1D Wave Equation

11 Key excerpts on "1D Wave Equation"

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  Classical Dynamics of Particles and Systems

  • 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).

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  Mathematics of Wave Propagation

  • 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.

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  Computing the Flow of Light: Nonstandard FDTD Methodologies for Photonics Design

  • Cole, James B., Banerjee, Saswatee(Authors)
  • 2017(Publication Date)
  • Society of Photo-Optical Instrumentation Engineers

    (Publisher)

  Chapter 4 The 1D Wave Equation In a uniform medium, Maxwell ’ s equations reduce to the wave equation, and analytic solutions are derived by solving the wave equation subject to the appropriate boundary conditions. Although the 1D Wave Equation may seem simple, when boundary and initial conditions are specified and when sources are included, its analytical solutions are far from trivial and sometimes do not even exist. Before moving on to FDTD for the wave equation, we develop some analytical solutions to which numerical solutions can be compared. Topics: • Derivation and various forms of wave equation • Analytical solutions • Extra topics: relativity from the wave equation • Green ’ s function and associated mathematics 4.1 The Homogeneous Wave Equation 4.1.1 General solution in an unbounded uniform medium The simplest form of the wave equation (see Appendix 4.1 for a derivation) is ð -2 t v 2 -2 x Þ c ð x , t Þ ¼ 0, (4.1) where n is the phase velocity. If no initial and boundary conditions are specified, it can be verified by direct substation that the general solution is c ð x , t Þ ¼ f ð x vt Þ þ g ð x þ vt Þ , (4.2) where f and g are arbitrary functions. Letting c ( x , t ) ¼ f ( x – n t ), 99 c ð x þ D x , t þ D t Þ ¼ f ½ x þ v D t v ð t þ D t ފ ¼ f ½ x vt þ ð D x v D t ފ : (4.3) If D x ¼ n D t , then c ( x þ D x , t þ D t ) ¼ c ( x , t ), and the shape of the disturbance described by f is the same at space – time points ( x , t ) and ( x þ n D t , t þ D t ); hence, the disturbance moves with velocity n in the þ x direction. Similarly, g ( x þ n t ) describes a disturbance traveling in the – x direction with velocity n . Thus, in Eq. (4.1), n is the velocity (technically the phase velocity) of the wave. It is often mathematically convenient to use another solution obtained by putting f ( x ) → f ( – x ) so that c ð x , t Þ ¼ f ð vt ∓ x Þ (4.4) describes a wave moving in the ± x direction.

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  Principles of Physical Optics

  • Charles A. Bennett(Author)
  • 2022(Publication Date)
  • Wiley

    (Publisher)

  To demonstrate that a given func- tion Ψ(x, t) describes a classical wave, it is necessary only to show that this function sat- isfies Equation 1.1. Conversely, any physical system that can be shown to be described by Equation 1.1 must necessarily involve classical traveling waves. Equation 1.1 is an example of a class of mathematical equations known as differential equations. When taking a partial derivative with respect to a given parameter, the remaining parameters are treated as constants. Example 1.1 Show that Ψ(x, t) = (x + 𝑣t) 2 is a solution to the differential wave equation. Assume that 𝑣 is a constant. Solution We must show that Ψ(x, t) solves Equation 1.1: 𝜕 2 Ψ(x, t) 𝜕x 2 = 1 𝑣 2 𝜕 2 Ψ(x, t) 𝜕t 2 To do this, we must take the indicated derivatives. Begin with the partial derivative with respect to x: 𝜕Ψ(x, t) 𝜕x = 𝜕 𝜕x (x + 𝑣t) 2 = 2(x + 𝑣t) 𝜕 𝜕x (x + 𝑣t) = 2(x + 𝑣t)(1) = 2(x + 𝑣t) Now take the second partial derivative with respect to x: 𝜕 2 Ψ 𝜕x 2 = 𝜕 𝜕x [ 𝜕Ψ 𝜕x ] = 𝜕 𝜕x [2 (x + 𝑣t)] = 2 Next, take the partial derivatives of Ψ(x, t) with respect to t: 𝜕Ψ 𝜕t = 2 (x + 𝑣t) 𝜕 𝜕t (x + 𝑣t) = 2 (x + 𝑣t) (𝑣) = 2𝑣 (x + 𝑣t) 𝜕 2 Ψ 𝜕t 2 = 𝜕 𝜕t [2𝑣 (x + 𝑣t)] = 𝜕 𝜕t [ 2𝑣x + 2𝑣 2 t ] = 2𝑣 2 Substitute these results into the differential wave equation 𝜕 2 Ψ(x, t) 𝜕x 2 = 2 1 𝑣 2 𝜕 2 Ψ(x, t) 𝜕t 2 = 1 𝑣 2 ( 2𝑣 2 ) = 2 1 The term classical refers to the physics that came before the modern theories of relativity and quantum mechanics. 1.3 General Solutions to the 1D Wave Equation 3 Thus 𝜕 2 Ψ(x, t) 𝜕x 2 = 1 𝑣 2 𝜕 2 Ψ(x, t) 𝜕t 2 Ψ(x, t) solves the differential wave equation, and thus we have shown that it is a travel- ing wave. 1.3 General Solutions to the 1D Wave Equation Consider a plot of Ψ(x, t) vs. x. Data for such a plot could be provided by an array of mea- suring devices, such as an array of pressure sensors arranged linearly to record the pressure amplitude of a passing acoustic wave.

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  Computational Acoustics

  Theory and Implementation

  • David R. Bergman(Author)
  • 2018(Publication Date)
  • Wiley

    (Publisher)

  3 Derivation of the Wave Equation

  3.1 Introduction

  The linear wave equation can be derived from several different starting points depending on whether the vibrations are traveling in a solid or fluid. In the case of vibrations in a solid, the form of the wave equation will depend on the symmetries of the solid structure and material properties [1]. Since the subject matter of this text deals with methods for modeling wave propagation in fluids, this will be dealt with in detail. Derivations of acoustic vibrations in solids for various models are presented briefly in the interest of demonstrating the connection of the wave equation, and generalizations of the wave equation, to the equations of the supporting medium. Reviewing the connection between the medium and the wave equation will serve two purposes: one is to provide a tangible connection between two paradigms, a micro description of the system and the macroscopic phenomena supported by the system, and the other is to show a direct connection between material properties, such as density, bulk modulus, and so on, and the observed properties of the wave as a separate entity, e.g. velocity.

  Sections appear as follows. The first section describes wave phenomenon without any reference to the medium. This is to establish basic vocabulary. The next section is a review of the coupled mass–spring system from classical mechanics along with a description of normal modes of vibration. The limit to a continuum is taken to arrive at an equation for wave propagation in one spatial dimension. Following the treatment of one‐dimensional waves is a presentation of the equations for wave propagation in three‐dimensional elastic solids. This presentation starts with the stress–strain relation, Hook’s law, and continuum mechanics and passes to a second‐order wave equation. The next section presents a detailed derivation of the linear first‐ and second‐order wave equations derived from the equations of fluid mechanics. This will be the foundation for the discussion of modeling methods for describing the acoustic field. This section introduces a novel way of looking at the differential operator in the wave equation, appearing in the acoustics literature in the late 1970s, which uses concepts from differential geometry and tensor analysis. Following waves in fluids is a very brief section on thin plates and rods, followed by a description of phonons from solid‐state physics. After a survey of the various manifestations of the wave equation, this chapter concludes with a brief section on tensors and manifolds to serve as preparation for later chapters.

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  Partial Differential Equations

  Classical Theory with a Modern Touch

  • A. K. Nandakumaran, P. S. Datti(Authors)
  • 2020(Publication Date)
  • Cambridge University Press

    (Publisher)

  CHAPTER 9 One-Dimensional Wave Equation 9.1 INTRODUCTION The wave equation □ c u ≡ u tt - c 2 u xx = 0, t > 0, x ∈ ℝ, (9.1) models many real-world problems: small transversal vibrations of a string, the longitudinal vibrations of a rod, electrical oscillations in a wire, the torsional oscillations of shafts, oscillations in gases, and so on. It is one of the fundamental equations, the others being the equation of heat conduction and Laplace (Poisson) equation, which have influenced the development of the subject of partial differential equations (PDE) since the middle of the last century. We shall now derive equation (9.1) in the case of transverse vibrations of a string. Physically, a string is a flexible and elastic thread. The tensions that arise in a string are directed along a tangent to its profile. We assume that the string is placed on the x-axis, with its end points at x = 0 and x = L (not shown in the figure); see Figure 9.1. We consider small transversal vibrations of the string, so that the motion of the points of the string is described by a function u(x, t), which gives the amount that a point of the string with abscissa x has moved at time t. We also assume that the length of element MM ′ of the string corresponding to x and x + Δx, Δx being very small, is equal to 1 Δx. We also assume that the tension of the string is uniform and denote it by T. Consider a small element of the string corresponding to the abscissa points x and x +Δx. Forces T act at the end points of this element along the tangents to the string. Let the tangents make angles  and  + Δ with x-axis, at M and M ′ , respectively. Then, the projection on the u-axis of the forces acting on this element will be equal to T sin( + Δ) - T sin . 1 This amounts to assuming u 2 x is very small and may be neglected. For, the length of element MM ′ of the string equals x+Δx ∫ x √ 1 + u 2 x dx ≈ x+Δx ∫ x dx = Δx. 252

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  Modern Physics

  • Kenneth S. Krane(Author)
  • 2020(Publication Date)
  • Wiley

    (Publisher)

  Although he disagreed with the probabilistic interpretation that was later given to his work, he developed the mathematical theory of wave mechanics that for the first time permitted the wave behavior of physical systems to be calculated. We can justify the form of the Schrödinger equation by examining the solu- tion expected for the free particle, which should give a wave whose shape at 5.3 The Schrödinger Equation 147 any particular time, specified by the wave function  (x), is that of a simple de Broglie wave, such as  (x) = A sin kx, where A is the amplitude of the wave and k = 2∕λ. If we are looking for a differential equation, then we need to take some derivatives: d  dx = kA cos kx, d 2  dx 2 = −k 2 A sin kx = −k 2  (x) Note that the second derivative gives the original function again. With the kinetic energy K = p 2 ∕2m = (h∕λ) 2 ∕2m = − h 2 k 2 ∕2m, we can then write d 2  dx 2 = −k 2  (x) = − 2m − h 2 K  (x) = − 2m − h 2 (E − U ) (x) where E = K + U is the nonrelativistic total energy of the particle. For a free particle, U = 0 so E = K ; however, we are using the free particle solution to try to extend to the more general case in which there is a potential energy U (x). The equation then becomes − − h 2 2m d 2  dx 2 + U (x) (x) = E (x) (5.5) Equation 5.5 is the time-independent Schrödinger equation for one-dimensional motion. The solution to Eq. 5.5 gives the shape of the wave at time t = 0. The mathe- matical function that describes a one-dimensional traveling wave must involve both x and t. This wave is represented by the function Ψ(x, t): Ψ(x, t) =  (x)e −it (5.6) The time dependence is given by the complex exponential function e −it with  = E∕ − h. (You can find a few useful formulas involving complex numbers in Appendix B.) We’ll discuss the time-dependent part later in this chapter. For now, we’ll concentrate on the time-independent function  (x).

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  Quantum Mechanics

  Foundations and Applications

  • Donald Gary Swanson(Author)
  • 2006(Publication Date)
  • CRC Press

    (Publisher)

  2 The Schr¨ odinger Equation in One Dimension In the previous chapter, the postulates helped us to use the wave function to find the expected results of experimental measurements by letting a dy-namical quantity F ( x, p, t ) → F (ˆ x, ˆ p, t ) and integrating over space. In the examples, however, the wave functions were typically given and not derived from first principles. We now endeavor to find the appropriate wave functions for a variety of physical systems in one dimension by solving the Schr¨ odinger equation where for time-independent cases, the total energy is an eigenvalue. In subsequent chapters, we will extend the method to three dimensions and develop the appropriate operators, eigenvalues, and eigenfunctions for angu-lar momentum. We will also develop methods for examining time-dependent systems. The latter chapters address special systems of interest that may be referred to as applications of quantum mechanics. 2.1 The Free Particle As our first example of solving the Schr¨ odinger equation, we choose the free particle in one dimension, which means that the particle is free of any forces. It thus has a constant potential, so the time-independent Schr¨ odinger equation in one dimension is written as ˆ H ψ = - 2 2 m d 2 ψ d x 2 + V ψ = Eψ , (2.1) where ψ = ψ ( x ) is the spatial part of the wave function, and the complete wave function is Ψ( x, t ) = ψ ( x ) exp ( -i Et/ ). Just as the time-dependent equation was easily solved in the previous chapter, so is the solution of Equation (2.1) easily solved, with solution ψ ( x ) = A exp ± i 2 m ( E -V ) x/ . (2.2) Problem 2.1 Show that the solution of the 1-dimensional Schr¨ odinger equa-tion for a free particle can be written as ψ ( x ) = A exp ( ± i kx ) where k is the classical momentum of a free particle of energy E . 41

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  Mathematical Analysis tools for engineering

  • 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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  Partial Differential Equations

  An Introduction with Mathematica and Maple

  • Ioannis P Stavroulakis, Stepan A Tersian;;;(Authors)
  • 1999(Publication Date)
  • WSPC

    (Publisher)

  65 Chapter 3 One-Dimensional Wave Equation 9. The Wave Equation on the Whole Line. D'Alembert Formula The simplest hyperbolic second-order equation is the wave equation u u - c 2 u xx = 0, (3.1) where x signifies the spatial variable or position, t the time variable, u = u(Xjt) the unknown function and c is a given positive constant. The wave equation describes vibrations of a string. Physically u (x, t) represents the value of the normal displacement of a particle at position x and time t. Using the theory of Section 7 the characteristic equation of (3.1) is {dxf ~ c 2 {dtf = 0 and J x -f ct = c X — Ct = C2 are two families of real characteristics. Introducing the new variables *-»7)/2 T?) /2c / t = x + ct i f x = {Z + rj)/2 r] = x-ct ' ■ ]_ t = (£ - j and the function 66 PDE+Mat/iemattca+MAPLE Ufofi) = u((t + T,)/2,(£-r,)/2e), the equation (3.1) reduces to Uto(£,v) = 0-(3.2) Therefore U (f.i,) = jF(Z)d4 + 5(77) = / ( 0 + g(v) , and in the original variables u (x, t) is of the form u (x, *) = / (x + ct) + g (x - ct) , (3.3) known as the general solution of (3.1). It is the sum of the function g(x — ct) which presents a shape traveling without change to the right with speed c and the function / (x -+-ct) -another shape, traveling to the left with speed c. Consider the Cauchy (initial value) problem for (3.1) { u tt -c 2 u xx = 0 iceR, t > 0, u{x,Q)=y{x) x G R , u t (x, 0) = ij) (x) x e R, where

  0}. Theorem 3.3. ( D'Alembert* formula ) . If

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  Separation of Variables for Partial Differential Equations

  An Eigenfunction Approach

  • George Cain, Gunter H. Meyer(Authors)
  • 2005(Publication Date)
  • Chapman and Hall/CRC

    (Publisher)

  Chapter 7 One-Dimensional Wave Equation 7.1 Applications of the eigenfunction expansion method Following the format of Chapter 6 we examine the application of the eigen-function expansion approach to the one-dimensional wave equation. Examples involve vibrating strings, chains, and pressure and electromagnetic waves. We comment on the convergence of the approximate solution to the analytic solution and conclude by showing that the equations to be solved in the eigenfunction expansion method are identical to those which arise when Duhamel’s principle is combined with the traditional product ansatz of separation of variables. Example 7.1 A vibrating string with initial displacement. The vibrating string problem served in Chapter 5 to introduce eigenfunc-tion expansions for the wave equation. Here we shall use this simple setting to examine the method in some detail. Suppose that a string of length L with fixed ends is displaced from its equilibrium position and let go at time t = 0. We wish to study the subsequent motion of the string. Newton’s second law and a small amplitude assumption lead to the follow-ing mathematical model for the displacement u ( x, t ) of the string at point x and time t from the equilibrium u ≡ 0: L u ≡ u xx − 1 c 2 u tt = 0 (7.1) u (0 , t ) = u ( L, t ) = 0 161 162 CHAPTER 7. ONE-DIMENSIONAL WAVE EQUATION u ( x, 0) = u 0 ( x ) u t ( x, 0) = 0 where the wave speed c is a known constant depending on the density of the string and the tension applied to it. For definiteness let us assume that u 0 ( x ) = x A , 0 < x < A L − x L − A , A < x < L so that the string is “plucked” at the point A . With this very problem began the development of the method of separation of variables two hundred and fifty years ago [2].

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