What’s seismology about?

• Seismology is the study of the generation, propagation and recording of elastic waves in the Earth (and other celestial bodies) and of the sources that produce them.

Elastic Moduli

• Young’s Modulus

• Poisson Ratio

• Bulk Modulus

• Shear Modulus

• Lamb’s constants

• Vp, Vs.

= tensile stress = (F /A)

= tensional strain = (L /L)… for uniaxial tension or compression

E =

Young’s modulus

= tensile stress = (F /A)

= tensional strain = (L /L)… for uniaxial tension or compression

E =

Young’s modulus

Poisson’s ratio = - (yy / xx)

Under a stress (xx) along the x-axis,

longitudinal strain = xx =

and transverse strain = yy =

L L

Then,

W W

Bulk Modulus = (P/)

… where = dilatation = V/V and P = pressure

A

shear stress = (F /A)

shear strain = (l /L)

L

=shear stressshear strain

shear modulus

From Lay & Wallace (1995)

Elastic Moduli and Densities of Some Common Materials

From Lay & Wallace (1995)

= k - =2 E3 ( 1 + ) ( 1 - 2 )

V = = =K + ( ) + 2

p4/3

V = =s

Where are the Lamé coefficients and is

S-wave

P-wave

dilatations

compressions

wavelengthamplitude

Body Waves

Body wave

Surface wave

Amp ~ 1/x

Amp ~ x-1/2

Late homework policy

• Late homework will not be accepted one week after the deadline.

• Late homework will be penalized: 90% (1 day late), 80% (2 day late), 60% (3 day late), 40 % (4 day late) and 20% (5-7 day late).

S-wave

P-wave

dilatations

compressions

wavelengthamplitude

Body Waves

Seismic Rays & Geometric Ray Theory

• Rays are defined as the normal to the wave fronts and thus point in the direction of propagation. Connecting the these successive directions forms a ray paths. In the case of plane wave, rays are a family of straight lines; in the case of spherical wave, rays are spokes radiating out from the seismic source. They provide a convenient means of tracking an expanding wave front.

• The seismic rays corresponding to P and S waves behave very much as light does in traveling through materials of varying indices of refraction. This leads to many parallels with optics: rays bend, focus, and defocus depending on the velocity distribution. Although geometric ray theory is an approximation, it, however, simplifies the analysis and yet gives the correct answer or a good approximation.

Fermat’s Principle

• Ray paths between two points are those for which travel time is an extremum, either a minimum or maximum, with respect to the nearby possible paths.

Snell’s Law and Fermat Principle

Fermat Principle: T(x) is either minimum or maximum

2

2

1

1 sinsin

v

i

v

i

Snell’s Law

Reynolds (1997)

R sin i

V= constant = p

Spherical Ray Parameter

What control reflection and transmission?

• Continuity of stress

• Continuity of displacement

I =

I =2

1

Reynolds (1997)