Elastic waves

The vibrations of the earth, felt as earthquakes, are elastic waves in soil and solid rock. These waves are similar to sound waves, which travel through the air, and sonic waves, which travel through water.

Elastic Behavior

Elastic waves are experienced frequently every day: Everything heard is an elastic wave in the air; every vibration felt, in the ground as a truck passes or in the floor from vibrations in a building, is from elastic waves in solid matter. Although the experience of elastic wave energy is familiar, the exact nature of this phenomenon is not something visible to the eye or easily described in a visual way. Ripples resulting from dropping an object in a still body of water move in ever-increasing circles away from the splash; such waves are not elastic, but rather are gravity waves. Yet, elastic waves are analogous to this example in that they originate from a disturbance and propagate outward and away in concentric circles or spheres.

An elastic wave moves in an elastic medium, which can be a solid, liquid, or gas. A substance is said to respond elastically if when it is compressed, stretched, bent, or submitted to shear forces, it deforms in proportion to the applied force and then immediately returns to its original unstressed state when the force is removed. A good illustration of this property is a spring scale. A 1-kilogram weight placed on the scale will cause the spring inside the scale to be stretched (deformed) into a longer length, such as dropping 1 centimeter. A 2-kilogram weight on the scale would cause it to move down 2 centimeters. Hence, the displacement of the spring is proportional to the force applied. When the weights are removed from the scale, it immediately returns to zero, its original unstressed length and shape. This is elastic behavior. If the spring of the scale stretched 1 centimeter for 1 kilogram and then stretched more or less than a centimeter for the second kilogram, it would not be elastic because it would not be a proportional response. Also, if the spring did not return to zero after the weight was removed but retained some permanent deformation, it would not be elastic.

In the case of wave propagation in the earth, consider the effect of striking the ground with a sledgehammer. When struck, the ground would be suddenly compressed, which would be transmitted to the neighboring soil and rock around and beneath the strike. Except in the immediate vicinity of the blow, where permanent deformation (non-elastic) may occur, the response of the neighboring soil and rock would be elastic. It would be temporarily compressed by the force of the blow and then immediately relax back into the former condition. A compression wave would irradiate spherically away from the blow, traveling across the surface like the ripples on a pond and down into the earth. In the passing of an elastic wave, the medium passing the wave is restored to its original unstressed state as if no wave had ever come through at all.

Elastic Wave Velocities

Elastic waves travel at certain velocities depending on the density and elastic stiffness or compressibility of the medium. If a substance is soft, elastic waves move more slowly; if it is very stiff, elastic waves move rapidly. In air, sound waves move at approximately 300 meters per second; in water, sonic waves move at roughly 1,200 meters per second. In rock, compressional waves move at a rate of from 3,000 to more than 10,000 meters per second, depending on the rock's hardness and how deeply it is buried.

If within a medium through which elastic waves can move the velocity is the same everywhere, that medium is called “homogeneous.” If, in addition, at any given point in that medium the velocities are the same in all directions, the substance is termed “isotropic.” In most rocks of the earth, which occur in layers, the deeper below the surface, the higher the velocity becomes. The increasing weight of the overburden acting on rocks found deeper in the earth causes their density and stiffness to change, and, generally, in the vertical direction the velocity of a seismic wave is different from its velocity in horizontal directions. Thus, many rocks of the earth are not isotropic; neither are they homogeneous. By analyzing seismograms from earthquakes, quarry blasts, and underground nuclear explosions, the inhomogeneities and anisotropies of the earth have been described to give a picture of what the earth's interior is like.

Elastic Wave Types

There are two basic kinds of elastic wave: P waves, or compressional waves, and S waves, or shear waves. P waves are sometimes called “push-pull” waves because they consist of a series of pushes (compressions) and pulls (rarefactions), where the motion of a particle of matter as the wave passes by is parallel to the direction the wave passed. S waves are sometimes called “shake” or “shear” waves, because they consist of shearing or shaking motions where the movement of a particle of matter as the wave passes by is transverse, or perpendicular, to the direction the wave passed. A “Slinky” toy spring, held in two hands, can provide an illustration of sending waves back and forth. The alternate stretched and compressed parts of the spring move from one end to the other. If a long rope is tied to a post and the end is shaken up and down, a wave will move from the shaken end to the post, but the motion of the particles of the rope are up and down, transverse to the wave motion. P waves can move in all substances, solid, liquid, or gas. S waves can move only in solids. Compressional and shear waves are the only types that can propagate anywhere interior to a solid, like the rocks of the earth. These are called “body waves.” Earthquakes generate both compressional and shear waves at the source where the fault moves.

There are two other important kinds of elastic waves, but these travel only parallel to free surfaces, like the surface of the earth, and have amplitudes that decay with depth. They are called “surface waves.” The two kinds are “Rayleigh waves” and “Love waves,” each named after the scientist who discovered and described it.

When a Rayleigh wave passes by on the surface of the earth, a particle of soil or rock is first moved forward, then up, then backward, and then down to its starting point in an elliptical path. For Rayleigh waves, when the displaced particle is at the top of its elliptical motion, it is moving in the opposite direction of the Rayleigh wave front. This is called an “elliptic retrograde” motion. When a Love wave passes by on the surface of the earth, a particle of soil or rock is moved from side to side perpendicular to the direction of the wave front. Love waves are horizontally polarized shear waves traveling parallel with the surface.

With regard to velocity, compressional waves are the fastest; next are shear waves, which move at roughly six-tenths the speed of the compressional wave; slowest are the surface waves, which move at approximately nine-tenths the speed of shear waves.

Elastic Wave Forms

One final aspect needs to be described in talking about elastic waves, and this is the form of the wave. Regardless of the type of wave (P, S, Rayleigh, or Love), they all consist of trains of disturbances that move through the earth. A wave that has only one vibration is a pulse. In elastic waves, even ones that sound like sharp pops or explosions, there is a train of several cycles of vibration—sometimes a few seconds in duration and sometimes for many minutes or even hours. (Rayleigh and Love waves can be recorded for an hour or more on seismographs when generated by a very strong earthquake.)

The form of a wave is described by its frequency and its amplitude, as well as by its particle motion. Frequency is merely the number of times per second that the vibrations occur as the wave passes. Earthquake waves have frequencies of from several cycles per second down to several seconds per cycle. In addition to the time between peaks of an elastic wave's passage, there is a distance that can be measured between peaks. This is called the “wavelength.” For waves in the earth, the wavelengths can vary from a few meters (for high-frequency P and S waves) to a kilometer or more (for low-frequency surface waves). Hence, in an earthquake one part of a railroad track can be under compression, being sheared to the left, while a few hundred meters away another part is under tension, being sheared to the right, all at the same instant.

Elastic Wave Propagation Complexity

Even though a source of elastic waves may be simple, generating only one kind of wave, as soon as boundaries between differing layers are encountered, other kinds of waves result, reflecting and refracting in many directions. Those that eventually find themselves back at the surface can be recorded. When P and S waves arrive at the surface, it is their complex interaction at the surface that produces the Rayleigh and Love waves. With regard to earthquake-generated waves, not only do waves reflect from the source back to the surface off the boundaries of crust, mantle, and core, but some waves can pass completely through the earth and be recorded on the other side. During an exceptionally strong earthquake, waves can refract through to the other side and then return again through the core and mantle to be recorded again on the original side. Surface waves generated by large earthquakes have also been known to circumnavigate the globe, sometimes circling several times before their amplitudes become too small to detect. In very large earthquakes (those of 8.6 or more on the Richter scale), these waves have been measured to complete as many as ten or more passages around the world, taking approximately 3 hours for each trip.

A wave train of a single type can change from one type to another repeatedly along its ray path. This is of great interest to seismologists. For example, a P wave may start from where it is generated at an earthquake fault and travel down until it hits the Mohorovičić, or Moho, discontinuity, the boundary between the earth's upper crust and mantle below. There it can refract through, turning into an S wave, bending its direction of travel slightly, and taking on a new velocity. As it propagates farther and farther downward, it speeds up until it hits the boundary of the outer core, where it must either change again or reflect back toward the surface. If it passes through the boundary, it must transform back into a P wave because the outer core of the earth acts like a plastic liquid and will not permit the passage of S waves. Continuing on past the center of the earth, it would strike the boundary between core and mantle on the other side and, again, it could change back into an S wave. As it traveled up toward the other side of the earth, it would gradually slow in speed until it hit the Moho on the other side. There it could turn into a P wave again and move through the crust until it emerged at the ground surface. There it would be reflected back toward the earth's interior or, perhaps, follow a curved ray path that would skip back to the surface at another location.

During this long and varied path, the ray would travel at P-wave velocities when in a compressional phase and at S-wave velocities (roughly 40 percent slower) when in a shear phase.

Seismographs and Seismic Stations

Elastic waves in the earth are measured and recorded by various kinds of seismographs. Some measure vertical motions only, some horizontal; some measure compressional waves only, such as those that move through water. To describe the particle motion of a train of passing waves requires a set of three seismographs: one for vertical motion and two for horizontal—one for east-west motion and one for north-south.

Seismic stations permanently installed to monitor earthquakes are built in a variety of ways, depending on what is to be measured. A given seismic sensor can detect only a given band of frequencies; outside that band it is insensitive. Since earthquakes generate a wide range of frequencies from high to ultralow, some stations measure high frequencies (also called “short periods”) while others measure low frequencies (called “long periods”). Moreover, seismic sensors respond only to a given range of amplitudes. Hence, some earthquake observatories have extremely sensitive instruments that detect and magnify even the tiniest vibrations 100,000 times or more. Other stations also have so-called strong motion instruments that do not record at all, unless a real jolt passes through. This diversity in equipment is necessary, because when strong high-amplitude seismic waves hit a high-magnification instrument, the readings go off the scale and cannot be deciphered. On the other end, strong-motion equipment is insensitive to smaller tremors.

Reflection Seismic Profiling

Because the interaction of seismic waves with the details (inhomogeneities) within the earth's interior enables what is there to be described, even though it is buried out of sight, artificially generated seismic waves are sometimes used to find oil and other things of interest belowground. “Reflection seismic profiling,” a method used by oil companies the world over, usually entails the use of an explosive to send elastic waves into the ground. Then, by an array of seismic sensors called geophones, deployed to catch the reflections at the land surface, geophysicists can deduce the structures of the subsurface. Some seismic profiling methods employ only P waves, while some have been successful with artificial S-wave sources.

When P and S waves hit a boundary between two rock layers, they each split into four parts. A P wave, for example, will reflect back both a P wave and an S wave but will transmit a portion of its energy through the boundary into the next layer down that also splits into a P wave and an S wave. The wave bouncing back is the “reflected” portion of the incident wave, while the part that passes through is the “refracted” portion. An incident S wave similarly splits into four parts, a reflected P and S and a refracted P and S.

Practical Applications

Understanding elastic wave propagation within the earth not only has been the means by which seismologists have been able to define the inner structure of the earth but also has enabled the discovery of almost all of the oil deposits found since the mid-twentieth century. Other practical applications include the monitoring of underground nuclear testing to verify that countries are living up to their treaties. Also, because submarine earthquakes are the cause of seismic sea waves, and because the seismic waves passing through the earth travel several times faster than the seismic sea waves (or tsunamis) do through water, these destructive waves from the sea can be predicted, sometimes hours before they strike a coastline, thus saving many lives.

Principal Terms

body wave: a seismic wave that propagates interior to a body; there are two kinds, P waves and S waves, that travel through the earth, reflecting and refracting off of the several layered boundaries within the earth

elastic material: a substance that, when compressed, bent, stretched, or deformed in any way, undergoes a degree of deformation that is proportional to the applied force and returns back to its original shape as soon as the force is removed

homogeneous: having the same properties at every point; if elastic waves propagate in exactly the same way at every point, they are homogeneous

ideal solid: a theoretical solid that is isotropic, is homogeneous, and responds elastically under applied forces, stresses, compressions, tensions, or shears

isotropic: having properties the same in all directions; if elastic waves propagate at the same velocity in all directions, they are isotropic

reflection: when an elastic wave strikes a boundary between two substances or between two rock layers of different seismic velocities, part of the incident ray bounces back (reflects)

refraction: when an elastic wave passes through a boundary between two rock layers of different seismic velocities, the rays passing through are bent (refracted) in another direction

surface wave: a seismic wave that propagates parallel to a free surface and whose amplitudes disappear at depth; there are two kinds—“Rayleigh waves” (first described in 1885) and “Love waves” (first described in 1911), that travel at the surface around the earth

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