Wave Energy

SOUND WAVES AND SEISMIC WAVES

What Are Waves?

Waves have cropped up in several contexts in this book. Chapter 7 described ocean waves—or more generally waves moving across a surface. Chapter 15 mentioned earthquake waves. Sound waves have appeared in a variety of contexts, for example, the roar of thunder and of landslides and the crash of breaking waves. Solar radiation, the chief source of energy at the earth's surface, has been touched on in contexts too numerous to list; likewise the infrared radiation emitted by sun-warmed land and sea. Electromagnetic radiation—light, radiant heat, and many other varieties—consists of electromagnetic waves; they will be considered in detail in chapter 18.

Waves of various kinds account for much of the energy of nature; indeed, they account for most of the energy we experience with our senses. All that we see comes to us as light waves and all that we hear as sound waves, and much of the warmth we feel comes as radiant heat. The output from radio and TV transmitters also comes into our houses uninvited, as imper-

ceptible electromagnetic waves ready to be converted to sound and light waves if we choose to listen and look. Waves (strictly speaking traveling waves) demand examination in any discussion of energy.1

The first questions to be considered are, What are the different kinds of waves, how do they differ from each other, and what, in spite of the differences, unites them all as waves? What are waves and what do they do?

Here are two definitions of the word "wave" from physics dictionaries: "a periodic change in some property or physical quantity through a medium or space," and "a disturbance which propagates from one point in a medium [or empty space] to other points without giving the medium [or the space] as a whole any permanent displacement."2 These definitions combined give us what we want. Thus water waves are displacements of a water surface to give regularly spaced crests and troughs that travel across the surface leaving the water unaltered and unmoved; in other words, leaving no trace. Similarly, sound waves are local pressure changes in a medium (usually air) that travel through the medium leaving no trace. Earthquake waves—seismic waves—often do leave a trace, but weak seismic waves—mild tremors—usually don't. Electromagnetic waves are very rapid changes in electromagnetic fields, moving at unimaginable speed through empty space and, once again, leaving no trace.

In all cases a physical quantity such as water level, air pressure, or the like varies regularly in a direction leading away from the source of the waves: the series of variations can be observed if you take an instant snapshot of it, in fact or in imagination. The same variations can be detected by a measuring device (or simply by eye if you're watching ocean swells) focused on a fixed spot for a length of time: the displacements or disturbances are then detected at regular intervals. To recap, for emphasis: a series of waves can be observed spread over space at one instant, or spread over time at one location. And waves are moving disturbances in a medium or in space: the medium itself (when there is one)does not shift in the direction the wave is traveling.

The preceding paragraph summarizes what waves are. Next, what do they do? They convey energy from one place to another. Equivalently, in waves "energy move[s] from one point to another but no material object makes that journey."3 You can confirm this by tossing a rock into a calm pond and contemplating the circles of waves that spread out: the waves carry a portion of the kinetic energy the rock possessed at the moment it touched the water, and they carry that energy outward in all directions away from the point. The passing waves leave the water almost unchanged by their passage. The word "almost" allows for the fact that an immeasurably small fraction of the waves'

energy is inevitably converted to heat because of viscous drag in the circling water under each wave (see fig. 7.3). The rest of the energy is carried to the margin of the pond (provided it's a small pond), where it sways water plants and shifts clods of mud. Notice that the energy the tossed rock once possessed has reappeared as movement at the margin of the pond: the waves have performed "action at a distance," without any horizontal displacement of the water.

The distances across which waves can transport energy are sometimes enormous. Earthquake waves can travel from their starting point to the other side of the earth (not necessarily in a straight line, as we shall see below). Giant sea waves—tsunamis—have been observed to travel 17,000 km (see chapter 7). Sound waves can travel 20,000 km in the sea, given the right conditions. Electromagnetic waves from quasars4 reach the earth from billions of light years away, action at a very long distance indeed.

But no wave can travel forever; all are eventually dissipated, their energy degraded to entropy. In the context of energy, every kind of wave inspires two questions: How do they originate? And how are they dissipated?

Sound Waves

Perhaps the most familiar waves, apart from water waves, are sound waves. Like all traveling waves, they are generated at one location and dissipated at another.

Take a simple example. An easy way to generate a sound—to make a noise—is to strike one object with another, for instance, to hit a nail with a hammer. Imagine a nail is being driven into a block of wood and focus on one particular hammer stroke: at the moment it hits the nailhead, the hammer has kinetic energy, KE; the amount of KE—the number of joules—depends on the weight of the hammer and its speed at the moment of impact. On hitting the nail, the hammer is brought to an abrupt stop; most of its KE is passed on to the nail, which penetrates the wood until friction stops it. The remaining energy is converted to other forms; some of it becomes waste heat at the point of contact of hammer and nailhead.

The rest of the energy sets both the hammer and the nail vibrating, making the air in contact with them vibrate too, so generating a sound. The vibrations start in the molecules of air touching the metal surfaces, which dislodge the molecules adjacent to them, which then dislodge molecules beyond them, and so on. In their vibrations, the molecules shift back and forth parallel with the direction in which the sound is traveling. Their movements are lengthwise (longitudinal), so sound waves are often called longitudinal waves. The vibrating air becomes successively compressed and rarefied, in layers that move steadily farther from the point where the hammer hit the nail. The vibrations reach the ears of anybody within range; rapid pressure changes in the air touching the eardrums are perceived as sound. In this way some of the energy of the original hammer stroke is carried far from its source while being partially absorbed, here and there, by solid objects it happens to touch.

Of course, the sound won't travel forever. As well as being absorbed by objects in their path, the vibrations of the air become attenuated as they spread out. At the same time, besides shifting back and forth through infinitesimal distances as they vibrate, the air molecules are also in a state of constant random motion in all directions, whose mean free path depends on the temperature (see chapter 3). Not surprisingly, repeated collisions among the air molecules ensure that pressure contrasts gradually become evened out between the compressed and rarefied layers, whereupon the sound fades away. All the original energy in the hammer stroke has now been dissipated: it has all become entropy.

A sudden sharp bang like the sound of a hammer stroke is difficult to analyze. Continuous sounds, like the roar of a waterfall or the hum of a quiet motor, are easier to deal with. The simplest sound of all is a prolonged pure musical note. Any continuous noise can be analyzed into a large number of component pure notes—or pure sounds as we'll call them—in the same way that the profile of a stormy sea can be analyzed into a large number of simple waves (see chapter 7). Recall that the energy of simple waves at sea is proportional to the square of the waves' heights. Analogously, the energy of any noise is proportional to the squares of the "heights" of all its component pure sound waves added together. What is meant by the height of a sound wave?

Consider figure 17.1, which shows two ways of portraying sound waves. Figure 17.1a shows sound pictorially: the varying density of the stippling represents the varying density of a representative "slice" of vibrating air; the sound is moving from left to right. Figure 17.1b translates the picture into the form of a wavy line showing how the air pressure varies. The speed at which the molecules move depends on the pressure differences, represented by the waves' height (the vertical distance from crests to troughs).To say that the energy of the waves is proportional to the square of their heights is therefore equivalent to saying that it is proportional to the square of the molecules' speed—which, when you recall how kinetic energy is defined (see chapter 3), is as it should be.5

Figure 17.1. Two representations of the same series of sound waves. (a) The density of the stippling represents the density (hence the pressure) of the air. (b) The curve shows how air pressure rises and falls.

The total energy of a continuous sound accumulates as the sound goes on and on, whereas its loudness at any moment depends on the rate at which sound energy is being produced—on its power. As always, power is measured in watts (recall that one watt equals one joule per second). The intensity of a sound is defined as the power of the sound traveling across one square meter at right angles to its path. The units of intensity are watts per square meter, or in symbols, W m-2.

It turns out that the human ear is astonishingly sensitive when you visualize watts in terms of a lightbulb's output. The quietest sound that the average human can hear—the threshold of hearing—has an intensity of 1 x 10-12 W m-2 (that is, one-trillionth of a watt per square meter). The intensity at which noise becomes painful is one trillion times as great, or 1 W m-2, equivalent to the sound of a loud indoor rock concert.6

A more convenient measure of intensity has been devised to allow for the fact that, for the listener, a change in the intensity of a quiet sound is much more noticeable than an equivalent change in a loud sound. The decibel scale of loudness corrects for this.7 The scale assigns a score of zero decibels (0 dB) to the threshold-of-hearing intensity and goes up from there, well past 120 dB, the threshold of pain. The rule is: If one sound is ten times more intense than another, then it is 10 dB louder; if it is 102 (or 100) times more intense, then it is 20 dB louder; if it is 103 (or 1,000) times more intense, then it is 30

dB louder; and so on.8 As an example, the loudness of ordinary speech is about 60 dB. This means that talk is a million times more intense than the softest perceptible sound.

Thus far we have concentrated on sound waves traveling through air, which they do at a speed of 0.343 kilometers per second. They also travel through liquids and solids, but at much higher speeds. In fresh water the speed is 1.48 km per second; in granite, it is 6 km per second.9 Converted to kilometers per hour (km/h) the speeds are: in air, 1,235 km/h; in fresh water, 5,328 km/h; in granite, 21,600 km/h. The more dense and the more rigid the medium, the higher the speed.10 Note that the speed of traveling waves of sound is not the same as the speed of the minute back-and-forth movements of the molecules creating the pressure changes within each wave.

Very low frequency sound waves of great amplitude travel through the body of the earth as one of the varieties of seismic waves.

Seismic Waves

Most of an earthquake's energy is dissipated at the site of the quake. The rest is carried away in seismic waves. Like all traveling waves, seismic waves convey energy from one place to another. They are set in motion by an earthquake and travel outward, to be dissipated throughout a volume of the earth's interior and over an expanse of its surface: the energy is eventually converted to entropy and the waves die away. Let's look at the details.

Some seismic waves are giant sound waves. By analogy with the hammer-and-nail example of sound generation, an earthquake shock corresponds to a collision between a hammer and a nailhead, and the emitted seismic waves correspond to the emitted sound waves. Unlike sound waves, however, seismic waves are not all alike; on the contrary, there are several varieties. There are surface waves, generated most efficiently by shallow quakes, with a focus less than 70 km deep; they stay close to the surface, and they cause nearly all the damage. About three out of four quakes are shallow. Deeper quakes generate body waves, which spread in all directions, some passing right through the earth's center. Body waves are of two kinds: primary waves or P-waves, and secondary waves or S-waves. This gives us three kinds of waves to consider or, what comes to the same thing, three mechanisms by which solids and highly compressed liquids can convey energy across a distance.

First, consider the two kinds of body waves. Primary waves are so called because their speed is greatest and they arrive at a distant site first; they are also

Figure 17.2. The single box on the left shows a cross section of a seismically "quiet" rock mass, just before seismic waves reach it; a fragment within the mass is colored black to show its changing shape as seismic waves pass. Three possible behaviors are shown, representing three types of waves; (a) a P-wave; (b) an S-wave; (c) an R-wave. (The distortions are exaggerated for clarity.)

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Figure 17.2. The single box on the left shows a cross section of a seismically "quiet" rock mass, just before seismic waves reach it; a fragment within the mass is colored black to show its changing shape as seismic waves pass. Three possible behaviors are shown, representing three types of waves; (a) a P-wave; (b) an S-wave; (c) an R-wave. (The distortions are exaggerated for clarity.)

called pressure or push-pull (compression) waves. Secondary waves travel more slowly; they come in second; they are also called shear waves. (Both kinds of waves have other descriptive names too, but they're not alliterative.) The two kinds of body waves operate in radically different ways.

P-waves are very low frequency sound waves; they entail the alternate expansion and contraction of each of the tiny fragments of rock that make up a large rock mass; the rock changes volume rhythmically, without any shearing. The reverse is true of S-waves: each fragment of rock changes shape rhythmically because of shearing, while its volume remains the same.

What happens is illustrated in figure 17.2. Each box in the figure represents the same vertical cross section of massive underground rock. The isolated box on the left shows the rock mass while the earth is "quiet," before the first jolt a b c of an earthquake. One particular block of rock, though continuous with its surroundings, has been singled out by coloring it black. Each of the three rows of boxes on the right shows how the block behaves as a seismic wave travels through the whole rock mass. The boxes in each row show the same scene at a sequence of times. Rows a and b show the two kinds of body waves, P-waves and S-waves respectively. Row c shows a surface wave, also known as a Rayleigh wave, or R-wave.11

In R-waves, the block we are concentrating on changes in both shape and location. Because the waves are shallow and the rock is less dense than at greater depth, the particle motion (all in the vertical plane) causes appreciable ground movement. The waves, aptly described as ground roll, resemble a swell at sea, but with a surprising difference. The rock particles, which trace out vertical ellipses, move backward relative to the direction of the wave: if the wave is going from left to right (as in the figure) the particles move counterclockwise, and vice versa. Compare this with the behavior of a particle of water circling within a water wave (see figs. 7.3 and 7.7).

Now to compare the speeds and frequencies of seismic waves with those of sound waves. (Note that we are not concerned with the tiny distances traversed by molecules and particles within the medium, which are much less in liquids, and very much less in solids, than they are in air. Here we are considering the speeds and sizes of entire waves.) We have already compared the speed of longitudinal waves in rock and in air: recall that in air, sound waves have a speed of 0.343 kilometers per second, whereas in rock, where the longitudinal seismic waves are P-waves, their speed near the surface averages about 6 km/s or more than seventeen times as fast. S-waves and R-waves are slightly slower than P-waves, with speeds of about 3.5 and 3.1 km/s, respectively. All these seismic wave speeds depend on the type of rock that the waves travel through, and they increase at progressively greater depths below the surface because the pressure increases.

The frequencies of sound waves determine their pitch; the deepest note the average person can hear is about twenty waves per second. The frequencies of seismic waves are much lower—far too low to be heard (the rumble of a quake is ordinary sound, a by-product of the quake). For P-waves, S-waves, and R-waves, only a fraction of a wave passes per second: the average frequencies are 0.1, 0.06, and 0.04 waves per second, respectively.12 These are averages: the output of a quake never consists of "pure notes." Rather, the waves of each type cover a band of frequencies, and low-frequency waves, which travel slightly faster than high-frequency ones, gradually pull ahead; this matches what happens to ocean waves when the whitecaps of a big "sea" travel away from the storm that caused them and become spaced out as an ocean swell (see fig. 7.5).

To make a long story short, there are always a variety of movements all happening simultaneously in the earth when seismic waves pass through. Measuring the energy output of a quake is correspondingly complicated. Exact and precisely timed measurements of earth movements (made with a seismometer) are unlikely to be obtainable anywhere near the epicenter of a big quake because seismometers there are likely to be broken, and even if they are not, they may give unreliable readings. The energy of a quake therefore has to be deduced from observations made far from the epicenter, in different directions and at different distances. A long list of factors has to be allowed for in the computations.

To begin with, the depth of the focus below the ground surface needs to be estimated. This is less than 100 km in the great majority of quakes and probably never exceeds 700 km, the depth at which the rock of the earth's mantle abruptly becomes more viscous and some of the subducting plates stop sinking (see chapter 15).13 To judge the magnitude of a quake, a seismologist then needs to discover the distance from each seismometer to the quake's focus, a task fraught with difficulties. Seismic waves do not travel in straight lines: they speed up as they descend, because of the increasing pressure; this causes their paths through the mantle to curve and to be deflected sharply if they reach the core. P-waves can continue through the core, but S-waves come to a halt as soon as they reach it, because the outer core is liquid. It is easy to see why P-waves travel through both solids and fluids (liquids and gases), whereas S-waves can travel only through solids. P-waves are compression waves, and fluids as well as (most) solids, when compressed, tend to spring back when the pressure is relaxed: that's what makes the waves. S-waves, however, are shear waves, and fluids do not spring back when a deforming (shearing) force is removed: they are limp.14

To sum up: Quakes produce a mixture of seismic waves of different types, with varying amplitudes and frequencies; the waves follow various paths through rocks of different densities, at a range of pressures. Discovering the extent of earth movement at the focus of a quake (the quake's magnitude) is unavoidably complicated. After making allowances for all the complexities, however, magnitudes can be computed approximately. For a big quake, the results are publicly announced as being "on the Richter scale" in honor of the American seismologist who, in 1935, first devised a formula for measuring earthquake magnitudes.15 His original formula has been improved and updated several times. Here we leap over all the difficulties, noting only that when magnitudes are measured at different observing stations, the results, though close, aren't necessarily identical. Moreover, the magnitudes of surface waves and body waves are seldom kept separate in newspaper accounts of an earthquake, although the difference is usually appreciable. For example, the British Geological Survey reported two magnitudes for the great earthquake that devastated Izmit, Turkey, and neighboring cities in August 1999: the magnitude of the body waves was 6.87, compared with 7.5 for the much more destructive surface waves.16 The magnitude scale for quakes behaves like the decibel scale for sounds: an increase of one on the magnitude scale for quakes indicates a tenfold increase in the extent of earth movement, just as an increase of one on the decibel scale for sound indicates a tenfold increase in sound intensity. Discovering the energy liberated by a quake requires yet another step beyond computing its magnitude. It turns out that if a quake has a magnitude that is one unit greater than that of another quake, then (provided both are greater than magnitude 5) the stronger quake liberates between twenty-seven and twenty-eight times as much energy as the weaker.17 For example, a quake of magnitude 6 liberates about 7.6 x 1010 kJ, and one of magnitude 7 liberates about 2.1 x 1012 kJ; in this case the energy of the stronger one is 27.6 times that of the weaker.

Estimates show that, over a study period of sixteen years (1977 to 1993) seismic waves transported earthquake energy through the earth at an average rate of about 4.7 million kilowatts.18 This represents the combined power of the waves emanating continually from earthquakes of all intensities happening in all the world's earthquake zones. The energy radiated by seismic waves is only about one-twentieth of the total energy produced by earthquakes, however.19 As remarked earlier, seismic waves radiating from a quake are analogous to sound waves radiating from the stroke of a hammer on a nailhead: they carry away only a small part of the energy generated. Seismic waves are, metaphorically, the background noise of the shifting tectonic plates, an inaudible noise, but one going on under our feet, in the earth's interior, all the time. They are carrying away the leftover energy of all earthquakes, after the initial, largest waves have done their damage if the quake was big.

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