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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 Earth’s surface before their amplitudes decay to the average level. For a large variety of materials the constants l and m are nearly equal, and Vp =Ö3.Vs, and the velocity of the Rayleigh type surface waves is nearly equal to 0.92 Vs. Velocity of the Love waves depends on the layering below the surface. These waves exist when more layers than one are present in the propagation path. The relation l=m, which facilitates many seismological interpretations, is called Poisson Relation.   Velocity of the surface waves is a function of the wavelength, and hence surface waves are dispersive. Generally, waves of longer period arrive first followed by those of shorter periods. This is called normal dispersion. Under certain circumstances shorter periods supersede longer periods. This is a case of what is called anomalous dispersion. Dispersion of surface waves plays an important role in investigating the seismic velocity structure.     

            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

Seismic Arrays

Seismic Source Discrimination