Seismic
Waves
The strain energy released during an earthquake travels from the
earthquake source to distant places in different directions in the form of
seismic waves. The Hookes law governs propagation of these waves in elasticity
and Snells law in optics. According to the Hookes law the stress p
and strain e (both tensors) are
related to each other. Both these quantities, p
and e, are tensors. Relationships
between tensors are similar to those followed by vectors. For example,
if a one dimensional vector Y of
n elements is related to another vector X (or X is transformed to Y) then the
two vectors are related as follows:
Yi
= å
CijXj
(i=1,2,3…n; j =1,2,3…n)
Here Cij is a 2x2
matrix operator having four elements, and the summation has been carried out
over the repeated subscript j. Generally, in vector mathematics, summation is
implied wherever a subscript is repeated, and the summation sign is omitted. The
transformation is, thus, written as:
Yi
= CijXj
(i=1,2,3…n; j =1,2,3…n)
The relationship
between p and e, each of which is a tensor, goes on the same lines:
plm = Cijlmeij
(i=1,2,3; j=1,2,3 )
Cijlm
are 81 constants of
proportionality. In an isotropic medium under symmetry and certain
thermodynamical conditions the number of constants reduces to two, so that the
equation above is written as :
pij
= qijdij
+
2meij;
q
= e11+ e22 + e33
Here e11, e22 and
e33 are the extensions in the directions of the 1,2 and 3 axes,
respectively (called the principal extensions) and e23, e31
and e12 are the angular deformations (shear components of strain).
The equation of motion in the medium is written as:
rfi
= (¶pij/¶xj)
+ rXi
Here
r
is the density of the material in motion; fi is the component of acceleration in
the direction of the ith axis (i=1,2,3) and Xi represent the body
force, which is neglected because of its very small effect.
After substitution and rearrangements the equation of motion becomes:
rd2u/dt2
= (l+m)(¶q/¶xi)
+ mÑ2u
This
equation represents propagation of longitudinal as well as transverse waves, and
governs the propagation of seismic waves during an earthquake. Seismic waves
travel along the surface of the Earth as Surface
Waves and in the Earth’s interior as Body waves. The velocities of the P
and S waves are given by:
Vp
=Ö[{k + (4/3)m}/r] ;
and Vs =Ö(m/r)
Here
k and m
are the bulk and shear modulii (The
constant l
is related to k). In P waves the particle motion is in the direction of
propagation, whereas in S waves it is at right angles to the direction of
propagation. For this reason, S waves are polarized into SV and SH waves. The P
and S waves undergo refraction and reflection according to the Snells law in
optics, namely:
Sin
i/Sin r =V1/V2
Here
i and r are the angles of incidence and refraction, and V1
and V2 are the
velocities in the first and second layers, respectively. A P or S type wave
gives rise to both P and S type waves, in the reflected as well as the refracted
wave, after it emerges from the boundary.
Surface
waves travel along the surface of the Earth. They arrive in the seismic wave
train after the S wave (which arrives after the P wave). Surface waves are of
two types, namely: the Rayleigh waves and Love
waves. In Rayleigh waves the particle motion is in the form of a retrograde
ellipse in the vertical plane, with major axis in the vertical plane as shown in
the figure. These are of the SV type. Love waves are surface waves of the SH
type. Surface waves have been seen travelling on
Earth up to great distances. In strong earthquakes these
waves do several rounds of the
It has been observed that during major earthquakes oscillations are set up in the Earth, so that the whole Earth oscillates as one body. The time periods of such oscillations are of the order of several minutes. These are called free oscillations. The largest period observed so far is 57 minutes.
Seismic waves, thus, cover a very wide range of frequencies, from several
tens of Hz to periods of several tens of minutes. In earthquake studies, P and S
waves are considered short period
waves (periods of the order of one seconds or less) and surface waves are
considered long period waves (10-20
seconds period). It was not possible, in the early days of instrumentation, to
record the entire range of the seismic spectrum using a single instrument.
Instruments of different types were used, in observatories, for recording
signals in the short period and long period ranges. Thus the instruments used
for recording earthquakes were divided into short and long period instruments.
Though both types of instrument record waves of both types magnification for the
two types varied. Another good reason for using two types of instruments to
record seismic signals was that at periods near 6 seconds the amplitude of the
ground noise is very large. This noise is generated by ocean waves hitting the
shores (microseisms), and
meteorological factors, winds etc. In the presence of this ground noise any
increase in magnification of the seismograph does not help in improving the
signal detection capability, because the noise also gets amplified. The
responses of the short and long period instruments were so chosen that they fell
steeply near the period of the microseisms. (The continuous vibrations of the
ground are some times called microtremors.
Their sources are not same every where, and remain, often, unidentified, These
are different from microseisms, which have a definite source. Again,
microtremors and microseisms are both different from microearthquakes, which are earthquakes having magnitude less than
3, and have a definite epicentre.
Additional Topics