Sound, dispersion

TOWARDS THE HYDRODYNAMIC LIMIT STRUCTURE FACTORS AND SOUND DISPERSION. The collective motions of water molecules give rise to many hydrodynamical phenomena observable in the laboratories. They are most conveniently studied in terms of the spatial Fourier ( ) components of the density, particle currents, stress, and energy fluxes. The time correlation function of those Fourier components detail the decay of density, current, and fluctuation on the length scale of the Ijk. 

Einstein showed that when a reversible reaction is present sound dispersion occurs at low frequency the equilibrium is shifted within the time of oscillation, the effective specific heat is at a maximum, and the speed of sound c0 is at a minimum. At high frequency the oscillations occur so rapidly that the equilibrium has no time to shift (it is frozen ). The corresponding Hugoniot adiabate (FHA) is shown in the figure. Here the effective heat capacity is minimal, the speed of sound c is maximal cx > c0. From consideration of the final state and the theory of shock waves it follows that C>c0. 

From sound-dispersion measurements in gases, it is possible to make estimates of the probability of tieactivation of vibrations per collision, and these turn out to be of the order of 10" to 10 . See, for example, H. S. W. Massey and E. II. S. Burhop, Electronic and Ionic. Impact Phenomena, Oxford University Press, New York, 1962. 

Solutions to equation (113) with a sinusoidal time dependence are of interest, since they illustrate sound dispersion resulting from finite reaction rates. Suppose one generates sound waves in a gas occupying the region X > 0 by oscillating a piston harmonically about the point x = 0 (see Figure 4.6). The velocity of the piston can be represented by the real part of... 

This result shows that the phase velocity of the disturbance is a = afo/ia cos (p) and that the disturbance attenuates as x increases with a characteristic decay length given by / = ayo/(cua sin (p). The consequent dependence of the phase velocity on the frequency [see equations (115) and (116)] represents sound dispersion, and the attenuation illustrates sound absorption. Various specific results follow directly from equations (114)-(116). As CD 0, a (i o attenuation) as cd oo, a ayo... 

Urick (1948) measured ultrasonic attenuation in aqueous kaolin as well as sound dispersion. The results were in good agreement with the losses predicted from viscous drag at the particle surface. Sound propagation in a suspension can also produce temperature gradients at the particle/suspending-fluid interface, and thus, results on attenuation via thermal diffusion. 

Except for Eq. (21), the acoustic wavelength A depends on the refractive index and hence on the measured sound frequency for all scattering geometries. In the case of Eq. (21), the refractive index enters the refraction process and the phase velocity in a compensating manner and therefore the sound wavelength becomes independent of n. Provided sound dispersion is absent, Eq. (21) can be used in combination with Eqs. (20), (22), or (24) to determine the refractive index from Brillouin data, e.g., as in Eqs. (25, 26). 

Erdem. R. Keskin, M. (2003). Sound dispersion in a spin-1 Ising system near the second-order phase transition point. Physics Letters A, 310, pp. 74. 

The same applies to the calculation of sound dispersion, i.e. frequency impact on sound speed. Even so, the spectrum can only be reproduced as sphere-spectram by assuming polydispersity. 

Emerging applications for lead foil or sheet include their use to contain or to shield against radioactive materials. Partitions and curtains (incorporating lead sheet) are also widely used to limit sound dispersion in noisy working environments. 

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