Physics
Wave Equations
Wave equations are mathematical descriptions that represent the behavior of waves, such as light, sound, and water waves. They typically involve partial differential equations and describe how waves propagate through a medium. The solutions to wave equations provide information about the wave's amplitude, frequency, and wavelength.
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10 Key excerpts on "Wave Equations"
eBook - PDF- Jerry B. Marion(Author)
- 2013(Publication Date)
- Academic Press(Publisher)
C H A P T E R I 5 The Wave Equation in One Dimension 15.1 Introduction In the preceding chapter we introduced the wave equation, one of the truly fundamental equations of mathematical physics. The solutions of this equation are in general subject to various limitations imposed by certain physical restrictions that are peculiar to a given problem. These limitations frequently take the form of conditions on the solution that must be met at the extremes of the intervals of space and time that are of physical interest. We must therefore deal with a boundary-value problem involving a partial differential equation. Indeed, such a description characterizes essentially the whole of what we call mathematical physics. We have already seen some indications of the solutions to this type of equation, and in this chapter we shall investigate in detail the solutions of the wave equation. We shall, however, confine ourselves to a discussion of waves in one dimension.* One-dimensional waves describe, for example, the motion of a vibrating string. On the other hand, the compression (or sound) waves which may be transmitted through an elastic medium, such as a gas, can * Three-dimensional waves are treated, for example, in Marion (Ma65b, Chapter 10). 478 15.2 SEPARATION OF THE WAVE EQUATION 479 also be approximately one-dimensional waves (if the medium is sufficiently large so that the edge effects are unimportant). In such a case, the condition of the medium is approximately the same at every point on a plane, and the properties of the wave motion are then functions only of the distance along a line normal to the plane. Such a wave in an extended medium is called a plane wave, and is mathematically identical to the one-dimensional waves which are treated here. We shall consider in this chapter only mechanical waves; an extensive discussion of electromagnetic waves is given in Marion (Ma65b).
eBook - PDFGravitational Waves
An Overview
- David M. Feldbaum(Author)
- 2022(Publication Date)
- Springer(Publisher)
PART I Review of Wave Motion 3 C H A P T E R 1 The Wave Equation 1.1 ONE SPATIAL DIMENSION From one point of view, gravitational waves (GWs) represent one particular example of general wave propagation. Mathematically wave propagation is described by a wave equation: a partial differential equation (PDE) in time and space variables, that has propagating solutions. Given initial conditions in the form of a disturbance at time zero, the wave-like solution should show a related disturbance at a different point in space at a later time. The non-dispersive wave equation @ 2 f @x 2 D v 2 @ 2 f @t 2 (1.1) is the prototypical simplest case of a wave equation. It possesses propagating disturbances which are unchanged with time: f .x; t/ D f .x ˙ vt; 0/; (1.2) where f .x; t/ is any differentiable function. Many other differential equations possess propagating solutions, and therefore may be characterized as Wave Equations. These include such well-known equations as: • transmission line with resistance R, inductance L, capacitance C , and leakage conductance G: @i @x C C @v @t D Gv @v @x C L @v @t D Rv • or the vibrating beam equation @ 4 y @x 4 C @ 2 y @t 2 D 0: Partial differential equations of order two may be classified into three classes (Elliptical, Parabolic, Hyperbolic), with the solutions behaving similar to Laplace, Heat, and Wave equation, respectively. The solutions of the hyperbolic partial differential equations (PDEs) have a wave- front which propagates with a finite speed (as opposed to both the parabolic PDEs, which lack sharp wavefront and exhibit infinite speed of propagation, and to the elliptical PDEs, which describe static situations). 4 1. THE WAVE EQUATION All of this is relatively irrelevant for the basics of GWs research. The more complicated Wave Equations are relevant when waves propagate in various dispersive media, which affect the speeds of propagation and/or amplitudes of the waves.
eBook - PDF- Charles A. Bennett(Author)
- 2022(Publication Date)
- Wiley(Publisher)
1 1 The Physics of Waves The solution of the difficulty is that the two mental pictures which experiment lead us to form — the one of the particles, the other of the waves — are both incomplete and have only the validity of analogies which are accurate only in limiting cases. Heisenberg 1.1 Introduction The properties of waves are central to the study of optics. As we will see, light (or more prop- erly, electromagnetic radiation) has both particle and wave properties. These complementary aspects are a result of quantum mechanics, and prior to the early 1900s, there were two schools of thought. Newton postulated that light consists of particles, while contemporaries Huygens and Hooke promoted a wave theory of light. The matter seemed settled with Young’s important double-slit experiment offering clear experimental evidence that light is a wave. Maxwell’s sweeping theory of electromagnetism finally provided a deep and complete description of electromagnetic waves that we consider in detail in Chapter 2. Although current theories of optics include both wave and particle descriptions, the wave picture still forms the bedrock of most optical technology. In this chapter, we will outline some general properties that apply to traveling waves of all types. 1.2 One-Dimensional Wave Equation Mechanical waves travel within elastic media whose material properties provide restoring forces that result in oscillation. When a guitar string is plucked, it is displaced away from its equilibrium position, and the mechanical energy of this disturbance subsequently propagates along the string as traveling waves. In this case, the waves are transverse, meaning that the displacement of the medium (the string) is perpendicular to the direction of energy travel. Acoustic waves in a gas are longitudinal, meaning that the gas molecules are displaced back and forth along the direction of energy flow as regions of high and low pressure are created along the wave.
- Yufeng Zhou(Author)
- 2015(Publication Date)
- CRC Press(Publisher)
5 2 Wave Equation Acoustic waves are the organized vibrations of molecules or atoms of a medium that is able to support the propagation of these waves. When the frequency of the vibration is above the audible range, the waves are known as ultrasonic waves. As their frequency increases, the wavelength of these waves gets progressively smaller, and this small size accounts for some of the unique resolution capabilities of ultrasound when compared with ordinary sound waves. The charac-teristics of the acoustic field are expressed by the acoustic pressure p , particle velocity of the medium v , and change of density ρ′ . For example, in acoustic wave propagation, at a certain time point, the acoustic pressure at different loca-tions has varying values; meanwhile, the acoustic pressure is a time-varying function at every point in space. In other words, the acoustic pressure is a temporally and spatially varying function. The relationship between acoustic pres-sure and its space and time variation is called the acous-tic wave equation. The theory is inherently approximate, and there are many different versions that differ slightly or greatly from each other depending on what idealizations are made at the outset. 2.1 FUNDAMENTAL EQUATIONS FOR AN IDEAL FLUID It has already been shown that, during an acoustic distur-bance, the acoustic pressure p , the velocity v of the particle in the medium, and density change ρ′ are all related. As a macroscopic phenomenon, acoustic vibration must satisfy fundamental physical laws, such as Newton’s second law and mass conservation law. For simplicity, the following assump-tions are made to derive the general expression of the acoustic wave equation: 1. The medium is an ideal fluid without viscosity, through which there is no energy loss during the acoustic wave propagation. 2. The medium is still (i.e., no initial velocity) and homogeneous in the equilibrium state, so the static pressure P 0 and the static density ρ 0 are both constants.
eBook - PDF- Walter Craig(Author)
- 2018(Publication Date)
- American Mathematical Society(Publisher)
Chapter 2 Wave Equations Windblown ocean waves at sunrise. Image credit: Lindsay imagery / iStock / Getty Images Plus. We will start the topic of PDEs and their solutions with a discussion of a class of Wave Equations, initially with several transport equations, which will lead us to the study of the standard second-order wave equation (1.9) in one space dimension. These transport equations are often called first-order Wave Equations for reasons that will become clear as we describe their solutions. We will use this study of Wave Equations as motivation to introduce the Fourier transform as well as to give examples of the concept of a field of characteristic curves. 9 10 2. Wave Equations 2.1. Transport equations: The Fourier transform The transport equation is a first-order equation that describes the time evolution of a quantity u , carried by a background flow field that possesses a (possibly) variable velocity c ( t, x ), namely (2.1) ∂ t u + c ( t, x ) · ∂ x u = 0 . This typically gives rise to an initial value problem where one asks that u (0 , x ) = f ( x ) . The function f ( x ) is called the initial data . The problem is to describe how the solution u ( t, x ) evolves given its initial “shape” f ( x ) at t = 0. Often these equations are known as first-order Wave Equations for the unknown function u ( t, x ). The coefficient c ( t, x ) is a vector known as the “speed of propagation”; we will see the reason for this when we describe the solution process. A first example is given by the case in which the coefficient c is constant. We will see in the solution why c is known as the speed of propagation . We will first solve this equation using the Fourier transform. For convenience we restrict ourselves to one space dimension. The process is elementary, but nevertheless it serves to introduce a surprisingly powerful technique for the study of PDEs.
eBook - PDF- Srinivasan Gopalakrishnan(Author)
- 2016(Publication Date)
- CRC Press(Publisher)
This procedure is adopted to determine the wavenumbers for a few 2D waveguides in the later chapters of this book. Introduction to Wave Propagation 119 5.4 GENERAL FORM OF Wave Equations AND THEIR CHAR- ACTERISTICS In this book, we will be looking at wave propagation in different material sys- tems. The characteristics of the waves in each of these systems will be different. Although the details of these differences will be highlighted in their respective chapters, in this section, among the many different material systems that we will be presenting in this textbook, we only outline briefly the characteris- tics of the wave in three different material systems—the anisotropic material system, inhomogeneous material system, and the nanostructure material sys- tems. First, the general form of the wave equation is presented followed by a general discussion of the characteristics of waves in the above three material systems. 5.4.1 General Form of Wave Equations The general form of the linear wave equation assuming the local theory of elasticity is given by ∂ 2 u ∂t 2 = X α,β u αβ , (5.31) where u(x, y, z, t) is the field variable governing the material system with u αβ being the derivatives of the field variable. The wave equation in the structural mechanics context is the conservation of momentum (dynamic equilibrium) equation ∇ · T = ρ ¨ u , (5.32) where T is the stress measure at any point in the body, which is in general a non-linear function of the displacement vector u = {u x , u y , u z }, (the wave). A non-linear relation between T and u results in a non-linear wave equa- tion. Most of the analysis presented in this book assumes a linear relationship between T and strain (that is, displacement gradient), either in the time do- main (linear elastic material) or in the frequency domain (visco-elastic mate- rial). However, the linear coefficient (of constitutive relation) can be direction (anisotropy) and/or position (inhomogeneity) dependent.
eBook - PDF- Mikhail Shubin, Maxim Braverman, Robert McOwen, Peter Topalov, Maxim Braverman, Robert McOwen, Peter Topalov(Authors)
- 2020(Publication Date)
- American Mathematical Society(Publisher)
Chapter 10 The wave equation 10.1. Physical problems leading to the wave equation There are many physical problems leading to the wave equation (10.1) u tt = a 2 Δ u, where u = u ( t, x ), t ∈ R , x ∈ R n , Δ is the Laplacian with respect to x . The following more general nonhomogeneous equation is also often encountered: (10.2) u tt = a 2 Δ u + f ( t, x ) . We have already seen that small oscillations of the string satisfy (10.1) (for n = 1), and in the presence of external forces they satisfy (10.2). It can be shown that small oscillations of a membrane satisfy similar equations, if we denote by u = u ( t, x ), for x ∈ R 2 , the vertical displacement of the membrane from the (horizontal) equilibrium position. Similarly, under small oscillations of a gas ( acoustic wave ) its parameters (e.g., pressure, density, displacement of gas particles) satisfy (10.1) with n = 3. One of the most important fields, where equations of the form (10.1) and (10.2) play a leading role, is classical electrodynamics. Before explaining this in detail, let us recall some notation from vector analysis. (See, e.g., Ch. 10 in [ 27 ].) Let F = F ( x ) = ( F 1 ( x ) , F 2 ( x ) , F 3 ( x )) be a vector field defined in an open set Ω ⊂ R 3 . Here x = ( x 1 , x 2 , x 3 ) ∈ Ω, and we will assume that F is sufficiently smooth, so that all derivatives of the components F j , which we need, are continuous. It is usually enough to assume that F ∈ C 2 (Ω) (which means that every component F j is in C 2 (Ω)). If f = f ( x ) is a sufficiently 215 216 10. The wave equation smooth function on Ω, then its gradient grad f = ∇ f = ∂f ∂x 1 , ∂f ∂x 2 , ∂f ∂x 3 is an example of a vector field in Ω. Here it is convenient to consider ∇ as a vector ∇ = ∂ ∂x 1 , ∂ ∂x 2 , ∂ ∂x 3 whose components are operators acting, say, in C ∞ (Ω), or, as operators from C k +1 (Ω) to C k (Ω), k = 0 , 1 , . . . . The divergence of a vector field F is a scalar function div F = ∂F 1 ∂x 1 + ∂F 2 ∂x 2 + ∂F 3 ∂x 3 .
eBook - PDF- Agustín Udías, Elisa Buforn(Authors)
- 2017(Publication Date)
- Cambridge University Press(Publisher)
5 Waves in an infinite elastic medium As mentioned in Chapter 2 , earthquakes produce waves that propagate through the Earth, cause damage at short distances, and are recorded and measured by seismographs, as seen in Chapter 3 . To understand the phenomenon of wave propagation in the Earth we begin with the simplest case of waves propagating in an in fi nite elastic medium. As we saw in Chapter 4 , in an in fi nite, homogeneous, isotropic, elastic medium, the basic equations of mechanics are the equation of motion and of continuity and propagation of energy. Under certain conditions these equations lead to the phenomenon of wave propagation that can be formulated in terms of Wave Equations , fi rst proposed in 1747 by Jean d ’ Alembert. In this chapter we present the solutions for the wave equation in terms of plane, spherical, and cylindrical waves in an in fi nite elastic medium, their characteristics, and relations with the observations. 5.1 Wave Equations for an elastic medium In order to study the propagation of waves in an in fi nite elastic medium, we begin with Navier ’ s equation which can be expressed in terms of the displacements ( 4.74 ) and of the cubic dilation and rotation vector ( 4.75 ) equations, which may be easily transformed into Wave Equations. We begin with application of the divergence operation in equation ( 4.75 ). The divergence of the gradient of θ is its Laplacian, that of the curl of ω is null, and the divergence of the displacement u is the cubic dilation θ . Thus, we obtain r 2 θ ¼ 1 α 2 ∂ 2 θ ∂ t 2 : ð 5 : 1 Þ To the same equation ( 4.75 ) we apply the curl operation. The curl of the gradient of the scalar function θ is null and that of the displacement u is the rotation vector ω . The curl of the curl of ω is equal to the gradient of the divergence, which is null minus the Laplacian (A1.30).
eBook - PDF- Julian L. Davis(Author)
- 2021(Publication Date)
- Princeton University Press(Publisher)
C H A P T E R T H R E E The Wave Equation PART I ONE-DIMENSIONAL WAVE EQUATION In chapter 1 we derived the one-dimensional wave equation for longitu-dinal stress waves in a bar and transverse waves in a vibrating string. We also derived expressions for the wave speed in a bar, in a vibrating string, and for sound waves. We showed that all solutions of the wave equation c 2 u xx — u tt = 0 must be of the form u(x, t) = fix — ct) + g(x + ct) for arbitrary functions / and g which depend on the ICs. In chapter 2 we investigated the method of characteristics for first- and second-order PDEs and used this method to classify PDEs into hyper-bolic, elliptic, and parabolic types. In part I of this chapter, we shall explore other methods of solving the one-dimensional wave equation, in addition to the method of characteristics, which will be more easily applicable to IV (Initial Value) and BV (Boundary Value) problems. In part II we shall investigate the wave equation in two and three dimen-sions. FACTORIZATION OF THE WAVE EQUATION AND CHARACTERISTIC CURVES In this section we cast the one-dimensional wave equation in the setting of chapter 2 where the concepts of the directional derivative and characteristic theory were discussed. The wave equation can be factored as follows: (cD x -D t )(cD x + D t )u = 0, (3.1) where the partial derivative operators are given by D X d = Sx' d D = — ' dt 85 CHAPTER THREE Set (cD x + D,)u = cu x + u t = v. (3.2) The directional derivative of u is du/dt. This is the total derivative of u with respect to t in the direction along the tangent to a curve in (x, t) space. This curve is called a characteristic curve. It plays an important role in the PDEs of wave propagation and will be described below. (See chapter 2 for more details.) We expand this directional derivative and obtain u = u x x + u t , (3.3) where du dx U = —— , X = ~~r . dt dt If eq. (3.3) for the directional derivative is correlated with the PDE given by eq.
eBook - PDF- Franco Tomarelli(Author)
- 2019(Publication Date)
- Società Editrice Esculapio(Publisher)
Chapter 10 Wave Equation This chapter resume and deepen the analysis ot the wave equation started in Section 2.5. First we solve the global Cauchy problem for the wave equation in one space dimension, achieving the complete d’Alembert’s formula. Second we study the global Cauchy problem for the wave equation in the physical three-dimensional space, providing the fundamental solution, stat-ing Huygen’s Principle and conservation of total mechanical energy. Eventually we deduce from Maxwell’s equations that the electrical and mag-netic field are solution of the wave equation with velocity of propagation re-lated to the dielectric constant and the magnetic permeability of the medium. 10.1 Global Cauchy problem for the wave equation ( n = 1) The global Cauchy problem for the wave equation in one space dimension reads as follows: 8 < : u tt -c 2 u xx = f ( x, t ) x 2 R , t > 0 , u ( x, 0) = u 0 ( x ) x 2 R , u t ( x, 0) = u 1 ( x ) x 2 R . (10.1) Here u 0 2 L 2 ( R ), u 1 2 L 2 ( R ) and f 2 C 0 ( [0 , + 1 ) , L 2 ( R n x ) ) are given, whereas u is the unknown. Since we want to deal with possibly singular data and we cannot expect solutions with better regularity than data (see Section 2.5), we make quite general assumptions: namely, we deal with u 0 and u 1 belonging to S 0 ( R ) and a distribution f on R 2 which is at least a tempered distribution in space at any time, say f ( · , t ) 2 S 0 ( R ) for every t . 398 10 Wave Equation Linearity property of the PDE allows to split the problem in 3 auxiliary problems that take into account various data separately, then to add their solutions to recover u : 8 < : v tt -c 2 v xx = 0 x 2 R , t > 0 , v ( x, 0) = u 0 ( x ) x 2 R , v t ( x, 0) = 0 x 2 R ; (10.2) 8 < : w tt -c 2 w xx = 0 x 2 R , t > 0 , w ( x, 0) = 0 x 2 R , w t ( x, 0) = u 1 ( x ) x 2 R ; (10.3) 8 < : z tt -c 2 z xx = f ( x, t ) x 2 R , t > 0 , z ( x, 0) = 0 x 2 R , z t ( x, 0) = 0 x 2 R .
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