In seismology

Seismology

Seismology is the scientific study of earthquakes and the propagation of elastic waves through the Earth or through other planet-like bodies. The field also includes studies of earthquake effects, such as tsunamis as well as diverse seismic sources such as volcanic, tectonic, oceanic,...

, the Zoeppritz equations describe how seismic waves are transmitted and reflected at media boundaries, which are boundaries between two different layers of earth.

The Zoeppritz equations, formulated by the German geophysicist Karl Bernhard Zoeppritz

Karl Bernhard Zoeppritz

Karl Bernhard Zoeppritz was a German geophysicist who made important contributions to seismology, in particular the formulation of the Zoeppritz equations. These equations relate the amplitudes of P-waves and S-waves at each side of an interface between two arbitrary elastic media as a function of...

, relate the amplitude

Amplitude

Amplitude is the magnitude of change in the oscillating variable with each oscillation within an oscillating system. For example, sound waves in air are oscillations in atmospheric pressure and their amplitudes are proportional to the change in pressure during one oscillation...

s of P-waves and S-waves at each side of an interface to the angle of incidence of the incoming wave.

For instance, a blast of dynamite

Dynamite

Dynamite is an explosive material based on nitroglycerin, initially using diatomaceous earth , or another absorbent substance such as powdered shells, clay, sawdust, or wood pulp. Dynamites using organic materials such as sawdust are less stable and such use has been generally discontinued...

 will create a seismic wave

Seismic wave

Seismic waves are waves of energy that travel through the earth, and are a result of an earthquake, explosion, or a volcano that imparts low-frequency acoustic energy. Many other natural and anthropogenic sources create low amplitude waves commonly referred to as ambient vibrations. Seismic waves...

 (P-wave) that will travel through the earth, be reflected off of various layers, and return to the surface where it can be detected. The time it takes to return is related to the depth of the various layers. By detecting the seismic waves at various points on the surface, one can see how the reflections change with angle of incidence. One can then use this information along with the Zoeppritz equations to learn more about the density

Density

The mass density or density of a material is defined as its mass per unit volume. The symbol most often used for density is ρ . In some cases , density is also defined as its weight per unit volume; although, this quantity is more properly called specific weight...

 and velocity of each layer. This is helpful for instance in locating underground reservoirs and deposits.

Approximations

Zeoppritz's equations are complex to use so approximations such as Bortfeld's 1961, and Shuey's 1985 are often used. Shuey's approximation is:



Where each term covers the reflection amplitude at greater angles. The first term gives the amplitude at normal incidence , the second term characterizes at intermediate angles, and the third term describes the approach to critical angles. Here is poisson's ratio

Poisson's ratio

Poisson's ratio , named after Siméon Poisson, is the ratio, when a sample object is stretched, of the contraction or transverse strain , to the extension or axial strain ....

, is the angle of incidence, and is a slowly changing value proportional to . This approximation is accurate to about 60 degrees of the critical angle and assumes that the change in density and velocity across the boundary is much less than 1.