Seismic wave

Seismic waves and seismic "rays"

In this article, forms of propagation (propagation) of seismic waves are described on the basis of approximate solutions, which could be called "seismic rays". The relation between "seismic rays" and "seismic waves" corresponds to that between ray optics and wave optics. A more precise description of the propagation of seismic waves is possible via partial differential equations, the so-called wave equations. These mathematical techniques correspond to those of earth spectroscopy.

Wave types

According to whether seismic waves propagate inside the earth's body or on its surface, a basic distinction can be made between body waves and surface waves. Further differences result from the type of oscillation, whether its plane is longitudinal or transverse to the direction of propagation or which form the particle movement has.

Space Waves

The designations of the primary waves (P waves) and secondary waves (S waves) described below refer to the fact that the former propagate faster: At a location distant from the quake focus, the P-waves are recorded first and only later the S-waves. From the time difference between the arrival of the P- and the S-waves, their travel time difference, the distance to the focus can be calculated. If the distance has been determined in this way at at least three different locations, the approximate position of the hypocenter can be given within the limits of measurement accuracy. The geographical location above it on the earth's surface is called the epicentre.

P-waves

P-waves, short for primary waves, are longitudinal waves, i.e. they oscillate in the direction of propagation. They can propagate in solid rocks, but also in liquids such as water or the quasi-liquid parts of the Earth's interior. They are compression waves (also: pressure or compression waves), similar to sound waves in air or water.

The propagation speed of the P-waves can be calculated with the following formula:

where K is the compressive modulus, μ is the shear modulus, and ρ is the density of the material through which the wave propagates.

In the Earth's crust, the speed of the P-waves is between 5 and 7 km/s, in the Earth's mantle and core it is over 8 km/s. The propagation velocity is highest in the lower mantle with almost 14 km/s, it decreases abruptly at the core-mantle boundary to about 8000 m/s (speed of sound for comparison: in air about 340 m/s, in water about 1500 m/s, in granite about 5000 m/s).

S-waves

The S-waves or secondary waves oscillate transversely to the direction of propagation (transverse wave). Since they lead to shearing of the propagation medium, they are also called shear waves. S-waves can propagate in solids, but not in liquids or gases, since the latter two have no (appreciable) shear resistance. One can therefore identify liquid areas in the earth's interior by the fact that no S-waves travel there.

The speed of propagation of the S-waves is calculated with the following formula:

With typical values of the elastic constants within the Earth, velocities of 3000 to 4000 m/s in the Earth's crust and about 4500 m/s in the upper mantle result for the S-waves. In the lower mantle the velocity increases further (see diagram of the IASP91 model in the figure). No shear waves exist in the liquid outer core of the Earth.

Surface waves

Besides the P- and S-waves as space waves, there are the surface waves. They are caused by P- or S-waves being refracted into the earth's surface. As with the S-waves, the particle movement or oscillation occurs perpendicular to the direction of propagation. However, they are distinguished by the fact that they travel along the surface and that the amplitudes of the waves decrease with depth. Moreover, the energy of the surface waves decreases with the distance r only by a factor 1/r, not like that of the space waves by a factor 1/r2 (in each case neglecting the damping). The surface waves propagate in vertical and horizontal oscillations.

Love waves

Love waves were named after the British mathematician A. E. H. Love, who in 1911 was the first to set up a mathematical model for the propagation of these waves. They are the fastest surface waves, propagating at around 2000-4400 m/s (depending mainly on the frequency and thus the depth of penetration into the Earth's crust), but slower than the S-waves. Ground motion occurs in a horizontal direction, perpendicular to the direction of propagation.

Rayleigh waves

Rayleigh waves were named after Lord Rayleigh, who mathematically proved the existence of these waves in 1885, before they were even observed. In Rayleigh waves, the ground rolls in an elliptical motion similar to ocean waves. On a homogeneous half-space, the polarization is always retrograde, i.e., the rolling motion occurs opposite to the direction of propagation of the Rayleigh wave. In the general case, prograde polarized Rayleigh waves also occur. This rolling moves the ground up and down as well as back and forth in the propagation direction of the wave. The propagation velocity is about 2000-4000 m/s, depending mainly on the wavelength. Most of the shaking felt during an earthquake is usually Rayleigh waves, whose amplitudes can become much larger than those of the other types of waves. The destructive effect of earthquakes is therefore largely due to this type of wave.

Scholte waves

Scholte waves are interface waves that propagate along the "liquid-solid" interface, for example on the sea floor. Like Rayleigh waves, they are of the P-SV type. This means that they are elliptically polarized in the radial-vertical plane. If the subsurface is stratified, the Scholte wave is dispersive, i.e., it then has frequency-dependent propagation velocities. In addition to the fundamental mode (with fundamental frequency), higher order modes (harmonics) are also formed.

P-waves (red) are recorded earlier than S-waves (green) by the seismograph (deployment times marked)


Change of propagation velocities of P- (black) and S-waves (grey) with depth in the Earth's interior according to the IASP91 reference model