Wave equation acoustic

Strittmater (S6) has presented a solution to the damped acoustic wave equation, and shown that the acoustic pressure has the form... 

In the absence of fluctuations in the heat release rate, dq/dt = 0, Equation 5.1.14 reduces to the standard wave equation for the acoustic pressure. It can be seen that a fluctuating heat release then acts as a source term for the acoustic pressure. 

Yasui et al. [29] have used solution of wave equation based on finite element method for characterization of the acoustic field distribution. A unique feature of the work is that it also considers contribution of the vibrations occurring due to the reactor wall and have evaluated the effect of different types of the reactor walls or in other words the effect of material of construction of the sonochemical reactor. The work has also contributed to the understanding of the dependence of the attenuation coefficient due to the liquid medium on the contribution of the vibrations from the wall. It has been shown that as the attenuation coefficient increases, the influence of the acoustic emission from the vibrating wall becomes smaller and for very low values of the attenuation coefficient, the acoustic field in the reactor is very complex due to the strong acoustic emission from the wall. 

The outline of this paper is as follows. First, a theoretical model of unsteady motions in a combustion chamber with feedback control is constructed. The formulation is based on a generalized wave equation which accommodates all influences of acoustic wave motions and combustion responses. Control actions are achieved by injecting secondary fuel into the chamber, with its instantaneous mass flow rate determined by a robust controller. Physically, the reaction of the injected fuel with the primary combustion flow produces a modulated distribution of external forcing to the oscillatory flowfield, and it can be modeled conveniently by an assembly of point actuators. After a procedure equivalent to the Galerkin method, the governing wave equation reduces to a system of ordinary differential equations with time-delayed inputs for the amplitude of each acoustic mode, serving as the basis for the controller design. 

The attenuation may be expressed by making the wavenumber complex (this would be k — ia in eqn (6.12)), and the velocity (= w/k) may also be written as a complex quantity. This in turn corresponds to a complex modulus, so that the relationship v - /(B/p) is preserved indeed the acoustic wave equation may be written as a complex-valued equation, without the need for the extra term in (6.11). Complex-valued elastic moduli are frequency-dependent, and the frequency-dependent attenuation and the velocity dispersion are linked by a causal Kramers-Kronig relationship (Lee et al. 1990). 

When the room is highly idealized, for instance if it is perfectly rectangular with rigid walls, the reverberant behavior of the room can be described mathematically in closed form. This is done by solving the acoustical wave equation for the boundary conditions imposed by the walls of the room. This approach yields a solution based on the natural resonant frequencies of the room, called normal modes. For the case of a rectangular room shown in figure 3.3, the resonant frequencies are given by [Beranek, 1986] ... 

Like in classical acoustics in cold flows [304], wave equations can also be derived for reacting flows but they are much more complex [270 340]. Af-... 

Wave Equation. The propagation of the displacement vector u of an acoustic excitation can be directly expressed in terms of materials parameters from the law of conservation of momentum and Equations (3) and (4) as a wave equation ( ). In vector notation,... 

The Eulerian finite difference scheme aims to replace the wave equations which describe the acoustic response of anechoic structures with a numerical analogue. The response functions are typically approximated by series of parabolas. Material discontinuities are similarly treated unless special boundary conditions are considered. This will introduce some smearing of the solution ( ). Propagation of acoustic excitation across water-air, water-steel and elastomer-air have been computed to accuracies better than two percent error ( ). In two-dimensional calculations, errors below five percent are practicable. The position of the boundaries are in general considered to be fixed. These constraints limit the Eulerian scheme to the calculation of acoustic responses of anechoic structures without, simultaneously, considering non-acoustic pressure deformations. However, Eulerian schemes may lead to relatively simple algorithms, as evident from Equation (20), which enable multi-dimensional computations to be carried out in a reasonable time. 

The simplest model of these waves is one based on acoustic principles. Assume that the earth can be treated as an acoustic medium and the influence of variations in density can be ignored. In this case the propagation of seismic waves in the earth can be described by the acoustic wave equation ... 

In general cases of elastic media, the seismic field equations will be much more complicated than acoustic wave equations. We will study these eejuations in Chapter 13. 

Let us assume that we are seeking a nonradiating source for an acoustic wave equation ... 

High frequency approximations in the solution of an acoustic wave equation The reader, familiar with the background of seismic exploration methods, should recall that many successful seismic interpretation algorithms are based on the simple principles of geometrical seismics, which resembles the ideas of geometrical optics. The question is how this simple but powerful approach is connected with the... 

Green s functions appear as the solutions of seismic field equations (acoustic wave equation or equations of dynamic elasticity theory) in cases where the right-hand side of those equations represents the point pulse source. These solutions are often referred to as fundamental solutions. For example, in the case of the scalar wave equation (13.54), the density of the distribution of point pulse forces is given as a product,... 

Second, if we assume that the process is linear (which can be seen immediately from the original wave equations), the sum of effects on an acoustic medium must be equal to the sum of its responses. In other words, the wavefield generated by the source presented as a linear combination of some other sources is equal to the linear combination (with the same coefficients) of the fields generated by the corresponding individual sources. 

Let us consider an acoustic medium. The propagation of acoustic waves can be described by the scalar wave equation... 

This is a wave equation in three-dimensional space, whose solutions we shall investigate later. It may make matters easier for the reader if we begin with corresponding problems in one and two dimensions and for the sake of perspicuity we shall take our examples from classical mechanics (acoustics). 

G. Cohen andP. Joly, Construction and analysis of fourth-order finite difference schemes for the acoustic wave equations in nonhomogeneous media, SIAM J. Numer. Anal., vol. 33, pp. 1266-1302, 1996.doi 10.1137/S0036142993246445... 

An amazing feature of shock compression is illustrated in Figs. Id-e. A driven shock front steepens up as it runs, in contrast to acoustic waves that disperse as they run [1]. Imagine a shock front that is not initially steep (Fig. Id). Think of the front as a higher pressure wave trailing a lower pressure wave. Equation (2) above shows the trailing wave moves faster. In an ideal continuous elastic medium, the shock front steepens until it becomes an abrupt discontinuity. The shock front risetime tr —> 0. 

Figure 6a shows the transmission hne representing a viscoelastic layer [64]. Every layer is represented by a T . The apphcation of the Kirchhoff laws to the Ts reproduces the wave equation and the continuity of stress and strain. The detailed proof is provided in [4]. To the left and to the right of the circuit are open interfaces (ports). These can be exposed to external shear waves. They can also be connected to the ports of neighboring layers (Fig. 6b). Alternatively, they may just be short-circuited, in case there is no stress acting on this surface (left-hand side in Fig. 6c). Finally, if the stress-speed ratio Zl (the load impedance, see below) of the sample is known, the port can be short-circuited across an element of the form AZl, where A is the active area (right-hand side in Fig. 6c). Figure 6c shows a viscoelastic layer which is also piezoelectric. This equivalent circuit was first derived by Mason [4,55]. We term it the Mason circuit. The capacitance, Co, is the electric capacitance between the electrodes. The port to the right-hand side of the transformer is the electrical port. The series resonance frequency is given by the condition that the impedance of the acoustic part (the stress-speed ratio, aju) be zero, where the acoustic part comprises all elements connected to the left-hand side of the transformer.

The variable k is called the scattering coefficient, (from wave scattering theory) and the equations are called scattering equations. They express the behavior of the wave equation at the boimdary between acoustic tube segments of different characteristic impedances, where part of the incoming wave is... 

Charlier J-P, Crowet F. Wave equations in linear viscoelastic materials. J Acoust Soc Am 1986 79 895-900... 

Upon taking the partial time derivative of Eq. 7, subtracting the divergence of Eq. 8 from the result and making use of the linearized equation of state (9), the acoustic pressure field is seen to satisfy the linear wave equation ... 

Also, the existence of the upper limit of the parameter K, which was explained in Chapter 1, solves one more problem. Namely, in Ref. [21], optical and acoustical frequencies were compared. Optical frequency is defined by Equation (27.27). The same procedure for linear wave. Equation (27.12), instead of for that nonlinear, Equation (27.13), would lead to a so-called acoustical frequency [21,31] ... 

Kim, Y.-H. 2010a. Acoustic Wave Equation and Its Basic Physical Measures. Sound Propagation, 69-128. NJ John Wiley Sons, Ltd. 

Perhaps the most general mathematical treatment of the surface mass loading effect on bulk shear wave resonators has been presented by Kanazawa (13). In this work, a wave equation was developed for acoustic wave propagation within the deposited layer, assuming the material had both elastic and viscous properties. Boundary conditions between crystal and deposited mass were established by assuming shear forces and particle displacements were equal for both materials at the interface plane. This approach results in a fairly complex mathematical model, but simplified relationships were derived for purely elastic and purely viscous behaviour. 

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