Propagation of sound

According to the theory of acoustics, the velocity of propagation of sound w through a gas is given by the equation... 

C. If the propagation of sound is assumed to occur isothermally, show that... 

The motion of atoms in the lattice can be depicted as a wave propagation (phonon). By dispersion we mean the variation in the wave frequency as reciprocal space is traversed. The propagation of sound waves is similar to the translation of all atoms of the unit cell in the same direction hence the set of translational modes is commonly defined as an acoustic branch. The remaining vibrational modes are defined as optical branches, because they are capable of interaction with light (see McMillan, 1985, and Tossell and Vaughan, 1992, for more exhaustive explanations). 

Ultrasound is the study and application of sound waves whose frequency is too high to be detected by the human ear, i.e., above about 16 kHz [1], This is a purely arbitrary cut-off point, determined by the limitations of the human ear. The physics describing the propagation of ultrasonic waves is the same as that describing the propagation of sound waves. 

Physically, the frozen sound velocity corresponds to the velocity of propagation of sound waves in a limit attained at high frequency, and the equilibrium sound velocity corresponds to the propagation velocity of sound waves in a limit attained at low frequency... 

The ultrasound wave is longitudinal in nature (i.e., the direction of propagation is the same as the direction of oscillation). Longitudinal sound waves cause compression and expansion of the medium at a distance of half the wavelength, leading to pressure variations in the medium. The resistance of the medium to the propagation of sound waves is dependent on the... 

Schwabl F (1985) Propagation of sound at continuous stractural phase transitions. J Stat Phys 39 719-737 Schwabl F, Tauber UC (1996) Continuous elastic phase transitions in pure and disordered crystals. Phil Trans Roy Soc Lond A 354 2847-2873... 

This equation describes propagation of sound waves, with wave speed Cs = jEjp, and is therefore usually referred to as the wave equation. In fact, the general solution may be expressed as u(x, t) =fR x — Cgt) +/l(x + Cgt), where the first term represents a wave traveling to the right and the second, a wave traveling to the left. Here, /r and are arbitrary functions of the indicated arguments. Although this particular derivation pertains to sound waves, all wave motions are in fact described by equations of the same form. 

Ultrasound imaging is based on reflection ofthe sound waves (1-3 MHz) at the borders between areas with different acoustic impedance, which is determined by the speed of propagation of sound and the density of the tissue. Because the interfaces between blood and other soft tissues, for example the heart or liver, do not... 

A prototype of a hyperbolic equation is the wave equation. The wave equation governs many physical phenomena such as the propagation of sound waves, water waves, vibration of a membrane in a two-dimensional setting, and vibration of a string in a one-dimensional setting. 

Propagation of sound is an established method of studying irreversible thermodynamics. Sound propagation is accompanied by heat production, viscous flow, relaxation phenomena and chemical reactions, each of which is determined by a particular relaxation time. 

Rahman A, Stillinger FH Propagation of sound in water. A molecular-dynamics study. Phys. Rev. A 1974, 10 368-378. 

Principle. The speed of propagation of sound waves in a gas is given by Laplace s equation, given below. 

Now we consider the wave equation for the propagation of sound in a cylinder of radius Tq and length 1. For cylindrical coordinates (r, 9, z) we obtain the homogeneous wave equation ... 

When p and (dp[Bp)T are known, the measurement of W provides a means for obtaining valuable information on thermodynamic properties which are accessible to direct measurement only with difficulty. For a liquid system ir equilibrium w ith its vapor, properties are mostly measured as a function of both temperature and pressure. As the propagation of sound is an adiabatic process, data on the sound velocity make it possible to solve this simultaneous dependence and to distinguish the separate contributions of p and T in the change of thermodynamic properties. 

A partial differential equation ( PDE) is a relation involving an unknown function of several independent variables and its partial derivatives with respect to those variables. Partial differential equations are used to formulate and solve problems that involve unknown functions of several variables, such as the propagation of sound or heat, electrostatics, electrodynamics, fluid flow, elasticity, or more generally any process that is distributed in space or distributed in space and time. (Definition taken from Wikipedia http //en.wikipedia.org/wiki/Partial differenti al equation)... 

An interesting phenomenon is the behaviour of fluids near the critical point. Approaching the critical state the time required to reach equilibrium increases with an inverse power of the difference between actual and critical temperature. In the experiments of Borisov et al. [7] for instance it took about 20 hours. As a consequence of large scale critical fluctuations the propagation of sound is severely impeded. Soundspeed measurements in the kHz and... 

The propagation of sound in the outer flow will be described by using the two dimensional wave equation. And the sound intensity outside the bubbly layer, defined as,... 

Here the velocity v(jc, y, z, t) of a liquid is measured in a particular fixed point in space x, y, z. It is not the velocity of a small unit volume of moving liquid. Sources mean the presence of sources and sinks in the volume discussed. If we are not interested in propagation of sound i.e. ignore a local compression and dilatation we may put 8p = 0 and p = const. Then, it is the case of incompressible liquid. ... 

The system of equations of horizontal motion [Eqs. (9) and (10)], hydrostatic equilibrium [Eq. (16)], mass continuity [Eq. (12)], thermodynamics [Eq. (8)], and the ideal gas law [Eq. (7)] is called the hydrostatic prediction model, or primitive equations. The hydrostatic assumption modifies the basic atmospheric prediction system in such a way as to eliminate the vertical propagation of sound waves. 

A process is adiabatic if the system considered does not exchange energy under the form of heat with the external surroundings. The propagation of sound through air is adiabatic (except at very high frequencies). 

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