D'alembert Wave Equation

4 Key excerpts on "D'alembert Wave Equation"

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  eBook - ePub

  Basic Partial Differential Equations

  • David. Bleecker(Author)
  • 2018(Publication Date)
  • Chapman and Hall/CRC

    (Publisher)

  Chapter 5 The Wave Equation Physical scientists, engineers and applied mathematicians regard the wave equation

  in dimension 3, as one of the most important PDEs, because this equation describes the vibrations of continuous mechanical systems, and the propagation of electromagnetic and sound waves. We have discussed some applications of the wave equation in Section 1.2 of Chapter 1 . In this chapter, we will study a special case of (*), the one-dimensional wave equation

  when u does not depend on y and z (cf. Chapter 9 for a study of (*)). For definiteness, we interpret u(x,t) as the transverse displacement (in a direction perpendicular to the x—axis) of a vibrating string at position x, at time t. With this model in mind, in Section 5.1 , we use Newton’s second law to derive (**), and solve this equation when the initial profile and velocity of the string is specified by finite Fourier sine series. We also establish the uniqueness of solutions by proving that the energy is conserved. In Section 5.2 , we derive D’Alembert’s formula for the solution of initial value problems for the infinite string. The method of images is used with D’Alembert’s formula to solve several problems for the semi-infinite and finite strings, and to prove certain maximum principles which are needed to analyze the error of a solution due to an error in the initial conditions. We begin Section 5.3 with a discussion of the various standard types of B.C.. The same techniques which were used in Chapter 3 to solve heat problems with inhomogeneous B.C. are shown to also work for wave problems. Moreover, a version of Duhamel’s method is motivated and used to solve problems for the inhomogeneous wave equation which results when external forces act on the string.

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  Advanced Engineering Mathematics with MATLAB

  • Dean G. Duffy(Author)
  • 2021(Publication Date)
  • Chapman and Hall/CRC

    (Publisher)

  It gives a representation of the solution in terms of known initial conditions. • Example 8.4.1 To illustrate d’Alembert's formula, let us find the solution to the wave equation, Equation 8.4.1, satisfying the initial conditions u (x, 0) = H (x + 1) — H (x — 1) and u t (x, 0) = 0, —∞ < x < ∞. By d’Alembert's formula, Equation. 8.4.17, u (x, t) = 1 2 [ H (x + c t + 1) + H (x − c t + 1) − H (x + c t − 1) − H (x − c t − 1) ]. (8.4.18) We illustrate this solution in Figure 8.4.1 generated by the MATLAB script: % set mesh size for solution clear; dx = 0.1; dt = 0.1; % compute grid X= [-10 : dx : 10] ; T = [0 : dt : 10] ; for j=1:length(T); t = T(j); for i=1:length(X); x = X(i); % compute characteristics characteristic_1 = x + t; characteristic_2 = x - t; % compute solution XX(i,j) = x; TT(i,j) = t; u(i,j) =. 0.5*(stepfun(characteristic_1,-1)+stepfun(characteristic_2,-1)... Figure 8.4.2 : The propagation of waves due to an initial displacement according to d’Alembert's formula. -stepfun(characteristic_1, 1)-stepfun(characteristic_2, 1)); end; end surf(XX,TT,u); colormap autumn; xlabel(’DISTANCE','Fontsize',20); ylabel(’TIME','Fontsize',20) zlabel(’SOLUTION’,’Fontsize’,20) In this figure, you can clearly see the characteristics as they emanate from the discontinuities at x = ±1. ❑ • Example 8.4.2 Let us find the solution to the wave equation, Equation 8.4.1, when u (x, 0) = 0, and u t (x, 0) = sin(2 x), —∞ < x < ∞. By d’Alembert's formula, the solution is u (x, t) = 1 2 c ∫ x − c t x + c t sin (2 τ) d τ = sin (2 x) sin (2 c t) 2. (8.4.19) In addition to providing a method of solving the wave equation, d’Alembert's solution can also provide physical insight into the vibration of a string. Consider the case when we release a string with zero velocity after giving it an initial displacement of f (x)

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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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  Applied Mathematics in Hydraulic Engineering

  An Introduction to Nonlinear Differential Equations

  • Kazumasa Mizumura(Author)
  • 2011(Publication Date)
  • WSPC

    (Publisher)

  Chapter 12 Wave Equations One of the most fundamental and common phenomena that occur in nature is the phenomenon of wave motion. When a stone is dropped into a pond, the surface of the water is disturbed and waves of displacement travel radially outward. When a bell is struck, sound waves propagate from the source of sound. We study wave equations such as the long wave equation and the Saint Venant equation. The method of solution is the separation of variables, series expansion, integral transform, and the method of characteristics. These are used for the computations of tidal waves, tsunamis, and flood propagations. 12.1. Classification of Partial Differential Equations of the Second-Order When independent variables are x and y and a dependent variable is φ , the partial differential equations of the second-order are written as A ∂ 2 φ ∂x 2 + B ∂ 2 φ ∂x∂y + C ∂ 2 φ ∂y 2 = f x, y, φ, ∂φ ∂x , ∂φ ∂y , (12.1) in which A , B , and C are smooth functions of x , y , φ , ∂φ/∂x , or ∂φ/∂y . Then, Eq. (12.1) is classified to the following three types: B 2 − AC > 0 · · · · · · hyperbolic type (wave equation) B 2 − AC < 0 · · · · · · elliptic type (potential equation) B 2 − AC = 0 · · · · · · parabolic type (diffusion or heat conduction equation) . 193 194 AMHE: An Introduction to Nonlinear Differential Equations As an example, the wave equation ∂ 2 φ ∂x 2 = ∂ 2 φ ∂y 2 (12.2) corresponds to A = 1, B = 0, and C = − 1 in Eq. (12.1). Since B 2 − AC > 0, Eq. (12.2) belongs to the hyperbolic type. The potential equation ∂ 2 φ ∂x 2 + ∂ 2 φ ∂y 2 = 0 (12.3) has A = 1, B = 0, and C = 1 in Eq. (12.1). Since B 2 − AC < 0, Eq. (12.3) belongs to the elliptic type. The equation of diffusion or heat conduction ∂φ ∂x = ∂ 2 φ ∂y 2 (12.4) has A = B = 0 and C = 1 in Eq. (12.1). Since B 2 − AC = 0, Eq. (12.4) belongs to the parabolic type. In the same way as ordinary differential equations, when we solve partial differential equations, boundary conditions are necessary.

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