UNIVERSIDADE
FEDERAL DA
BAHIA
CENTRO DE PESQUISA
EM
GEOF~SICA
E GEOLOGLA
SEISMIC SURFACE WAVES
Oldrich Novotny
Lecture notes for post-graduate studies
Instituto
de
Fisica
Instituto de Geociencias
Salvador, Bahia,
1999
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2) Love waves. The particle motion in these waves is transverse and parallel to the surface (Fig. 1.2). As opposed to Rayleigh waves, Love waves cannot.
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UNIVERSIDADE FEDERAL DA BAHIA
CENTRO DE PESQUISA EM GEOF~SICAE GEOLOGLA
SEISMIC SURFACE WAVES
Oldrich Novotny
Lecture notes for post-graduate studies
Instituto de Fisica
Instituto de Geociencias
Salvador, Bahia, 1999
Contents
- Contents
- Preface
- 1 Main Types of Elastic Waves and Their Properties - 1.1 Body waves - 1.2 Surface waves - 1.3 Main differences between seismic body waves and surface waves - 1.4 Dispersion of waves ..............................................................................1
- Seismic Surface Waves 2 Historical Development of the Theory of Elasticity and of the Theory of
- 2.1 Theory of elasticity in the seventeenth and eighteenth centuries
- 2.2 Propagation of light and the theory of elasticity
- 2.3 Mathematical theory of elasticity ........................................................1
- 2.4 Beginnings of seismology ....................................................................-
- 2.5 Studies of other types of surface waves - 2.5.1 Channel waves and higher modes - 2.5.2 PL waves and leaking modes ........................................................- - 2.5.3 Microseisms ..................................................................................-
- 3 Principles of Continuum Mechanics - 3.1 Mathematical models in physics ...........................................................- - 3.2 Displacement vector .............................................................................- - 3.3 Strain tensor - 3.3.1 Tensor of finite strain ....................................................................- - 3.3.2 Other strain measures ....................................................................- - 3.3.3 Physical meaning of the components of the tensor of finite strain - 3.3.4 Principal axes of strain - - 3.3.5 Tensor of infinitesimal strain ........................................................- - 3.3.6 Volume dilatation - 3.4 Stress vector and related problems - 3.4.1 Body forces and surface forces - 3.4.2 Stress vector - 3.4.3 Conditions of equilibrium in integral form - 3.4.4 Equations of motion in integral form - 3.4.5 One property of the stress vector - 3.5 Stress tensor - 3.5.1 Components of the stress tensor - 3 S.2 Cauchy's formula - 3.5.3 Conditions of equilibrium in differential form - 3.5.4 Equations of motion in differential form - 3.6 Stress-strain relations - 3.6.1 Rheological classification of substances - 3.6.2 Generalised Hooke's law - 3.7 Equations of motion - 3.7.1 Equations of motion for a homogeneous isotropic medium - 3.7.2 Wave equations - 3.8 A review of the most important formulae - Medium ......................................................................................................- 4 Separation of the Elastodynamic Equation in a Homogeneous Isotropic - 4.1 Wave equations in terms of potentials - 4.2 Expressions for the displacement and stress in terms of potentials - C.m esian coordinate 4.3 Special expressions for wave fields which are independent of one - 4.3.1 P-SV problems ..............................................................................- - 4.3.2 SH problems - 4.4 Plane waves ..........................................................................................- - 4.5 Surface waves as superpositions of body waves
- 5 Rayleigh Waves in a Homogeneous Isotropic Half-Space - 5.1 Potentials for a plane harmonic Rayleigh wave - 5.2 Displacement and stress components - 5.3 Boundary conditions - 5.4 Velocity of Rayleigh waves - 5.5 Polarisation - - 5.6 Non-existence of Love waves in a homogeneous half-space
- 6 Love Waves in a Layer on a Half-Space
- 6.1 Expressions for displacements ..............................................................-
- 6.2 Boundary conditions
- 6.3 Dispersion equation and its solutions - interference 6.4 Derivation of the dispersion equation from the condition of constructive - 6.4.1 Reflection and transmission of SH waves - 6.4.2 Condition of constructive interference
- 6.5 Methods of computing the group velocity
- 6.1 Expressions for displacements ..............................................................-
- 7 Rayleigh Waves in a Layer on a Half-Space - 7.1 Expressions for potentials - 7.2 Displacements and stresses - 7.3 Boundary conditions - 7.4 Dispersion equation - 7.5 Another form of the dispersion equation
- 8 Matrix Methods for Love Waves in a Layered Medium - 8.1 Model of the medium - 8.2 Matrix for one layer - 8.3 Matrix for a stack of layers .................................................................1 - 8.4 Expressions for the half-space - 8.5 Dispersion equation ............................................................................1 - 8.5.1 Traditional formulation - 8.5.2 Formulation in terms of the inverse matrices - 8.6 Comments on some numerical problems - 8.7 Other foms of the dispersion equation. Thomson-Haskell matrices
- 6 Love Waves in a Layer on a Half-Space
- 9 Matrix Methods for Rayleigh Waves in a Layered Medium - 9.1 Basic notations and formulae - 9.2 Matrix for one layer - 9.3 Boundary conditions and the matrix for a stack of layers - 9.4 Expressions for the half-space and the dispersion equation - 9.5 Matrices of the sixth order .................................................................1 - 9.5.1 Associated matrices - 9.5.2 Associated matrices in the Rayleigh-wave problem - 9.6 Historical remarks and other formulations of the dispersion equation - 9.6.1 Thomson-Haskell matrices and their modifications - 9.6.2 Knopoff s method - 9.6.3 Computing reflection and transmission coefficients
- 10 .Matrix Formulations of Some Body-Wave Problems - wave 10.1 Motion of the surface of a layered medium caused by an incident SH - zone 10.2 Reflection and transmission coefficients of SH waves for a transition - 10.3 Spectral ratio of the horizontal and vertical components of P waves - transition zone 10.4 Reflection and transmission coefficients of P and S V waves for a - 10.5 Some other studies
- 11 Wave Propagation in Dispersive Media - medium 11.1 Superposition of two plane harmonic waves in a non-dispersive - 11.2 Superposition of two plane harmonic waves in a dispersive medium - 11.3 Propagation of a plane wave with a narrow spectrum - 11.3.1 Form of a wave with a narrow spectrum - from observations 11.3.2 Simple methods of determining the phase and group velocities - 11.4 Propagation of a plane wave with a broad spectrum - 11.4.1 Asymptotic expressions for large distances - 11.4.2 Properties of the asymptotic solution - observations 11.5 The peak and trough technique for estimating group velocities from - observations .........................................................................................- 11.6 The peak and trough technique for estimating phase velocities from - 11.7 Determination of phase velocities from Fourier spectra - 1 1 .8 Time-frequency analysis ...................................................................1
- 10 .Matrix Formulations of Some Body-Wave Problems - wave 10.1 Motion of the surface of a layered medium caused by an incident SH - zone 10.2 Reflection and transmission coefficients of SH waves for a transition - 10.3 Spectral ratio of the horizontal and vertical components of P waves - transition zone 10.4 Reflection and transmission coefficients of P and S V waves for a - 10.5 Some other studies
12. Examples of Structural Studies by Surface Waves ............................. 147 12.1 Short-period surface waves generated by explosions and their
interpretation ......................................................................................... 147
12.2 Surface waves generated by earthquakes and their application in studies
of the Earth crust and upper mantle ...................................................... 148
References ....................................................................................................-
the application of computers which, e.g., have made it possible to solve transcendental dispersion equations for surface waves quickly. Since then, surface waves have been used to treat many specific problems, such as: to study the existence and structure of the so-called low-velocity channel in the upper mantle; to distinguish between the continental and oceanic type of the Earth's crust; to determine the mean parameters of the Earth's crust in extended regions, including regions which are difficult to access (mountains, oceans, polar regions); to study lateral inhomogeneities in the Earth's crust, e.g., the position of faults. Another very promising application of surface waves seems to be the computation of complete synthetic seismograms by summirig surface-wave modes. Surface waves also find technical applications in non-destructive testing of materials, electro-mechanical transducers and many others. Moreover, the mathematical methods used in the theory of surface waves are also applicable to some problems of propagation of elastic body waves, electromagnetic waves (e.g., waves in the ionosphere), temperature waves, in the physics of thin layers, etc. Although surface waves are important from the scientific and practical points of view, less attention is usually paid to them in physics textbooks than to body waves. This is caused mainly by the more complicated physical character of surface waves. For example, it is dificult to imagine them in terms of rays propagating from the source. On the other hand, surface waves do not represent a principally new type of wave, but only an interference phenomenon of body waves. Therefore, the theory presented in these lecture notes can also be used in studies of other types of interference waves we encounter in seismology, physics and technical practice. These lecture notes have been written for the purposes of post-graduate studies in geophysics, in particular for the corresponding part of the course of lectures on the attenution and dispersion of seismic waves, organised by the Universidade Federal da Bahia, Salvador, Brazil. I would like to thank the Centro de Pesquisa em Geofisica e Geologia (CPGGRJFBA), Departamento de Geofisica Nuclear do Instituto de Fisica, and the Instituto de Geociencias for their support in preparing this text. I wish to express my thanks especially to the CNPq (Conselho Nacional de Desenvolvimento Cientifico e Tecnol6gico) for providing me with the fellowship which made my stay at the Universidade Federal da Bahia possible. I thank the MinistCrio de Educaqiio e Cultura, and the CPGG for their subsequent fellowships which helped me to extend and complete this text. I would also like to express my gratitude to the students and professors whose advice and comments contributed to improving the text. I thank RNDr. Jaroslav Tauer, CSc for the language revision of the text with an understanding of the subject. My thanks are also due to my wife, Mrs. h k a Novotni, for the technical preparation of the text. I shall also be grateful to every reader for any critical comments and remarks concerning these notes.
Salvador, 1999 (^) Oldrich Novotny
Chapter 1
Main Types of Elastic Waves and Their
Properties
In physics, waves are usually divided into progressive and standing waves. Seismic waves are also of these two types. Progressive seismic waves propagate away from seismic sources. In these lecture notes, we shall deal only with this type of waves. Standing seismic waves, known as the free oscillations of the Earth, represent vibrations of the Earth as a whole. These oscillations are generated by strong earthquakes. From the point of view of the spatial concentration of energy, waves can be divided into body waves and surface waves. Body waves can propagate into the interior of the corresponding medium, whereas surface waves are concentrated along the surface of the medium. Acoustic waves in air, or electromagnetic waves in vacuum are examples of body waves. Examples of surface waves have already been mentioned in the Preface. Note that, instead of surface waves, we should rather speak of a broader category of guided waves. Guided waves propagate along the surface of a medium (surface waves), along internal discontinuities, or other waveguides. Since seismic surface waves represent the most important type of seismic guided waves, we shall speak only of surface waves. In these lecture notes we shall deal with various types of elastic waves. In order to obtain an initial idea of them, we shall give a brief review here; see Tab. 1.1. The corresponding derivations will be given in the following chapters.
Table 1.1. Principal types of progressive elastic waves.
longitudinal waves
/ body waves transverse waves elastic waves Rayleigh waves surface waves
<Love waves
1.1 Body Waves
It follows from the theory of elasticity that there are two principal types of elastic body waves:
- Longitudinal waves, also called compressional, dilatational or irrotational waves. In seismology, they are also called P waves (primary waves), because they represent the first waves appearing on seismograms. These
1.2 Surface Waves
Only longitudinal and transverse waves can propagate in a homogeneous, isotropic and unlimited medium. If the medium is bounded, another type of waves, surface waves, can be guided along the surface of the medium. These waves usually form the principal phase of seismograms. There are two types of surface elastic waves:
- Rayleigh waves. These waves are elliptically polarised in the plane which is determined by the normal to the surface and by the direction of propagation (Fig.- 1.2). Near the surface of a homogeneous half-space, the particle motion is a retrograde vertical ellipse (anticlockwise for a wave travelling to the right).
- Love waves. The particle motion in these waves is transverse and parallel to the surface (Fig. 1.2). As opposed to Rayleigh waves, Love waves cannot propagate in a homogeneous half-space. Love waves can propagate only if the S-wave velocity generally increases with the distance from the surface of the medium. (a) Rayleigh wave
,. tij)] Love wave
Fig. 1.2. The particle motion for surface waves: (a) Rayleigh waves and (b) Love waves. (After Fowler (1994)).
The simplest medium in which Rayleigh waves can propagate is a homogeneous isotropic half-space. The velocity of Rayleigh waves in this
medium, c, is slightly less than the transverse wave velocity, c = 0.9P, and is
independent of frequency. Thus, Rayleigh waves in this simple model of the medium are non-dispersive. The simplest model in which Love waves can propagate consists of a homogeneous isotropic layer on a homogeneous isotropic half-space. Both the Rayleigh and Love waves in this model are already dispersive, i.e. their velocities are dependent on frequency. As we have already mentioned, elastic surface waves do not represent principally new types of waves, but only interference phenomena of body
waves. Therefore, in principle, we could attempt to construct the wave field of surface waves (and of other guided waves) by summing body waves. However, this approach would be inconvenient if a large number of waves is to be taken into account (thin layers, large distances from the source). Therefore, a more appropriate mathematical description must be sought for surface waves, as well as for the other interference waves. We shall emphasise the interference character of surface waves in many places of these lecture notes, in order to gain a deeper insight into the formation of these waves, to understand their specific properties, such as their dispersion and polarisation better, and to be able to apply the same mathematical approaches (e.g., matrix methods) both to surface-wave and body-wave problems.
1.3 Main Differences between Seismic Body Waves and Surface Waves
Let us summarise the main properties of seismic body waves and surface waves, as observed on seismograms of distant earthquakes:
- Records of a seismic event begin with longitudinal waves, followed by transverse waves, and finally by surface waves.
- Surface waves usually have larger amplitudes and longer periods.
- Surface waves display a characteristic dispersion and polarisation. An example of a seismogram in shown in Fig 1.3. Other examples will be given below.
Fig 1.3. The China earthquake of November 13, 1965, recorded at Kiruna, Sweden. The higher-mode Rayleigh waves are exceptionally pronounced (the waves with higher fiequencies at the beginning of the surface-wave group). (After BAth (1979)).
1.4 Dispersion of Waves
In these lecture notes we shall pay much attention to the dispersion of surface waves.
Chapter 2
Historical Development of the Theory of
Elasticity and of the Theory of Seismic Surface Waves Science is the knowledge of many, orderly and methodically digested and arranged, so as to become attainable by one. (J.F.W. Herschel)
The theory of seismic waves is based on the theory of elasticity. In this chapter we shall deal with the historical development of these theories, especially with those aspects of the theory of elasticity which are closely related to the development of seismology. The theory of elasticity studies the behaviour of bodies subjected to forces, both as to their deformation as well as to their ultimate disruption under sufficiently large stresses. In preparing this chapter we have drawn mainly on the books by Love (1927), Love (191 I), Bith (1979), and the very comprehensive treatise by Todhunter and Pearson (1886).
2.1 Theory of Elasticity in the Seventeenth and Eighteenth Centuries
The modem theory of elasticity may be considered to have originated in 1821, when Navier first presented the equations for the equilibrium and motion of elastic solids. To understand the evolution of our modern conceptions, it is necessary to go back to the research of the seventeenth and eighteenth centuries, when experimental knowledge of the behaviour of strained bodies was gained and some special principles were formulated. The first memoir requiring notice is the second dialogue of the Discorsie Dimostrazioni matematiche by Galileo Galilei (1638). This dialogue not only gave the impulse, but also determined the direction which was subsequently followed by many researchers. Galileo formulated conditions with regard to the fracture of solids (rods, beams and hollow cylinders). The noteworthy feature of his discussion is his assumption that the fibres of a strained beam are inextensible. He endeavoured to determine the resistance of a beam, one end of which is built into a wall, at the moment it tends to break under its own or an applied weight. He found that, with increasing load, the beam bends around an axis perpendicular to its length and situated in the plane of the wall. The problem to determine this axis is referred to as Galileo's problem. Although Galileo did not give any mathematical relations between load and deformation, his work was pioneering in the theory of elasticity. Undoubtedly the next great landmark in the theory, initiated by Galileo's question, is the discovery of Hooke's law. This law provided the necessary experimental foundation for the theory. Hooke discovered this law in 1660, but
did not publish until 1676. In 1678 he formulated this law as follows: "Ut tensio sic vis", i.e. "The power of any spring is in the same proportion to the tension thereof'. By "tension" Hooke understood, as he proceeded to explain, that which we now call "extension". Hooke did not probably apply this law to solving Galileo's problem. This application was made by Mariotte, who discovered the same law independently in 1680. Hooke in England and Mariotte in France then appropriated the experimental discovery of what we now term stress-strain relations. In the interval between the discovery of Hooke's law and that of the general differential equations of elasticity by Navier, the attention of the researchers in the elasticity theory was chiefly directed to the solution and extension of Galileo's problem, and the related theories of the vibration of bars and plates, and the stability of columns. Many famous mathematicians and physicists, such as James Bernoulli, Daniel Bernoulli, Euler, Lagrange, Coulomb, Young, took part in these investigations. Although many special problems were solved during this period, these investigations did not lead to broad generalisations. The situation was complicated mainly by unresolved problems concerning the constitution of bodies. According to the Newtonian conception, material bodies are made up of small parts which act on one another by means of central forces. Newton regarded the "molecules" to have finite sizes and definite shapes. However, his successors gradually simplified the "molecules" into material points. The conception of material points was found to be very useful in many branches of mechanics. However, its application to the problems of elasticity often led to oversimplified results. In particular, the conception of material points, between which central forces act, leads to a smaller number of elastic constants than those which are actually necessary to describe real media. Navier was the first to investigate the general equations of the theory of elasticity. He presented his memoir to the Paris Academy in 1821. He set out from the Newtonian conception of the constitution of bodies, and assumed that elastic reactions arise from the variations in the intermolecular forces which are due to changes in the molecular configuration. He regarded the molecules as material points, and assumed that the force between two molecules, whose distance is slightly increased, is proportional to the product of the increment of the distance and some function of the initial distance. His method consisted in forming an expression for the forces that act upon a displaced molecule, which then yielded the equations of motion of the molecule. The equations were thus obtained in terms of the displacements of the molecule. Navier assumed the material to be isotropic, and the equations of equilibrium and vibration to contain a single constant. We now know that an isotropic medium is characterised by two elastic constants. This demonstrates the simplifications arising from the conception of material points and central forces acting between them.
2.2 Propagation of Light and the Theory of Elasticity
In the same year, 1821, in which Navier's memoir was read to the Paris Academy, the study of elasticity received a powerful impulse from an
at the time), Cauchy found 15 true elastic constants; actually he found 21 independent constants, but 6 of these constants expressed the initial stress and vanished identically if the initial state was one of zero stress. Cauchy also applied his equations to the question of the propagation of light in crystalline as well as in isotropic media. The first memoir by Poisson dealing with the problems of elasticity was read before the Paris Academy in 1828. Poisson obtained the equations of equilibrium and motion of isotropic elastic solids which were identical with those of Navier. The memoir is very remarkable for its numerous applications of the general theory to special problems. Cauchy and Poisson, as well as other researchers, applied the theory of elasticity, the former two had developed on the basis of material points and central forces, to many problems of vibrations and of statical elasticity. It provided the means for testing its consequences experimentally, but adequate experiments were made much later. Poisson (1831) used his theory to investigate the propagation of waves through an isotropic elastic solid of unlimited extent. He proved that two kinds of waves with different velocities could propagate in such a medium. He found that, at a large distance from the source of disturbance, the motion transmitted by the quicker wave was longitudinal, and the motion transmitted by the slower wave was transverse. This theory indicated that the ratio of these velocities was
&:l. Poisson also considered the vibration of a sphere.
Afterwards Stokes (1849) proved that the quicker wave was a wave of irrotational dilatation, and the slower wave was a wave of equivoluminal distorsion,characterized by differential rotation of the elements of the body. He also derived the well-known formulae for the velocities of the two waves,
dm and m, where p denotes the density, ,LL the rigidity, and A + (213)~the modulus of compression. These two velocities will be denoted
here by a: and P. This is the first time that we have come across waves P and S,
now so well known in seismology. Stokes also proved that the two waves were separated completely at a sufficiently large distance from the initially disturbed region. At shorter distances they are superposed for part of the time. Note that the "dilatational wave" is now also called the "longitudinal wave" or "compressional wave". Analogously, the "distortional wave" is also termed the "transverse wave" or "shear wave". Green (1839) was dissatisfied with the hypothesis of material points and central forces on which the theory was based, and he sought a new foundation. Starting from what is now called the principle of the conservation of energy he propounded a new method of obtaining the equations of elasticity. He derived the potential energy of the strained elastic body, expressed in terms of the components of strain, and then applied the methods which are used in analytical mechanics. Green stated that this approach "appears to be more especially applicable to problems that relate to the motions of systems composed of an immense number of particles mutually acting upon each other". He deduced the equations of elasticity, in the general case containing 21 constants. In the case of isotropy there are two constants, and the equations are
the same as those of Cauchy's first memoir. The revolution which Green effected in the elements of the theory is comparable in importance with that produced by Navier's discovery of the general equations. Kelvin supported the existence of Green's strain-energy function on the basis of the first and second laws of thermodynamics. The methods of Navier, of Poisson, and of Cauchy's later memoirs lead to equations of motion containing fewer constants than occur in the equations obtained by the methods of Green, and of Cauchy's first memoir. The questions in dispute are as follows: Is elastic anisotropy to be characterised by 21 constants or by 15, and is elastic isotropy to be characterised by two constants or by one? The two theories were called the "multi-constant" theory and the "rari-constant" theory, respectively. The importance of the discrepancy was first emphasised by Stokes in 1845. He made the observation that resistance to compression and resistance to shearing are the two fundamental kinds of elastic resistance, and he definitely introduced the two principal moduli of elasticity. The two parameters are now called the modulus of compressibility and the modulus of rigidity. Much attention was also paid to the ratio of lateral contraction to longitudinal extension of a bar under tractive load. This ratio is often called "Poisson's ratio". From his theory Poisson deduced that this ratio must be 114. However, experiments on some materials did not support this result. The experimental evidence led Lam6 to adopt also the multi-constant equations, and after the publication of his book in 1852 they were generally adopted. We have already mentioned Poisson's discovery of longitudinal and transverse waves which can propagate through the interior of a solid elastic body. This theory takes no account of the existence of a boundary. When the waves from a source reach the boundary, they are reflected, but in general the longitudinal wave, on reflection, gives rise to both kinds of waves, and the same is true of the transverse wave. Any subsequent state of the body can be represented as the result of superposing waves of both kinds reflected one or more times at the boundary. Without mathematical analysis it is not easy to see what the properties of the resulting wave will be. In 1887, Lord Rayleigh discovered that a specific wave can be formed near the fi-ee surface of a homogeneous body. The wave has the following main properties:
it propagates along the surface at a certain velocity, less than both a
and p;
it does not penetrate far beneath the surface because its amplitude decreases exponentially with distance from the surface; it is elliptically polarised in the plane determined by the normal to the surface and by the direction of propagation. In Lord Rayleigh's work the surface was regarded as an unlimited plane, and the waves could be of any length. Gravity was neglected, and it was found that the wave velocity was independent of the wavelength. Such waves have since been called Rayleigh waves, after the person who had discovered them theoretically. (Note that Love (191 1) called them "Rayleigh-waves", but the hyphen was later omitted). These waves belong to the category of so-called
Very soon it was noticed that the records of distant earthquakes displayed two very distinct stages, the first characterised by a very feeble motion, the second by a much larger motion. These stages were called the "preliminary tremor" and the "main shock" (the "main shock" was also often described as the "large waves" or sometimes as the "principal portion"). The idea that these two waves might be dilatational and distortional waves, emerging at the surface, took firm root among seismologists for a time. In the light of increasing knowledge this idea had to be abandoned. As mentioned above, Rayleigh suggested that the surface waves he had investigated might play an important part in earthquakes. This suggestion was not, at first, well received by seismologists, mainly because the records did not show a preponderance of vertical motion in the main shock. It was first systematically applied to the interpretation of seismic records by Oldham (1900). He recognised two distinct phases in the preliminary tremors, and showed that their travel times to distant stations correspond to the propagation through the body of the Earth of waves travelling with practically constant velocities. On the other hand, the main shock is recorded at times which correspond to the propagation over the surface of the Earth of waves travelling with a different nearly constant velocity. Oldham, therefore, proposed to identify the first and second phases of the preliminary tremors respectively with dilatational and distorsional waves, travelling along nearly straight paths through the body of the Earth, and he proposed to regard the main shock as Rayleigh waves. The suggestion that the first and second phases of the preliminary tremors should be regarded as dilatational and distorsional waves, transmitted through the body of the Earth, was generally accepted. However, the proposed identification of the main shock with Rayleigh waves was less favourably received for two reasons: partly on account of the difficulty already mentioned with regard to the ratio of the horizontal and vertical displacements; partly because observation showed that a large part of the motion transmitted in the main shock was a horizontal motion at right angles to the direction of propagation (these waves are now called Love waves). Lamb (1904) considered in detail the waves produced by impulsive pressure suddenly applied at a point of the surface. The motion recorded at a distant point begins suddenly at a time corresponding to the arrival of the longitudinal wave. The surface rises rather sharply, and then subsides very gradually without oscillation. At the time corresponding to the arrival of the transverse wave a slight motion occurs. This is followed, at the time corresponding to the arrival of the Rayleigh wave, by a much larger motion, after which the motion gradually subsides without oscillation. The subsidence is indefinitely prolonged. Lamb's theory easily accounted for some of the most prominent features of seismic records, namely the first and second phases of the preliminary tremors and the larger disturbance of the main shock. However, it did not account for the existence of horizontal motion at right angles to the direction of propagation. Such motions are observed both in the second phase of the preliminary tremors and in the main shock. The existence of such motions in the second phase of the preliminary tremors could be accounted for easily by
assuming a different kind of initial disturbance, for example by a sudden horizontal shearing motion, or by a couple applied locally. But no assumption as to the nature of the disturbance at the source was able to account for the relatively large horizontal displacements in the main shock which were transverse to the direction of propagation. Moreover, the theory did not account for the approximately periodic oscillations which were a prominent feature in all seismic records. Lamb suggested that these might be due to a succession of primitive shocks. Nevertheless, such an explanation seemed to be rather artificial. All the controversies between theory and observations were resolved in an excellent way by Love (1911). Instead of a homogeneous half-space, which was considered by Rayleigh and Lamb, Love considered an elastic medium consisting of a layer on a half-space. The main properties of surface waves in this medium already agreed with observations. In particular, he found that a new type of surface waves can propagate in a layer on a half-space. These waves are polarised in the horizontal plane perpendicularly to the direction of propagation, so that give a good explanation of the transverse motion in the main shock. Such waves have since been called Love waves. The propagation of Rayleigh waves in a layer on a half-space has been studied in many papers, starting with those by Bromwich (1898) and Love (1911). Love found that the ratio of the horizontal and vertical components of these waves was already close to the observed values. For a review we refer the reader to Ewing et al. (1957); see also the papers by Bolt and Butcher (1960), and by Money and Bolt (1966). Rayleigh and Love waves in a layer on a half-space, and in all more complicated models of the medium, are dispersive. The dispersion equation for Love waves in one layer on a half-space was derived by Love (191 I), for two layers on a half-space by Stoneley and Tillotson (1928), and for three layers on a half-space by Stoneley (1937). We shall deal with Rayleigh and Love waves in the simplest models of the medium in Chapters 5 to 7, after explaining the necessary principles of continuum mechanics in Chapter 3 and of the theory of elastic waves in Chapter 4.
2.5 Studies of Other Types of Surface Waves
We have seen that the main shock (now called the "main phase" of a seismogram) was originally interpreted as a body wave, but later' it was found to be formed by surface waves. In particular, this seismic phase is formed by the fundamental modes (fundamental tones) of Rayleigh and Love waves. Similarly, also other waves on seismograms were at first interpreted as body waves, but later identified with surface waves.
2.5.1 Channel waves and higher modes
Press and Ewing (1952) found two short-period large-amplitude waves on the records of surface waves crossing North America. The existence of these waves