Ultrasonic wave equation

The ultrasonic relaxation loss may involve a thermally activated stmctural relaxation associated with a shifting of bridging oxygen atoms between two equihbrium positions (169). The velocity, O, of ultrasonic waves in an infinite medium is given by the following equation, where M is the appropriate elastic modulus, and density, d, is 2.20 g/cm. ... 

Ultrasonic waves also bring about a reduction in the viscosity of nitrocellulose. The effect is more marked the higher the initial viscosity, and thus the longer the nitrocellulose chains (Sollner [93] Schmidt and Rommel [94]). For instance, nitrocellulose with a molecular weight of 123,000, estimated by means of Staudinger s viscosity equation, subjected in solution form to the action of ultrasonic waves suffered a degradation to a molecular weight of 70,000-80,000. 

Bis(alkylamino)-6-chloro-l,3,5-triazines are important herbicides which undergo dealkylation as the major detoxication process in animals. In vitro studies have shown that thermal dealkylation occurs via a Chugaev-like mechanism (equation 15). The dealkylation is also effected by sonication (irradiation with ultrasonic waves), but the mechanism has not been elucidated (71MI22000, 77JCS(P1)1257). It is also possible to dealkylate such 1,3,5-triazines by using hydroxyl radicals (71MI22001). 

A mathematical description of an ultrasonic wave must describe the dependence of the particle displacement on distance and time, and the reduction of its amplitude with distance traveled through the material. For plane sinusoidal waves the following equation is appropriate ... 

Here A is the amplitude of the wave, and x is the distance traveled. The attenuation coefficient is determined by measuring the dependence of the amplitude of an ultrasonic wave on distance and fitting the measurements to the above equation. The attenuation is often given in units of decibels per meter (dB nr1) where 1 Np = 8.686 dB. 

CSIRO Minerals has developed a particle size analyzer (UltraPS) based on ultrasonic attenuation and velocity spectrometry for particle size determination [269]. A gamma-ray transmission gauge corrects for variations in the density of the slurry. UltraPS is applicable to the measurement of particles in the size range 0.1 to 1000 pm in highly concentrated slurries without dilution. The method involves making measurements of the transit time (and hence velocity) and amplitude (attenuation) of pulsed multiple frequency ultrasonic waves that have passed through a concentrated slurry. From the measured ultrasonic velocity and attenuation particle size can be inferred either by using mathematical inversion techniques to provide a full size distribution or by correlation of the data with particle size cut points determined by laboratory analyses to provide a calibration equation. 

Let us begin our discussion with a description of a plane ultrasonic wave in terms of the displacement of a particle, from its equilibrium position as a function of the distance that wave has traveled. Let us assume that the wave propagates along the x coordinate. We can then write the general wave equation as... 

To obtain the absolute sound attenuation in the coal slurry, the diffraction loss, the acoustic mismatch loss, the attenuation due to the Teflon window, and the oil coupling must be calculated. Thus, it is difficult to accurately determine the absolute attenuation. In practice, one measures the relative attenuation with respect to a standard. The attenuation of ultrasonic waves in a solid suspension is attributed to three major factors, namely, scattering, viscosity, and thermal effects. Although the presence of particles affects the fluid viscosity and thermal conductivity, the primary source of attenuation may be due to particle scattering. Hence, one may define the relative attenuation of the HYGAS coal slurry by comparing the slurry attenuation with that of the carrier fluid, i.e., the toluene/benzene mixture. This can be expressed by the equation... 

An important improvement of reconstructions can be obtained by selecting a more appropriate model and abandon the straight pathway approximation . A more sophisticated model gives a description of the propagation of ultrasonic waves in a medium based upon the wave equation ... 

Elastic Coefficients The elastic stiffness coefficients Cy can be calculated from the measured velocity of propagation of bulk acoustic ultrasonic waves, according to the Papadakis method (quartz transducer with center frequency of 20MHz) (Papadakis, 1967), on differently oriented bar-shaped samples using the equations given by Truell et al. (1969) and corrected for the piezoelectric contributions (Ljamov, 1983 Ikeda, 1990). The samples were oriented in axial directions XYZ, and 45° rotated against the X- and Y-axes, respectively. In order to obtain optimized values for the elastic materials parameters, the elastic stiffness coefficients Cy were used to calculate and critically compare the results of surface acoustic wave (SAW) measurements. 

Several equations for polymer reaction in solution caused by ultrasonic waves have been reported. These studies are discussed in Volume 2, Section VIIB.D. Van der Hoff and Glynn [56-58] studied the rate of polystyrene... 

The velocity of elastic ultrasonic waves in solution is strongly influenced by solute-solvent and solute-solute interactions which, in turn, are determined by the chemical structure of the solute and solvent molecules. Adiabatic compressibility P for a system having the density p, is related to the measured ultrasonic velocity u by the following equation ... 

In Equation 2.10 B is a constant and the A/s represent the relaxation amplitudes that are proportional to the square of the isentropic volume change 0/° given by Equation 2.9 and that contains both 07 and AH°. Indeed, the propagation of ultrasonic waves in fluids gives rise to harmonic changes of p and T. The investigated equilibria are shifted periodically by these two perturbations, thus the ultrasonic relaxation amplitude dependence on... 

This is a consequence of the wave equation for ultrasonic and acoustic waves. In one-dimension the general form is... 

The velocity v of the ultrasonic wave in a material is given by the following equation ... 

Table 1 shows relations between the elastic constants of materials of a hexagonal symmetry and the velocity,direction of propagation and plane of polarisation of the ultrasonic waves. The relations between the material constants (Young"s modulus, stiffness modulus and Poisson number) and the elastic constants, derived from the equations quoted in Table 1 are as follows ... 

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