Sa(k),,R(k)dk.
P(u,) =
(2.46)
-0)
As R(k) is only appreciable in the range c = 0.915 to 0.925, corresponding to a 1
per cent change in c and k (as k = w/c), the integral can be approximated to a first
order by taking the product of the mean values of Sa(k), and R(k) over this interval.
Let c, = 0.928 (equation (2.47)) and k, = w/c, (equation (2.48)), then the change
in k
= 6k = 0.01 k,
(2.49)
and so
(2.50)
P(u3) = 0.01 k,{SaO,}
{Rm}.
The variation in Sa(k), will be slight so that
(2.51)
but
R(k) = +R(k,) approximately,
(2.52)
where R(k,) is the maximum value of R(k), and = 160.58 from Table 1. Thus
P(u3) =
*’
k,-’ ~ , - ~ ( 8 0g2
- 3pw2)(0.01k,){Sa(k,),}
= +a2k,-’ ps-2{0*803g2 pw2Sa(k,,,),} (km, s units).
(2.53)
As P(u,) gives the power of an oscillation of constant frequency and with a variation
in wave number of only 1 per cent it can be regarded as the power of a monochromatic wave train u3 sin (wt- k, x) and thus it is equal to 4.~4~’. Hence
(2.54)
u3’ = 0.803 g2 pw2k,-’ pS-’ a’ Sa(k,), (km, s units).
In general, however, we do not have monochromatic wave trains but a wave system
covering a wide range of frequencies and represented by a frequency power spectrum
Sa(w) where Sa(w) is the power per unit frequency. Similarly in the case of u3 we
) so to generalize we have:
have S U ~ ( Wand
su3(0)
-- - 0.803 g2 pw2 pS-’ k,-’ Sa(k,), (km,s units).
Sdw)
(2.55)
3. The generation of secondary microseisms in shallow water
The Longuet-Higgins theory is adapted to the case of two progressive waves of
unequal amplitude and nearly equal frequencies moving in opposite directions
a, cos (wlt - k , x) and a, cos (0, t + k , x). The surface elevation is given by qo
and the elevation at any depth z is given by q
= U, cos (01 t - k , x)+u,
cos (
t+
2 k , x).
(3.1)
Now the total vertical momentum of a column of water of cross-sectional area
dx dy is, if the depth is h:
~
h
The force on the bottom will be given by the time derivative of this expression and
the pressure by dividing this by dxdy. The first term represents the first-order contribution and the oscillatory part of it vanishes as the depth h gets very large, so we
71
A study of primary and secondary microseisms
shall only consider the second term. Again we are concerned with an average taken
over a large distance much greater than Ilk, or l/k2. The second term is proportional
to:
qorjo = (a1 cos (0,t-k, x)+a, cos ( 0 2 t + k , x)}
{-w, a, sin (0,t-k, x)--wz
sin2(0, t--k,x)-+o,azZ
=
a2 sin (02t + k 2 x)}
sin2(co2t+k,x)
-~2a,a,cos(o,t--k,x)sin(02t+k2x)
a, a, sin (0, t-k, x) cos (a2t+k,x).
(3.3)
The first two terms represent waves of double the primary frequency and will average
out over a wavelength. The last two terms give:
-0,
-+uza, u2[sin ((q+ m y ) t+ (k, -k,) x} - sin {(a,
- 02)t - (k,-k,)x}]
-30,
a, a2[sin ( ( 0 ,+a,) t +
(k,-kl)x}+sin
( ( 0 ,-02)t-
(k2+k,) x}].
The terms involving (k,+k2)x have a short wavelength and would also average out
over a large distance, so we are left with
Vorjo =
-!L~a,a,b2
sin{(o,+o,)t+(kz-k,)x}
+w, sin ( ( 0 ,+a2)t + (k,-kl) x}]. (3.4)
Then the bottom pressure meaned over a large distance is given by:
= Pcos((o,+02)t-(k,-k2)x},
where
P
= -+pw(O,
+0,)2a,
a,.
(3 * 6 )
(3.7)
From equation (3.7)it follows that
+P2 = + p W 2 ( 0 l+ w ~ ) ~ + I ~ ~ + u ~ ~ .
(3.8)
If we extend equation (3.8) to waves covering a wide band of frequencies and use
power spectra, then we have:
+~0,)~Su,(o)do Sa,(o) do.
S P ( w ) du = +pw2(ol
(3.9)
Resonance will be obtained if the phase velocity of the modulated wave (or the
modulation velocity), 20/(k, -k,) equals the velocity of Rayleigh waves along the
surface of the ground. If /? is taken to be 2-8 km s-', then as c,,, = 0.928,
c,,, = 2.6 km s-' and k = 02/g for gravity waves in deep water, the condition for
resonance becomes, using km, s units:
(3.10)
and so
01-02
= 0 b = -
9
2.6 '
(3.11)
As in equation (2.49), the permissible change in the value of 2-6km s-' is only
f0.5 per cent so the range allowed in cob is
60, = 0.01 wb.
(3.11)
72
J. Darbydh and E. 0. Okeke
We shall only consider microseisms caused by reflection of swell off the coast,
so that the initial assumption of a shallow layer of water (shallow in the seismic
sense but deep as far as gravity water waves are concerned), still holds. To simplify
matters further, the reflection is assumed to take place at normal incidence. Thus
if Sa(w) is the power spectrum function of the incident waves, R2Sa(w) will be
that of the reflected waves, R being the reflection coefficient.
To estimate the power used to produce microseisms, then, two frequency bands
of width Am are taken from both the incident and reflected wave spectra, the extreme
limits of the band on one spectrum being separated by a frequency difference of
g/c, from the corresponding ones on the other spectrum. The bands are sub-divided
into n divisions of width 60,. If the values of the power for each sub-division on the
incident set are given by h,, h,, ..., hn, and the corresponding ones on the reflected
set by g l , g , , ..., g., then the contribution to the power of the pressure wave that is
n
in resonance is proportional to C h , g , as products with different subscripts will
1
not contribute as the frequency difference would not be right. If the h values are
all equal, then
h = Sa(U)dwb
(3.12)
and
(3.13)
g = hR2.
Thus from equation (3.8), and assuming 2 0 = (w,+w,)
S P ( 0 ) . 2A0 = 8pw2w4R2[sa(w)]2(6wb)2.
n
= 8pw2w4[Sa(w)]zAw60, R2.
(3.14)
2Aw is used on the left-hand side of equation (3.14) as the gravity wave energy
contained within the frequency interval Am is spread over an interval 2Aw in the
pressure wave. From equation (3.11)
(
SP(w) = 4pw2w4[Sa(w)l2RZ )gi:O
-
(3.15)
From equations (2.40)-(2.42),
-u3
=-
p
u3
agpw
and
u3, - u3,
P2 a 2 g 2 p w 2
and thus at resonance, from Table 1 (using km, s units)
Su3(0) = 0.01 SP(0){80*29k,,,-,p,-,}
(3.16)
and therefore from equations (3.15) and (3.16)
(3.17)
SU,(O) = 1.23 pw2 p S - , k,,,-,R2 g w4[Sa(o)l2 (km,s units).
Assuming p w / p , = 1/2.7 and g = 0.01 km s-,, k,,,= 2w/2.6 km-', equation (3.17)
simplifies to :
Su3(w)= 0.0028 w 2R2[Sa(w)12(km, s units).
(3.18)
R, the reflection coefficient, is very difficult to determine, but work by Savarenski
et al. (1958) at Lake Yussi-Kul showed that it was about 1/30 for a steep rocky coast.
4. The generation of primary microseisms
4.1 Derivation of wave number spectrum for waves approaching the shore
The wave bottom pressure becomes measurable when the depth gets less than
half the wavelength and then increases as the depth decreases. When the waves
start to break, however, this increase will be halted and the wave pressure variation
A
study of primary and mxmdary microseisms
73
will start to decrease until it becomes zero at the water's edge. At a depth h, with a
wave number k,, the pressure is proportional to sech k, h but it can be shown empirically that this is closely approximated by exp -0.08 k, h or if the beach gradient
is constant and equal to a', exp -0.8 a' ko x where x is the distance from the shore.
It will, however, be more convenient to take the point where the breaking starts as
the origin and a unit amplitude wave train will be assumed to approach the shore
from the right. For the breaker zone to the left of the origin, then, the wave bottom
pressure will be assumed to vary as 1 +x/d where d is the width of the zone, and to
the right of the origin, the formula given above will still be assumed to apply.
-I
9Pw
- (exp -ax)
=
cos ( ot +ko x) for x > 0, a = 0.8 a'ko,
(l++)cos(ot+kox)
(4.1)
for - d < x < ~ .
The cosine term can be expanded to:
cos k, x cos ot- sin k, x sin ot
and the Fourier transforms of these terms with respect to k can be found so we
have :
p33
- - {gl(k)dk cos kx+g,(k)dk sin kx} cos wt
SPW
+ {h,(k)dk cos kx +h,(k) dk sin kx} sin ot,
where
1( I + +)
J ( + 5)
0
2ng1(k) =
m
cos ko x sin kx dx
1
2nh1(k) =
s
(1
-d
+ +) sin ko x cos kxdx+
S, + +)
0
2nh2(k) =
-
(1
sin ko x sin k x d x +
On integration we have:
2ng1(k) = a{a' + (ko-k)'}-'
exp (-ax) cos ko x sin kx dx
0
-d
0
exp(-ax)cosk,xcoskxdx
0
0
2ng2(k) =
i
+s
J'
i
cosk,xcoskxdx+
-d
(4.2)
exp (-ax) sin ko x cos kxdx
0
exp (-ax) sin ko x sin kxdx.
0
+a{a2 + (k,+k)'}-'
+{1-cosd(ko-k)}{d(ko-k)2}-'+{1 -cosd(ko+k)}(d(ko+k)'})-'
+ (ko-k)'}-' + (k,+k){a2 + (k,+k)'}-' + (ko-k)-'
- (ko+k)-' - {sin d(ko-k} {d(ko-k)'}-' + {sind(ko+k)} {d(ko+k)'}-'
2nh1(k) = (ko-k){a2 + (k0-k)'}-' + (ko+k){a2+ (k,+k)'}-' - (k,-k)-'
2ng,(k) = - (ko-k){a2
- (ko+k)~'+{sind(ko-k)}{d(ko-k)2}~'+{sind(ko+k)}~d(ko+k)2~~'
(4.9)
2nh2(k) = a{a2+ (k0-k)'}-' --{a2+ (k,+k)'}-'
+ { 1-COS
d(k0 -k)} {d(ko -k)'}-'
- { 1 -cos d(k0 + k)} {d(ko+k)'}-'.
(4.10)
74
J. Darbyshire and E. 0. Okeke
These terms give the amplitude spectral densities. To obtain the power density
with respect to wave number, Sa(k),, we use a similar argument to that used in
Section 2.4. The power or variance of a unit amplitude wave train cos (ot+k, x )
is 3 but this is spread over a range of wave numbers with no loss of power. In the
range dk the power is shared between four uncorrelated stationary wave systems of
amplitudes gl(k)dk, g2(k)dk, h,(k)dk, h2(k)dk. The total energy must be proportional to
{gl(~)d~}2+{g2(k)d~}2+{hl(k)dk}2 +{h2(k)dkl2
and so
Sa(k),dk = const dk[{gl(k)}2+{g2(k)}Z+{hl(k)}2+{h2(k)}Zldk (4.11)
W
Sa(k),,,dk =
-m
const dk = #
=
constdk
/i
J’
[{g,(k)}2+{g2(k)}2+{h1(k)}2+{hz(k)}21d~
-m
[{91(k)}2+{g2(k)}2+{hl(k)}2+{h2(k)}21dk
-w
and so from equation (4.11) therefore
The integral in the denominator comes out to be:
In the cases considered in this paper, k, varies from 20 to 100 km-’, a‘ is about
0.01, k, varies for the ground waves from 0.1 to 0.4 km-I and so a < k, and also
k, 4 k, and so in these cases the formulae (4.7)-(4. lo), for g,(k), g2(k), h,(k)
and h2(k)reduce to:
2
27rg,(k) = 7
k0
+ 2(1-coskOd)
dko2
2nh1(k) =
2(sin k , d )
dkO2
h,(k) = 0.
(4.13)
(4.15)
(4.16)
Hence from equations (4.12) and (4.12a)
a
S@)<U =
(l-coskod))2+ ((sink,d)]2]
dko2
ko2d
D
(4.17)
We are, however, concerned only with progressive waves on the ground which reach
the seismograph station beyond the shore line, so only positive values of k matter.
Hence only half of the power represented by Sa(k), is used. Let
(4.18)
A study of primary and secondary miamisms
75
This simplifies to:
Ha(k), = ~ { ~ 2 d 2 + 2 ( 1 - c o s k , d ) ( l + a dD-'
)} ko-4d-2
or expressing it in full:
If ud
Q
1 as is often the case, this simplifies to:
Ha(k), = 0-065(1-cos k, d ) u2d-' k,-4
= 0~042(1-cosk,d)a'2d-'k,-2.
(4.21)
(4.22)
This shows that the pressure effect in range dk is inversely proportional to kO2or w4.
This contrasts with the pressure due to wave interference given by equation (3.18).
The primary pressure effect also increases with the slope of the coast and decreases
as the breaking zone width increases.
If the power spectrum function of the waves with respect to frequency is Sa(o),
then we have from equation (2.55)
-su3(0)
Sa(w)
- 0.803{Ha(k),}
g2 pw2pS-'
k,-
(km,s units).
(4.23)
Because of the cosine term in the numerator in equation (4.20), Su,(o) should tend
to oscillate as k,d goes through multiples of and as k, is a function of o,the
power spectrum will oscillate also. Thus if 6ko is the change in k, between two
successive maxima,
Skod = 271, but as k,, = 0 2 / g , dk, = 2w6w/g
and thus
d = Rg/Q60,
(4.24)
where 6w is the frequency difference between successive maxima.
4.2 Primary microseisms generated in the Irish Sea
Equation (4.23) shows that the generation of primary microseisms is favoured
by long wave periods (low k,) values, steep coasts, and a narrow breaking zone.
For microseisms generated in the Irish Sea and recorded at Anglesey, as described
in Section 5 and shown in Fig. 2, the first two conditions are not often present as
the Irish Sea is relatively flat and shallow. Waves with periods of 7 s or less, however,
do not affect the bottom at depths greater than about 30m and so we are only concerned then with the coastal gradient down to this depth. The gradient in this case
is often quite large and can be of the order of 0.01. To investigate this matter more
fully, it was decided to construct wave refraction diagrams for waves approaching
Anglesey from the SW with periods of 20, 15, 12, 9 and 7s. Those for 20, 12 and
7s are shown in Figs 3-5. The relative wave heights were obtained by taking them
to be inversely proportional to Wc, where W is the orthogonal spacing and c, the
wave group velocity. The wave pressure would be inversely proportional at the
sea bottom to Wc, cosh k, h and this was calculated from point to point. The effect
of bottom friction, assuming a coefficient 0.01 with a quadratic velocity law, was
negligible. The curves shown in Fig. 6 for 7,9, 12, 15 and 20 s giving relative pressure
variations against distance from the coast were then obtained. It was assumed that
76
J. Darbyshire and E. 0. Okeke
CAERNARVON
CAERNARVONSH IRE
BAY
0 Position of
X Position of
seismograph
wave recorder
FIG.2. Map of Anglesey showing position of wave recorder and seismograph.
FIG. 3. Wave refraction diagram for 20 s waves entering the Irish Sea from the
south-west direction.
77
A study of primary and secoadary microseisms
4'
I
I
I
FIO.4. Wave refraction diagram for 12 s waves entering Irish Sea from the south-west.
FIO.5. Wave refraction diagram for 7 s waves entering Irish Sea from the southwest.
78
-10
9s
-8
-6
6
15 s
4
2
0
I
< , ,
400
,
,
,
300
,
.
,
I
,
200
Miles
I
I
I
I
,
I
I
I
.
100
.
10
0
FIG. 6. Variation of wave bottom pressure with distance from west coast of
Anglesey, for 7, 9, 12 and 20 s waves.
waves of all periods broke at a depth of 2 m which would be approximately 0.33 km
from the shore. It is rather difficult to fit these variations into the exp-ax and
1 + x / d formulae used in equation (4.1) but the closest approximation was obtained
by using the values for a and d shown in the fourth and fifth columns of Table 2.
With these values and using equation (4.23), the response values shown in the
second and third columns of Table 2 would be obtained to an applied power spectrum
of uniform spectral density.
5. Comparison of theory with-observations
5 . 1 Recording of waves and microseisms in]Anglesey
A horizontal and vertical seismograph of the type described by Tucker (1958)
was installed in the grounds of the Marine Science Laboratories, Menai Bridge,
Anglesey. Because of the rocky nature of the ground, it was not possible to dig
79
A study of prhnary and secoadary microseiSms
Table 2
Variation of response with wave period for waves approaching the coast of Anglesey
from the south-west
Period
6)
5
6
7
9
12
15
20
Power
response
x 10- l4
Linear.
response
~10-7
2.75
4.13
1a 6 4
2.15
0,173
0.214
0.134
d
7.6
17.1
2.68
4.61
0.0302
0.0745
0.0180
a
(W
2.5
1 *6
0.33
0-33
0.33
0.33
0.33
0.33
560.0
0.40
0.18
0.0042
0.0134
0.0134
foundations for the seismograph hut any deeper than two feet but with good thermal
insulation and extra precautions to keep the mains voltage constant, very satisfactory records were obtained.
Corresponding wave records were obtained by using an N.I.O. frequency modulation type wave pressure recorder, described by Harris & Tucker (1963) just below
the low water line at Rhosneigr beach (see Fig. 2). Whenever possible, simultaneous
records of waves and microseisms were taken. Records of waves in the Atlantic,
taken by weather ships with a ship-borne wave-recorder at either station Juliett,
India or Alpha were obtained from the National Institute of Oceanography.
All the records were eventually obtained on punched tapes and from them a
computer worked out the power frequency spectrum using a programme based on
the work of Blackman & Tukey (1958). The spectra are shown in Figs 7-13.
5.2 Types of microseisms
The microseisms shown on the spectra could be due to four different causes.
(1) Primary microseisms caused by Atlantic waves being reflected off steep coasts
such as those of Ireland and Iceland.
(2) Secondary microseisms caused by interference of waves in the Atlantic in
storm areas or by reflection off the coasts of Ireland and Iceland.
16.11.66
Id
0
I
I
01
I
1
I
0.2
,
0.3
I
1
0.4
I
Frequency (s-'I
FIG.7. Spectra of microseisms recorded at Menai Bridge and waves recorded in
the Atlantic Ocean, 1966 November 16. Broken line, waves. Full line, microseism. Frequency in s-
80
J. Darbysbire and E. 0. Okeke
19.1.67
1100
Frequency (s-')
FIG.8. (a) Spectra of microseisms recorded at Menai Bridge and waves recorded
in the Atlantic Ocean, 1967 January 19. (b) Spectra of microseisms recorded at
Menai Bridge and waves recorded at Rhosneigr, 1967 January 19. Broken line.
waves. Full line, microseism. Frequency in s-'.
(3) Secondary microseisms caused by waves in the Irish Sea.
(4) Primary microseisms caused by waves in the Irish Sea.
There seems to be very little evidence of microseisms of type 4 on the spectra
but there is some evidence of the presence of the other types. Those of type 1 stand
out because of their relatively high period but types 2 and 3 tend to cover the same
range of periods and are often difficult to distinguish from one another but there are
some examples, particularly those of 1966 September 5 to 6 where this is possible.
5.3 Examples of microseism generation
Several examples are shown in Figs 7-13 and in most cases simultaneous spectra
of microseisms, Atlantic waves as recorded at the weather ship stations, and Irish
Sea waves as recorded at Rhosneigr are shown. The power density per unit frequency in m2 (c/s)-' for waves and p2 (c/s)-' for microseisms is shown on the
81
A study of primary and aeamdmy microseisms
lo
1
0
21.2.67
0.2
0.1
03
0.4
12 00
4
10
,
0
0.1
m
l
0.2
n
,
0.3
9
,
8
0.4
Frequency(s-')
FIG.9. (a) Spectra of microseisms recorded at Menai Bridge and waves recorded
in the Atlantic Ocean, 1967 February 21. (b) Spectra of dcroseisms recorded at
Menai Bridge and waves recorded at Rhosneigr, 1967 February 21. Broken line,
waves. Full line, microseisrn. Frequency in s-'.
vertical scale. Frequency in c/s on the horizontal scale. The wave spectra are shown
by the dashed curves. For the examples of 1966 November, 1967 February, and
1967 March, the Atlantic waves were recorded at station Juliett (52+'", 20"W)
and for 1967 January, at station A (61.8"N, 19"W).
The Atlantic weather charts for all these times are shown in Figs 14 and 15. In
the case of all the examples but those for 1966 September, the original records were
split into 1100 equally spaced observations and the power spectrum estimates were
computed with 130 lags. This gave 17 degrees of freedom which allows a 95 per cent
confidence range of from 0-62to 1.80 times the observed value.
The records for 1966 September varied somewhat in length and so the number
of lags used in the power spectrum analysis varied between 80 and 140 so that there
were about 10 degrees of freedom corresponding to a 95 per cent confidence range
of 0.35 to 2.2 of the observed value.
The various types of microseisms will be considered in turn.
5.3.1 Primary microseisms caused by Atlantic waves
(a) 1966 November 16 (Figs 7 rmd 14). The Atlantic waves show a marked peak
at 0-05s-' frequency. There is a corresponding maximum at the same fre uency
on the microseism spectrum the ratio of the power densities being 1.8 x 10-
x.
82
J. Ikrbysbhapnd E. 0. Okeke
I
21.3.67
\
'Ol
I
12 00
FIG. 10. (a) Spectra of microseisms recorded at Menai Bridge and waves recorded
in the Atlantic Ocean, 1967 March 21. (b) Spectra of microseisms recorded at
Menai Bridge and waves recorded at Rhosneigr, 1967 March 21. Broken line,
waves. Full line, microseism. Frequency in s-'.
(b) 1967 January 19 (Figs 8 and 14). In this case the waves were recorded at
station A which is south of Greenland. A fast strong storm had passed over this
area during January 18-19 westwards towards Iceland and the north of Scandinavia.
Both the wave and the microseism spectra show a peak at 0.077s-', the ratio of
the power densities being 1 . 2 0 ~
(c) 1967 February 21 (Figs 9 and 14). The ocean waves were caused in this
example by a storm WSW of the British Isles. The ocean wave and microseism
Comparison
spectra both show a peak at 0464s-', the ratio being 7 . 0 ~
with the Irish Sea wave spectrum shows that there is a subsidiary maximum at
0-064s- ' here also due to the penetration of Atlantic swell.
(d) 1967 March 20-21 (Figs 10 and 14). The waves in this example were caused
by a storm to the north-west of the British Isles and were much lower in height than
in the other examples. The ocean wave spectrum shows a maximum at 0-062s-'
frequency and the microseism spectrum two significant peaks at 0-058 and 0.073 s-'.
The power ratio being 4.2 x
The Atlantic swell has also penetrated into the Irish Sea and Fig. 1O(b) shows the
microseism spectrum one hour later than before compared with the corresponding
Irish Sea wave spectrum. There is now a peak at 0*073s-' on both spectra.
A study of primary and seeaadary microseisms
5.9.66
0.5
0
"1
I100 I
18 00
83
2300
A
0.05
040
0.15
-
o
Fraqurncy (s-')
FIG.i l . Spectra of waves recorded at weather ship station India, 1966 September
5-6.
5 . 3 . 2 Secondary niicroseisnis caused by Atlantic waues
(a) 1966 Nouernber 16 (Figs 7 and 14). The Irish Sea spectra are not available
in this case and the ratio between the frequencies of the main peaks for microseisms
and waves is rather high, being 2.8.
(b) 1967 January 19 (Figs 8 and 14). The main peak on the wave spectrum is at
0.074 s-' and on the microseism spectrum 0-139s-' in good agreement with the
The main
wave interference theory. The power density ratio is 1 0 . 4 ~lo-".
microseism secondary peak is 100 times greater than the subsidiary primary peak.
(c) 1967 February 21 (Figs 9 and 14). In this example the ratio between the periods
of the main peaks on ocean wave and microseism spectra, 0.064 and 0-139s-' is
The main microseism peak is
2.18. The ratio of the power densities is 1 . 9 ~
a4
J. Darbysbire and E; 0. Okeke
I
1
r'
FIG.12(a). Spectra of microseisms recorded at Menai Bridge, 1966 September 5.
Vertical
2.d
3
Microseisms
6.9.66
--
N-S
-
1
0
0300
5
8
or1
-
62
'
Frequency Is-')
FIO.12(b). Spectra of microseisms recorded at Menai Bridge, 1966 September 6.
A
85
study of primary and seunuhy miaoseismS
-0
5
0.10-
4
3
v1
P
$
$
e ! -
c
c0.05-
v1
2
1
0
Frequency (s-?)
FIG.13. Comparison of observed spectra for waves recorded at Rhosneigr and
microseisms recorded at Menai Bridge 1966 &ptembex 5-6.
30 times greater than the subsidiary primary peak. It is not possible to rule out
completely the effect of the Irish Sea waves however, as they would give primary
microseisms with a peak at 0.147 s-' if allowance is made for the response to different
periods as shown in Table 2. This would, however, give a much broader distribution
than is actually observed. It may be significant in this example that the storm is from
WSW where the effect of refraction of the microseisms (Darbyshire & Darbyshire
1957) causes convergence of the microseism wave rays rather than divergence as in
the case of NW storms.
(d) 1967 March 20-21 (Figs 10 and 14). The correspondence between the main
peaks of the ocean wave and microseism spectra is not very good, the two frequencies
being 0.062 and 0.154s-' a ratio of 2.5. However, the frequency at the weather
ship station may not be typical of the swell nearer the coast as the Irish Sea spectrum
shows a subsidiary maximum at 0.075s-' due to extraneous swell, and which is
much nearer half the microseism frequency.
5.3.3 Secondary microseisms caused by Irish Sea waves. As some of the Atlantic
swell penetrates into the Irish Sea as shown by the examples of 21.2.67 and 21.3.67
the possibility exists that this extraneous swell causes the 0.139 and 0.154s-' frequency microseisms observed. This effect does not seem very likely, however, as
from equation (3.18) the intensity of secondary microseisms increases with the
square of the frequency of the waves generating them and one would thus expect
secondary microseisms, corresponding to the local waves 0.14-0.20 s-' frequency.
These do not appear to exist at any appreciable intensity.
86
J. Darbyshire and E. 0. Okeke
15.11.66
16.11.66
19.11 .67
20.2.67
.21 .2’ -67
21.3 -67
FIG.14. 1200 G.M.T.meteorological charts for The North Atlantic Ocean, 1966
November 15-16, 1967 January 19, February 20 and 21, and March 21.
A study of primary and secondary microseisms
3.9.66
4.9.66
5.9.66
6 . 9 . 66
87
FIG. 15. 1200 G.M.T.meteorological charts for the North Atlantic Ocean, 1966
September 3-6.
5 . 3 . 4 Primary microseisms caused by Irish sea waves
(a) 1967 March 21 (Figs 10 and 14). If allowance is made for the response factors
given in Table 2, then the Irish Sea wave spectra fit very well with that of the microseisms. The correspondence between the North Atlantic wave spectra and the microseism spectra is not very good in this case as has been already stated. If the microseisms were caused by Irish Sea waves, the ratio of the wave and microseism power
spectrum densities is 1 x lo-" which is much higher than the value of given by Table
3 for waves of this period.
(b) 1966 September 5-6 (Figs 11-14). This appears to be a very good example
as an intense storm of tropical origin was moving very fast across the Atlantic and
reached the position of station India at about 1966 September 5, 21b00giving immediately a sharp rise in the wave amplitude and period as shown in Figs 12, 15 and
16. Although these waves were recorded by a weather ship at the India position,
from the track of the storm one could expect similar waves to arrive at about the
same time as at the west coast of Ireland. The microseisms (Fig. 16) however, do
not follow the trend of the waves very well but just show a gradual rise until 1966
September 6, 09b00by which time the Atlantic waves had decreased considerably in
height. Moreover a plot of half the Atlantic dominant wave period values against the
microseism periods shows (Fig. 17), very little correspondence and the ratio is considerably more than 2.0. Unfortunately no record of Irish Sea waves are available
88
J. Darbysbire and E. 0. Okeke
-------
5
I\\
I
\\
I
\
I
I
I
Atlantic waves
N-S
microseisms
-.-.-.-.-.-.- Square of
predicted
wave height
\
\
\
\
'\
'\
'X
0'
I
I
06
24
18
5.9.66
1
t
12hours
6.9.66
Fro. 16. Variation against time of maximum value of power spectrum density on
spectra for Atlantic waves and N-S microseisms and square of predicted wave
heights for Irish Sea, 1966 September 5-6.
107
-0
In
g
5-
------.-.-.-.-.0'
1
Microseism period
Half related wave period
Predicted wave period
1
18
I
24
5.9.66
I
I
06
i
l2hours
6.9.66
FIG. 17. Plot of Atlantic wave period, Irish Sea predicted wave period, and
microseism period, 1966 Scptember 5-6.
A study of primary and secondary microseisms
89
until 1966 September 6, 09"00 but predictions from the wind speeds and directions
recorded at Valley, Anglesey (see Table 3), were obtained of the maximum wave
heights and dominant wave periods by the method given by Darbyshire & Draper
(1963). The tropical storm passed through the north of the British Isles and its
only effect on the Valley records was to change the direction of the Force 6-7 wind
from S to SSW and thereby greatly increasing the fetch. The predicted (heights)2
are shown in Fig. 16 also. Clearly the correspondence in each case is much better.
The predicted periods are shown in Fig. 17 and again show much better agreement
with the microseism periods. The one observed wave period available, that for 1966
September 6,09"00 agrees very well with both the predicted value and the microseism
period.
The observed spectra for the Atlantic waves and for microseisms are shown in
Figs 11 and 12. A comparison of the observed spectrum for Irish Sea waves at 1966
September 6,09"00 and microseisms at 1966 September 6, 08h00is shown in Fig. 13.
There is very good correspondence at the high frequency end of the spectrum, the
ratio of the spectral densities coming out to be 4.2 x lo-" which, as in the last
example, is much higher than the predicted ratio. The wave spectrum shows that a
considerable amount of 0.05 s-l swell from the Atlantic storm had penetrated into
the Irish Sea but this does not produce any microseisms of this frequency and confirms the low response factor for waves of this frequency given in Table 2. It is
possible that the subsidiary peak at 0.109 s-l is due to this swell causing secondary
microseisms in the Irish Sea, or more likely to the west coast of Ireland.
As given by equation (4.24), the power spectra of the microseisms should show
a periodic variation with the frequency. If d is the width of the breaker zone and
60 the angular frequency difference between successive maxima on the spectra
Table 3
Wind records, R.A.F. Valley, Anglesey
1966 September 5
12
13
14
15
16
17
18
19
20
21
22
23
24
Speed (knots)
Direction x 10"
28
31
31
32
33
30
31
31
30
31
31
28
27
20
19
19
20
20
19
20
20
21
21
21
21
22
1966 September 6
01
02
03
Speed (knots)
Direction X 10"
23
25
23
28
25
24
23
24
23
19
19
20
23
23
22
23
25
24
24
25
25
23
23
23
04
05
06
07
08
09
10
11
12
90
J. Darbyshire and E. 0. Okeke
^1
7
I *34
1
I
I
18
21
24
5.9.66
03
OGhaurs
6.9.66
F I ~18.
. Variation of spectral densities with time.
Points corresponding to maxima are shown in Fig. 18 for the NS spectra from 1966
September 5, 18WO to 1966 September 6, 06h00. The periods corresponding to the
maxima on the spectra for 18h, 21h,ooh, 03h and 06" are shown and show very little
variation until 03h00,the time of high tide, when there could be a change in d. The
mean value of ST the period between the maxima is 0-375s and if T = 5-5 s, this
gives d = 0.35 km which is just the distance of the one fathom contour from the
water-line.
6. Conclusions
The power density ratios for primary and secondary microseisms given in the
frequency examples are tabulated in Table 4. Predicted ratios based on equations
(3.18) and (4.23) are also tabulated. In the case of the secondary microseisms, the
reflection coefficient R was assumed to be 1/30. In the case of primary microseisms,
generated in the Atlantic, the coastal gradient was assumed to be 0.01 and the width
of the break zone, 1 km. For primary microseisms thought to be generated in the
Irish Sea, the response values given in Table 2 were used. Table 4 shows that within
the uncertainty of the data, the theory developed for the generation of microseisms
by a resonance mechanism, assuming damping gives good quantitative agreement
Table 4
Comparison of predicted and observed values of Su,(o)/Sa(o)
Date
16.11.66
19.1.67
21.2.67
21.3.67
5-6.9.66
Atlantic primary
Predicted
ratio
(assuming
Observed
o' = 0.01
ratio
d = 1 km)
x 1013
x 1013
1.8
17.0
1.35
17.0
0.7
17.0
4.2
17.0
Atlantic secondary
Irish Sea primary
Predicted
ratio
(assuming
reflection
Observed
Observed
Predicted
ratio
cwff = 1/30) ratio
ratio
x 10l2
x 10"
x 10"
x 1014
0.5
0.87
10.4
0.13
1 -9
0.22
1
17.0
9.8
0.027
s o
4.2
17.0
A study of primary and secondary microseisms
91
in the case of primary microseisms generated in the Atlantic Ocean. In the case of
secondary microseisms generated in the Atlantic Ocean, predicted ratios based on
coastal reflection are rather on the low side and suggest that reflection off the coast
may not be the main mechanism. Longuet-Higgins (1950) also came to this conclusion.
Primary microseisms also appear to be formed in the Irish Sea. The theory shows
that in this case for waves off Anglesey, shorter period waves down to about 5 s are
favoured and this is borne out by the observations. The ratio of the spectral densities
in these cases, is, however much higher than the theory would predict, being about
lo-", two hundred times too big. It is difficult to explain this anomaly, the discrepancy can be partly explained by the waves at the position of recording being
already reduced in height due to breaking, and also the width of the breaker zone
might at some headlands be much less than 0.35 km. When the coasts are curved
concavely towards the recording station there can also be some focussing effects
but all of these seem insufficient to explain the discrepancy.
Acknowledgments
We wish to thank the Natural Environment Research Council and the Federal
Nigeria Government Scholarship Scheme for grants to carry out the work described
in this paper, the Director of the National Institute of Oceanography for the loan
of the wave records from the weather ships, the R.A.F. station at Valley, Anglesey
for supplying meteorological data, and to Mrs Eileen Pritchard for preparing the
diagrams.
Marine Science Laboratory,
University College of North Wales,
Menai Bridge,
Anglesey.
1968 May.
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