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Concluding Examples and Observations
Published in Ronald B. Guenther, John W. Lee, Sturm-Liouville Problems, 2018
Ronald B. Guenther, John W. Lee
We should expect that the energy of a damped string modeled by (8.15) decreases with time. To confirm this, differentiate the total energy E in (8.11) and use the damped wave equation ρutt+ρkut=τuxx to find that dEdt=∫0lut(ρutt−τuxx)dx=−∫0lρkut2dx≤0. Thus, the total mechanical energy of the damped string decreases.
Boundary Element Methods for Heat Conduction
Published in W.J. Minkowycz, E.M. Sparrow, Advances in Numerical Heat Transfer, 2018
In this section, we present BEM models for the non-Fourier mode of heat conduction modeled by the Hyperbolic Heat Conduction Equation (HHCE). This damped wave equation has been proposed by many researchers (see Catteno[57] and Morse and Feshbach[10]) to model a finite speed of propagation of energy in heat conducting media. An excellent review of the subject is found in the text by Tzou[58]. Numerical solutions of the 1-D HHCE were undertaken by Glass et al.[59] using the MacCormak finite difference method, by Tamma and Railkar[60] using specially tailored transfinite element methods, and by Yang[61] (who presents a comparative study of various schemes) using characteristics based finite- volume methods. Yang[62] also solved the 2-D HHCE using highly accurate finite-volume based TVD schemes. Tzou[63] presents a review of recent research on the HHCE and moving heat sources and their effects on crack propagation. Kaminski[64] experimentally determined the relaxation time, τ, for several materials such as sand and glass ballotini. There is, however, a dearth of thermophysical data for τ. Vedarvarz et al.[65] present a regime map indicating the characteristic length and time scales for which the HHCE is applicable and indicate some applications in which the Fourier model for the heat conduction equation may be in error. Kassab and Nordlund[66] use the Laplace transform BEM to solve two-dimensional HHCE problems, while Priedeman and Kassab[67] and Vicki and West[68] use time-dependent fundamental solutions and a convolution marching scheme to resolve the 1-D damped wave propagation. The first authors consider application to heat transfer, while the second consider application to damped propagating waves in a rod. Lu et al.[69] develop a dual reciprocity BEM to solve the HHCE with applications to thermal wave propagation in biological tissue. In this section, the BEM solution of the HHCE is considered using two approaches: time-dependent fundamental solutions and Laplace transform method.
Powder and Granular Flow
Published in Efstathios E. Michaelides, Clayton T. Crowe, John D. Schwarzkopf, Multiphase Flow Handbook, 2016
By eliminating the uctuating density in Equations 10.52 and 10.54, a damped wave equation is found for v as 2 v 2 2 v 9k 3 v = c0 2 + t 2 y 5r0 ty 2 When the attenuation is weak, a longitudinal wave solution is found readily as follows: v(x,y,t) = v b (x)cos(xy - w// t)e - Vy with x = w// /c 0 and V = 9kw2 /10r0c 0 // (10.57) (10.56) (10.55)
An inverse problem for the transmission wave equation with Kelvin–Voigt damping
Published in Applicable Analysis, 2023
Apart from the acoustic wave equation, the damped wave equation is also very important in applied disciplines such as exploration geophysics since the real media is all some degree of attenuation or damping. In [23], a semi-discrete coefficient inverse problem for the strongly damped wave equation with the continuous coefficients is investigated. In this paper, we consider the transmission strongly damped wave equation and the coefficients in the equation are discontinuous.
A hybrid computational method for local fractional dissipative and damped wave equations in fractal media
Published in Waves in Random and Complex Media, 2022
Ved Prakash Dubey, Jagdev Singh, Ahmed M. Alshehri, Sarvesh Dubey, Devendra Kumar
A wave equation is a linear PDE of second order which provides the general characteristics of mechanical or light waves that arise in classical physics. The wave equation informs about the propagation of oscillations at a constant speed in some quantity. The problem related to a vibration of a string was initially studied by d’ Alembert [22]. A classical 1D wave equation was discovered by d’ Alembert [22] and was described as , where denotes the displacement of the vibration of the string in a vertical direction or along the direction of motion and denotes the speed of propagation of the wave. It is notable that the string can be fixed at both ends or just at one end or an infinite string. In each case, different boundary conditions will be obtained for the wave equation. The equation for the damped wave is given by , where symbolizes the coefficient of friction arises from displacement of string. The wave equation describes how the displacement of a string changes as a function of time and position. A damped wave is that which possesses successive cycles of progressively diminishing amplitudes. It is notable that the amplitude decreases exponentially with distance due to energy loss. On the other hand, dissipation is used to mark out the course of actions in which energy is wasted. A dissipative wave is that which loses its amplitude. The dissipative wave equation indicates the loss of wave energy and consequent decline in wave height due to turbulence, wave breaking and viscous effects, and in case of shallow water waves by virtue of bottom friction. The damped wave equation appears in various disciplines of engineering and science. The normal supposition is that the damping mechanics is proportional to the velocity of propagation. The quantitative magnitude of damping is frequency-dependent and its behavior mostly follows a power law. This phenomenon usually occurs in various kinds of applications, for instance, viscoelasticity, acoustics, seismic wave propagation, and structural vibration.