DOCSLIB.ORG

Music of the Spheres: Teaching Quantum Field Theory at The

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Music of the Spheres: Teaching Quantum Field Theory at The Music of the Spheres: Teaching Quantum Field Theory at the Introductory Level Ian H. Redmount∗ Department of Physics Saint Louis University St. Louis, MO 63103–2010 USA (Dated: August 6, 2019) Abstract Quantum field theory has formed the conceptual framework of most of physics for more than sixty years. It incorporates a complete revision of our conception of the nature of matter and existence itself. Yet it is rarely taught, or even mentioned, in introductory physics—from high school, college, and university survey courses through upper-division “modern physics” courses. This omission is not necessary: This paper describes an approach through which the fundamental concepts and consequent insights of quantum field theory can be grasped, building upon familiar notions from classical and quantum mechanics. arXiv:1908.01579v1 [physics.hist-ph] 27 Jun 2019 ∗[email protected] 1 I. INTRODUCTION Quantum field theory, or the theory of quantized fields, forms the current conceptual framework of almost all of physics, and has ever since Richard Feynman, Julian Schwinger, and Tomonaga Shin-ichiro discovered in 1948 how to make the theory give finite answers to physical questions. Only gravitation lies outside the theory’s purview; this is the mo- tivation for searches for a “quantum theory of gravity” or “Theory of Everything,” from Schr¨odinger in the 1940’s through today’s string theory, M-theory, and loop quantum grav- ity. Einsteinian quantum field theory is the foundation of elementary-particle physics, while the Newtonian limit undergirds our understanding of solid-state physics, superfluidity, and superconductivity.[8] The theory affords a sweeping syntheses of many features of physics, and a profound new understanding of the nature of existence itself. Given the centrality and scope of the theory, it is remarkably difficult to find an introduc- tory physics text—whether a high-school text, an algebra/trig-based college physics text, a calculus-based university physics text, or even a text for a higher-level “modern physics” course—which even mentions quantum field theory, much less explains it. (Recent texts by Hobson [1] and Redmount [2] are notable exceptions). There are semi-popular books which describe it, but its absence from introductory physics texts and courses is striking. As a result, for example, intrductory treatments of elementary particles—which do appear in all the texts—read like classical botany, all terminology and taxonomy. It has been proposed to teach string theory at the high-school level, but this is surely putting the cart before the horse, as all the motivations for string theory are rooted in issues which arise in quantum field theory; indeed, string theory is a quantum field theory. Recently Hobson [3–6] and Huggins [7] have argued forcefully for a more central role for quantum field theory in the teaching of introductory physics, in particular to clarify many aspects of quantum mechanics. Of course it would be difficult even to set up the quantum-field-theoretic calculation of anything at the introductory level. But the fundamental ideas and breathtaking insights of the theory are readily understandable. They can be explained by building on concepts from classical and quantum physics which are accessible in elementary treatments. My purpose in this paper is to sketch out one such approach. The necessary foundations are described in Secs. II and III. The emergence of quantum field theory from these roots, and the features and implications of the theory, are detailed in Sec. IV. 2 II. QUANTUM MECHANICS Quantum field theory is the quantum mechanics of fields—extended systems—e.g., fluids, or electromagnetic fields, or the quantum waves of elementary particles. Quantum mechanics is built on two principles, both inferred from observation in the early decades of the 20th century: Electromagnetic waves, described classically as continuous fields, actually come in discrete “packets,” “lumps,” “quanta,” or “photons.” And conversely, classical particles of matter, such as electrons, protons, and neutrons, must be described via continuous wave fields. The particulate nature—quantization—of electromagnetic waves was introduced by Max Planck in 1900, in order to get the spectrum of radiation from hot bodies right. (This is usually considered the opening shot of the quantum revolution; classical electromagnetic theory fails miserably here.) Albert Einstein invoked it in 1905 to explain the photoelectric effect, and Arthur Holly Compton demonstrated it with his x-ray scattering experiments in 1923. The mechanical properties of the particles or photons, viz., energy E and momen- tum p, are related to the properties of the corresponding waves, viz., frequency ν, angular frequency ω =2πν, wavelength λ, and wave number k =2π/λ, via the familiar relations E = hν = ~ω , (1a) and h p = = ~k , (1b) λ where h = 2π~ = 6.626 ... 10−34 J s is the “fundamental quantum of action” intro- × duced by Planck. These equations are not as radical as they might appear: They accord with the classical relations for the energies and momenta of electromagnetic wave packets which follow from the Maxwell equations. They do not, however, represent a throwback to Newton’s corpuscular theory of light. The phenomena which reveal the wave nature of light—interference and diffraction—remain, even when the experiments are performed with one photon at a time. Wave and particle natures coexist, the famous wave-particle duality of quantum mechanics. The converse principle, that matter particles exhibit behaviors which must be described with waves, was revealed during the same period by experiments akin to those that originally revealed the wave nature of light. The scattering of electrons and neutrons from crystals, 3 e.g., in the experiments of Max von Laue and Davisson and Germer, displayed interference patterns. Schr¨odinger tunneling of particles through potential barriers they do not have the energy to surmount is simply the counterpart of a phenomenon in wave optics, where it is known as Frustrated Total Internal Reflection. The relationships connecting matter-wave properties with particle properties were formulated by Louis de Broglie in 1921: h p λ = i.e., k = (2a) p ~ and E E ν = i.e., ω = , (2b) h ~ analogous to the photon relations above. Incorporating the wave nature of particles into mechanics fundamentally alters the nature of the science; the questions which are asked and answered are changed. A basic problem in classical mechanics is to determine the trajectory x(t) of a particle, given its initial position and momentum x0 and p0. A wave description does not allow this—that is the content of the Heisenberg Uncertainty Principle. In quantum mechanics a basic problem is the deter- mination of a particle’s wave function ψ(x, t), given its initial configuration ψ(x, 0). The squared magnitude of the wave function is associated with the probability density for finding the particle, in the standard or Copenhagen interpretation of the theory. Alternative inter- pretations have been and continue to be proposed and debated, but all are interpretations. They change the outcome of no calculation or prediction of the theory. The basic “equation of motion” in ordinary quantum mechanics, the Schr¨odinger wave equation, is obtained from the Newtonian energy-momentum relation p2 + U = E, (3a) 2m for a particle of mass m with potential energy U, by imposing the de Broglie relations (2) in the form of differential operators applied to the wave function. Calculus speakers will recognize the resulting form: ~2 ∂ψ 2ψ + U ψ = i~ , (3b) − 2m ∇ ∂t with i = √ 1 the imaginary unit, and the time-independent form − ~2 2ψ + U ψ = E ψ , (3c) − 2m ∇ 4 for wave functions describing particle states of definite energy E. Despite the formidable appearance of these equations, their physical content is simply that of the energy-momentum relation (3a). There are only five exactly, analytically soluble problems in quantum mechanics: the free particle, the “particle in a box,” the free rotor, the hydrogen atom, and the harmonic oscillator. (If this seems rather limited, it may be recalled that there are not that many more exactly, analytically soluble problems in classical mechanics.) The last of these is the key to understanding quantum field theory. For an oscillator in one dimension, with mass m and “spring constant” k = mω2, the time-independent Schr¨odinger equation (3c) for the energy levels of the system takes the form ~2 dψ + 1 mω2 ψ = E ψ . (4a) − 2m dx2 2 The resulting wave functions are 1 mω mω 1/4 mω − x2 n √ −1/2 2 ~ ψn(x)= ~ (2 n! π) Hn ~ x e , (4b) r with Hn the Hermite polynomials discovered in the 19th century. The corresponding energy levels are 1 ~ En =(n + 2 ) ω , (4c) where n is a non-negative integer. The oscillator has zero-point energy 1 ~ E0 = 2 ω (4d) in its lowest-energy or ground state. And the excitation energies of all the levels above this come in equal increments ∆E = ~ω. These features will prove crucial in what is to follow. III. DYNAMICS OF SYSTEMS: NORMAL MODES The other physical principle underlying quantum field theory is a result from classical mechanics: The dynamics of any system, however complex, obeying a linear equation of motion—including almost any system slightly perturbed from any equilibrium—can be de- scribed via a collection of independent harmonic oscillators. The most general motion of the system can be represented as a combination of patterns called normal modes or harmonics. (The harmonics of the vibrating strings or air columns of musical instruments are normal 5 k m k m FIG. 1: The not-quite-simplest system of two coupled harmonic oscillators. modes.) And each normal mode behaves as an individual harmonic oscillator, uncoupled from the others. Consider, for example, the system illustrated in Fig.
Load more

Recommended publications
  • Resonance Beyond Frequency-Matching
    Resonance Beyond Frequency-Matching Zhenyu Wang (王振宇)1, Mingzhe Li (李明哲)1,2, & Ruifang Wang (王瑞方)1,2* 1 Department of Physics, Xiamen University, Xiamen 361005, China. 2 Institute of Theoretical Physics and Astrophysics, Xiamen University, Xiamen 361005, China. *Corresponding author. [email protected] Resonance, defined as the oscillation of a system when the temporal frequency of an external stimulus matches a natural frequency of the system, is important in both fundamental physics and applied disciplines. However, the spatial character of oscillation is not considered in the definition of resonance. In this work, we reveal the creation of spatial resonance when the stimulus matches the space pattern of a normal mode in an oscillating system. The complete resonance, which we call multidimensional resonance, is a combination of both the spatial and the conventionally defined (temporal) resonance and can be several orders of magnitude stronger than the temporal resonance alone. We further elucidate that the spin wave produced by multidimensional resonance drives considerably faster reversal of the vortex core in a magnetic nanodisk. Our findings provide insight into the nature of wave dynamics and open the door to novel applications. I. INTRODUCTION Resonance is a universal property of oscillation in both classical and quantum physics[1,2]. Resonance occurs at a wide range of scales, from subatomic particles[2,3] to astronomical objects[4]. A thorough understanding of resonance is therefore crucial for both fundamental research[4-8] and numerous related applications[9-12]. The simplest resonance system is composed of one oscillating element, for instance, a pendulum. Such a simple system features a single inherent resonance frequency.
    [Show full text]
  • Normal Modes of the Earth
    Proceedings of the Second HELAS International Conference IOP Publishing Journal of Physics: Conference Series 118 (2008) 012004 doi:10.1088/1742-6596/118/1/012004 Normal modes of the Earth Jean-Paul Montagner and Genevi`eve Roult Institut de Physique du Globe, UMR/CNRS 7154, 4 Place Jussieu, 75252 Paris, France E-mail: [email protected] Abstract. The free oscillations of the Earth were observed for the first time in the 1960s. They can be divided into spheroidal modes and toroidal modes, which are characterized by three quantum numbers n, l, and m. In a spherically symmetric Earth, the modes are degenerate in m, but the influence of rotation and lateral heterogeneities within the Earth splits the modes and lifts this degeneracy. The occurrence of the Great Sumatra-Andaman earthquake on 24 December 2004 provided unprecedented high-quality seismic data recorded by the broadband stations of the FDSN (Federation of Digital Seismograph Networks). For the first time, it has been possible to observe a very large collection of split modes, not only spheroidal modes but also toroidal modes. 1. Introduction Seismic waves can be generated by different kinds of sources (tectonic, volcanic, oceanic, atmospheric, cryospheric, or human activity). They are recorded by seismometers in a very broad frequency band. Modern broadband seismometers which equip global seismic networks (such as GEOSCOPE or IRIS/GSN) record seismic waves between 0.1 mHz and 10 Hz. Most seismologists use seismic records at frequencies larger than 10 mHz (e.g. [1]). However, the very low frequency range (below 10 mHz) has also been used extensively over the last 40 years and provides unvaluable information on the whole Earth.
    [Show full text]
  • 22.51 Course Notes, Chapter 9: Harmonic Oscillator
    9. Harmonic Oscillator 9.1 Harmonic Oscillator 9.1.1 Classical harmonic oscillator and h.o. model 9.1.2 Oscillator Hamiltonian: Position and momentum operators 9.1.3 Position representation 9.1.4 Heisenberg picture 9.1.5 Schr¨odinger picture 9.2 Uncertainty relationships 9.3 Coherent States 9.3.1 Expansion in terms of number states 9.3.2 Non-Orthogonality 9.3.3 Uncertainty relationships 9.3.4 X-representation 9.4 Phonons 9.4.1 Harmonic oscillator model for a crystal 9.4.2 Phonons as normal modes of the lattice vibration 9.4.3 Thermal energy density and Specific Heat 9.1 Harmonic Oscillator We have considered up to this moment only systems with a finite number of energy levels; we are now going to consider a system with an infinite number of energy levels: the quantum harmonic oscillator (h.o.). The quantum h.o. is a model that describes systems with a characteristic energy spectrum, given by a ladder of evenly spaced energy levels. The energy difference between two consecutive levels is ∆E. The number of levels is infinite, but there must exist a minimum energy, since the energy must always be positive. Given this spectrum, we expect the Hamiltonian will have the form 1 n = n + ~ω n , H | i 2 | i where each level in the ladder is identified by a number n. The name of the model is due to the analogy with characteristics of classical h.o., which we will review first. 9.1.1 Classical harmonic oscillator and h.o.
    [Show full text]
  • Normal Modes (Free Oscillations)
    Introduction to Seismology: Lecture Notes 22 April 2005 SEISMOLOGY: NORMAL MODES (FREE OSCILLATIONS) DECOMPOSING SEISMOLOGY INTO SUBDISCIPLINES Seismology can be decomposed into three representative subdisciplines: body waves, surface waves, and normal modes of free oscillation. Technically, these domains form a continuum, each pertaining to particular frequency bands, spatial scales, etc. In all cases, these representations satisfy the wave equation, but each is subject to different boundary conditions and simplifying assumptions. Each is therefore relevant to particular types of subsurface investigation. Below is a table summarizing the salient characteristics of the three. Boundary Seismic Domains Type Application Data Conditions Body Waves P-SV SH High frequency travel times; waveforms unbounded dispersion; group c(w) & Surface Waves Rayleigh Love Lithosphere phase u(w) velocities interfaces spherical Normal Modes Spheroidal Modes Toroidal Modes Global power spectra earth As the table suggests, the normal modes provide a framework for representing global seismic waves. Typically, these modes of free oscillation are of extremely low frequency and are therefore difficult to observe in seismograms. Only the most energetic earthquakes are capable of generating free oscillations that are readily apparent on most seismograms, and then only if the seismograms extend over several days. NORMAL MODES To understand normal modes, which describe the modes of free oscillation of a sphere, it’s instructive to consider the 1D analog of a vibrating string fixed at both ends as shown in panel Figure 1b. This is useful because the 3D case (Figure 1c), similar to the 1D case, requires that Figure 1 Figure by MIT OCW. 1 Introduction to Seismology: Lecture Notes 22 April 2005 standing waves ‘wrap around’ and meet at a null point.
    [Show full text]
  • Normal Mode Analysis ©David Ronis Mcgill University
    Chemistry 365: Normal Mode Analysis ©David Ronis McGill University 1. Quantum Mechanical Treatment Our starting point is the Schrodinger wav e equation: N −2 ∂2 − h + → → Ψ → → = Ψ → → Σ → U(r1,...,r N ) (r1,...,r N ) E (r1,...,r N ), (1.1) = ∂ 2 i 1 2mi ri where N is the number of atoms in the molecule, mi is the mass of the i’th atom, and → → U(r1,...,r N )isthe effective potential for the nuclear motion, e.g., as is obtained in the Born- Oppenheimer approximation. If the amplitude of the vibrational motion is small, then the vibrational part of the Hamil- tonian associated with Eq. (1.1) can be written as: N −2 ∂2 N ↔ → → ≈− h + + 1 ∆ ∆ Hvib Σ →2 U0 Σ Ki, j: i j,(1.2) i=1 2mi ∂∆ 2 i, j=1 i → ∆ ≡ → − → → where U0 is the minimum value of the potential energy, i ri Ri, Ri is the equilibrium posi- tion of the i’th atom, and 2 ↔ ∂ ≡ U Ki, j → → (1.3) ∂ ∂ → ri r j → r k = Rk is the matrix of (harmonic) force constants. Henceforth, we will shift the zero of energy so as to = make U0 0. Note that in obtaining Eq. (1.2), we have neglected anharmonic (i.e., cubic and higher order) corrections to the vibrational motion. The next and most confusing step is to change to matrix notation. We introduce a column vector containing the displacements as: ∆≡ ∆x ∆y ∆z ∆x ∆y ∆z T [ 1 , 1, 1,..., N , N , N ] ,(1.4) where "T"denotes a matrix transpose.
    [Show full text]
  • Waves & Normal Modes
    Waves & Normal Modes Matt Jarvis February 2, 2016 Contents 1 Oscillations 2 1.0.1 Simple Harmonic Motion - revision . 2 2 Normal Modes 5 2.1 Thecoupledpendulum.............................. 6 2.1.1 TheDecouplingMethod......................... 7 2.1.2 The Matrix Method . 10 2.1.3 Initial conditions and examples . 13 2.1.4 Energy of a coupled pendulum . 15 2.2 Unequal Coupled Pendula . 18 2.3 The Horizontal Spring-Mass system . 22 2.3.1 Decouplingmethod............................ 22 2.3.2 The Matrix Method . 23 2.3.3 Energy of the horizontal spring-mass system . 25 2.3.4 Initial Condition . 26 2.4 Vertical spring-mass system . 26 2.4.1 The matrix method . 27 2.5 Interlude: Solving inhomogeneous 2nd order di↵erential equations . 28 2.6 Horizontal spring-mass system with a driving term . 31 2.7 The Forced Coupled Pendulum with a Damping Factor . 33 3 Normal modes II - towards the continuous limit 39 3.1 N-coupled oscillators . 39 3.1.1 Special cases . 40 3.1.2 General case . 42 3.1.3 N verylarge ............................... 44 3.1.4 Longitudinal Oscillations . 47 4WavesI 48 4.1 Thewaveequation ................................ 48 4.1.1 TheStretchedString........................... 48 4.2 d’Alambert’s solution to the wave equation . 50 4.2.1 Interpretation of d’Alambert’s solution . 51 4.2.2 d’Alambert’s solution with boundary conditions . 52 4.3 Solving the wave equation by separation of variables . 54 4.3.1 Negative C . 55 i 1 4.3.2 Positive C . 56 4.3.3 C=0.................................... 56 4.4 Sinusoidalwaves ................................
    [Show full text]
  • The 1D Schrödinger Equation for a Free Particle
    Lecture 3 Phys 3750 Two Coupled Oscillators / Normal Modes Overview and Motivation: Today we take a small, but significant, step towards wave motion. We will not yet observe waves, but this step is important in its own right. The step is the coupling together of two oscillators via a spring that is attached to both oscillating objects. Key Mathematics: We gain some experience with coupled, linear ordinary differential equations. In particular we find special solutions to these equations, known as normal modes, by solving an eigenvalue problem. I. Two Coupled Oscillators Let's consider the diagram shown below, which is nothing more than 2 copies of an harmonic oscillator, the system that we discussed last time. We assume that both oscillators have the same mass m and spring constant k s . Notice, however, that because there are two oscillators each has it own displacement, either q1 or q2 . ks m m ks q1 q2 q1 = 0 q2 = 0 Based on the discussion last time you should be able to immediately write down the equations of motion (one for each oscillating object) as ~2 q&&1 +ω q1 = 0, and (1a) ~ 2 q&&2 + ω q2 = 0 , (1b) ~ 2 where ω = k s m . As we saw last time, the solution to each of theses equations is harmonic motion at the (angular) frequency ω~ . As should be obvious from the D M Riffe -1- 1/4/2013 Lecture 3 Phys 3750 picture, the motion of each oscillator is independent of the other oscillator. This is also reflected in the equation of motion for each oscillator, which has nothing to do with the other oscillator.
    [Show full text]
  • What Is a Photon? Foundations of Quantum Field Theory
    What is a Photon? Foundations of Quantum Field Theory C. G. Torre June 16, 2018 2 What is a Photon? Foundations of Quantum Field Theory Version 1.0 Copyright c 2018. Charles Torre, Utah State University. PDF created June 16, 2018 Contents 1 Introduction 5 1.1 Why do we need this course? . 5 1.2 Why do we need quantum fields? . 5 1.3 Problems . 6 2 The Harmonic Oscillator 7 2.1 Classical mechanics: Lagrangian, Hamiltonian, and equations of motion . 7 2.2 Classical mechanics: coupled oscillations . 8 2.3 The postulates of quantum mechanics . 10 2.4 The quantum oscillator . 11 2.5 Energy spectrum . 12 2.6 Position, momentum, and their continuous spectra . 15 2.6.1 Position . 15 2.6.2 Momentum . 18 2.6.3 General formalism . 19 2.7 Time evolution . 20 2.8 Coherent States . 23 2.9 Problems . 24 3 Tensor Products and Identical Particles 27 3.1 Definition of the tensor product . 27 3.2 Observables and the tensor product . 31 3.3 Symmetric and antisymmetric tensors. Identical particles. 32 3.4 Symmetrization and anti-symmetrization for any number of particles . 35 3.5 Problems . 36 4 Fock Space 38 4.1 Definitions . 38 4.2 Occupation numbers. Creation and annihilation operators. 40 4.3 Observables. Field operators. 43 4.3.1 1-particle observables . 43 4.3.2 2-particle observables . 46 4.3.3 Field operators and wave functions . 47 4.4 Time evolution of the field operators . 49 4.5 General formalism . 51 3 4 CONTENTS 4.6 Relation to the Hilbert space of quantum normal modes .
    [Show full text]
  • Normal Modes and Quality Factors of Spherical Dielectric Resonators: I – Shielded Dielectric Sphere
    PRAMANA °c Indian Academy of Sciences Vol. 62, No. 6 | journal of June 2004 physics pp. 1255{1271 Normal modes and quality factors of spherical dielectric resonators: I { Shielded dielectric sphere R A YADAV and I D SINGH Spectroscopy Laboratory, Department of Physics, Banaras Hindu University, Varanasi 221 005, India E-mail: [email protected] MS received 16 July 2002; revised 27 October 2003; accepted 1 April 2004 Abstract. Electromagnetic theoretic analysis of shielded homogeneous and isotropic di- electric spheres has been made. Characteristic equations for the TE and TM modes have been derived. Dielectric spheres of radii of the order of ¹m size are found suitable for the optical frequency region whereas for the microwave region radii of the order of mm size are found suitable. Parameters suitable for their application in the optical and microwave frequency ranges have been used to compute the frequencies corresponding to the normal modes for the TE and TM modes. Expressions for the quality factors for realistic res- onators, i.e., for a dielectric sphere with a non-zero conductivity and a metal shield with a ¯nite conductivity have also been derived for the TE and TM modes. Computations of the quality factors have been made for resonators with parameters suitable for the optical and the microwave regions. Keywords. Eigenmodes; spherical resonators; spherical dielectric resonators; quality factors. PACS No. 42.50.Dv 1. Introduction The earliest reported study on spherical resonators seems to be that of Debye [1] wherein he has studied normal modes of a conducting sphere embedded in a perfect dielectric medium.
    [Show full text]
  • Calculation of Molecular Vibrational Normal Modes
    Calculation of Molecular Vibrational Normal Modes Benjamin Rosman 0407237H September 4, 2008 Supervisor Dr Alex Welte Abstract Normal mode analysis provides a vital key to understanding the dynamics of a complicated system. In this case, this is the motion and vibrations of the atoms in a molecule. It is shown in several test cases that the algorithm successfully detects every normal mode of the molecule, as well as all rigid body rotations and translations. The frequencies of these vibrations are also computed to within 6% of the values reported in the literature. 1 Introduction The energy and wave function describing the states of a molecule are computed by solving the time- independent Schrodinger¨ Equation. However, for large molecules this can involve finding the solution to partial differential equations of several hundred variables, which can become completely infeasible. This calculation can be simplified by use of the Born-Oppenheimer approximation, which allows for the electronic and nuclear problems to be solved independently [Griffiths 2005]. Research is currently being conducted into different ways of solving the electronic problem. The work presented in this document serves to complement that, by discussing the development of tools which would allow for solving the nuclear component of the molecular energy problem, assuming that the electronic problem had already been resolved. The output from the electronic problem is a grid of potentials, which is not something that is directly measurable in a molecule. Instead observables such as frequencies are calculated by this software. Section 2 provides a discussion of normal modes and their particular relevance to quantum chemistry, including the method used in their computation.
    [Show full text]
  • 8 Surface Waves and Normal Modes
    8 Surface waves and normal modes Our treatment to this point has been limited to body waves, solutions to the seismic wave equation that exist in whole spaces. However, when free surfaces exist in a medium, other solutions are possible and are given the name surface waves. There are two types of surface waves that propagate along Earth’s surface: Rayleigh waves and Love waves. For laterally homogeneous models, Rayleigh waves are radially polarized (P/SV) and exist at any free surface, whereas Love waves are transversely polarized and require some velocity increase with depth (or a spherical geometry). Surface waves are generally the strongest arrivals recorded at teleseismic distances and they provide some of the best constraints on Earth’s shallow structure and low-frequency source properties. They differ from body waves in many respects – they travel more slowly, their amplitude decay with range is generally much less, and their velocities are strongly frequency dependent. Surface waves from large earthquakes are observable for many hours, during which time they circle the Earth multiple times. Constructive interference among these orbiting surface waves, to- gether with analogous reverberations of body waves, form the normal modes,or free oscillations of the Earth. Surface waves and normal modes are generally ob- served at periods longer than about 10 s, in contrast to the much shorter periods seen in many body wave observations. 8.1 Love waves Love waves are formed through the constructive interference of high-order SH surface multiples (i.e., SSS, SSSS, SSSSS, etc.). Thus, it is possible to model Love waves as a sum of body waves.
    [Show full text]
  • Normal Mode and Surface Wave Observations
    1.04 Theory and Observations: Normal Mode and Surface Wave Observations G Laske, Scripps Institution of Oceanography, University of California San Diego, La Jolla, CA, USA R Widmer-Schnidrig, Black Forest Observatory, Wolfach, Germany ã 2015 Elsevier B.V. All rights reserved. 1.04.1 Introduction 117 1.04.1.1 Free Oscillations 117 1.04.1.2 Surface Waves 118 1.04.1.3 What We See in Seismograms: The Basics 119 1.04.2 Free Oscillations 120 1.04.2.1 Modes of a Spherically Symmetric Earth 120 1.04.2.2 Modes of a Heterogeneous Earth 122 1.04.2.2.1 Mode splitting 122 1.04.2.2.2 Mode coupling 126 1.04.2.3 Measuring Mode Observables 128 1.04.2.3.1 Multiplet stripping and degenerate mode frequencies 130 1.04.2.3.2 Singlet stripping and receiver stripping 132 1.04.2.3.3 Retrieving the splitting matrix with the matrix autoregressive technique 133 1.04.2.3.4 Retrieving the splitting matrix with iterative spectral fitting 133 1.04.2.3.5 Observed mode coupling 135 1.04.2.4 Example of a Mode Application: Inner-Core Rotation 136 1.04.2.5 Example of a Mode Application: Earth’s Hum 139 1.04.3 Surface Waves 141 1.04.3.1 Standing Waves and Traveling Waves 141 1.04.3.2 The Measurement of Fundamental Mode Dispersion 144 1.04.3.2.1 Group velocity 148 1.04.3.2.2 The study of ambient noise 149 1.04.3.2.3 Phase velocity 149 1.04.3.2.4 Time-variable filtering 151 1.04.3.3 Other Surface Wave Observables 152 1.04.3.4 Higher Modes 154 1.04.3.4.1 Higher-mode dispersion and waveform modeling 154 1.04.3.4.2 Love waves and overtones 157 1.04.3.5 Surface Waves and Structure at Depth 157 1.04.3.6 The Validity of the Great-Circle Approximation and Data Accuracy 160 1.04.4 Concluding Remarks 160 Acknowledgments 161 References 161 1.04.1 Introduction Valdivia, Chile, earthquake.
    [Show full text]