Wave equation elastic displacement

Let us analyze the space and time structure of the elastic displacement field in detail. We will demonstrate that equation (13.26) describes the propagation of two types of body waves in an elastic medium, i.e., compressional and shear waves travelling at different velocities and featuring different physical properties. To this end, let us recall the well-known Helmholtz theorem according to which an arbitrary vector field, in particular an elastic displacement field U(r), may be represented as a sum of a potential, Up(r), and a solenoidal, Us(r), field (Zhdanov, 1988) ... 

As we can see here, both the potential and solenoidal components of the elastic displacement field satisfy wave equations and therefore represent waves traveling in space at velocities Cp and Cg respectively. Let us examine them in detail. 

According to the linearity of the wave equation, the vector field of an arbitrary source can be represented as the sum of elementary fields generated by the point pulse sources. However, the polarization (i.e., direction) of the vector field does not coincide with the polarization of the source, F . For instance, the elastic displacement field generated by an external force directed along axis x may have nonzero components along all three coordinate axes. That is why in the vector case not just one scalar but three vector functions are required. The combination of those vector functions forms a tensor object G" (r, t), which we call the Green s tensor of the vector wave equation. 

Any disturbance W [e,g, the displacement of an elastic string or the height of a water wave) which is propagated along the r-direction with velocity u satisfies a simple partial differential equation, the wave equation ... 

When an ac voltage is applied between the electrodes, the motion of the AT cut quartz crystal can be described by a system of two coupled dilferential equations, which constitute the wave equation for elastic displacements, u z,t) = u z,03) exp(ioit), and the equations that establish... 

Nonuniform distribution of species results in nonuniform distribution of the properties of liquid near the vibrating surface of the resonator. The properties change with distance from the interface, until the values corresponding to the bulk of solution have been reached. In order to simplify the description of this nonuniformity on the QCM, it is assumed that a thin film of liquid, having different values of >/yand pf, exists at the interface [61]. To calculate the effect of this film on the frequency shift, one has to solve the wave equation for the elastic displacements in the quartz... 

Ultrasound in Liquids. Sound passes through an elastic medium as a longitudinal wave, ie, a series of alternating compressions and rarefactions. This means that hquid is displaced parallel to the direction of motion of the wave. The maximum displacement velocity together with the density (p) and the speed of sound in the mediiun (y) determine the ultrasound intensity and the maximum acoustic pressure (PA,max) (8). The variation of the acoustic pressure (Pa) as a function of time t) at a fixed frequency (/ ) is described by equation 1 ... 

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. 

Certainly, the system of equations for the attraction field is much more complicated than that for the potential. Before we continue it may be appropriate to make the following comment. In all geophysical methods the fields, such as the particle displacement caused by elastic waves, the constant and time-varying electric... 

Equation (14.19) is applicable to the case where a thick layer of a liquid is placed atop the resonator surface. Physically, it predicts that only a thin layer of liquid will undergo displacement at the surface the bulk wave device, and device response will be a function of the mass of this layer. Bruckenstein has observed that the response of a resonator which has both an elastic solid deposited layer and liquid atop the surface will be a linear sum of the responses expected for each individual perturbation (12). 

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