Sound wave theory

In Eq. (4-29) jc is the distance traveled by the wave, and a is the absorption coefficient. Sound absorption can occur as a result of viscous losses and heat losses (these together constitute classical modes of absorption) and by coupling to a chemical reaction, as described in the preceding paragraph. The theory of classical sound absorption shows that a is directly proportional to where / is the sound wave frequency (in Hz), so results are usually reported as a//, for this is, classically, frequency independent. 

The harmonic oscillator is an important system in the study of physical phenomena in both classical and quantum mechanics. Classically, the harmonic oscillator describes the mechanical behavior of a spring and, by analogy, other phenomena such as the oscillations of charge flow in an electric circuit, the vibrations of sound-wave and light-wave generators, and oscillatory chemical reactions. The quantum-mechanical treatment of the harmonic oscillator may be applied to the vibrations of molecular bonds and has many other applications in quantum physics and held theory. 

The pioneering work on the chemical applications of ultrasound was conducted in the 1920 s by Richards and Loomis in their classic survey of the effects of high frequency sound waves on a variety of solutions, solids and pure liquidsQ). Ultrasonic waves are usually defined as those sound waves with a frequency of 20 kHz or higher. The human ear is most sensitive to frequencies in the 1-5 kHz range with upper and lower limits of 0.3 and 20 kHz, respectively. A brief but useful general treatment of the theory and applications of ultrasound has been given by Cracknel 1(2). 

Analytical and empirical correlations for droplet sizes generated by ultrasonic atomization are listed in Table 4.14 for an overview. In these correlations, Dm is the median droplet diameter, X is the wavelength of capillary waves, co0 is the operating frequency, a is the amplitude, UL0 is the liquid velocity at the nozzle exit in USWA, /Jmax is the maximum sound pressure, and Us is the speed of sound in gas. Most of the analytical correlations are derived on the basis of the capillary wave theory. Experimental observations revealed that the mean droplet size generated from thin liquid films on... 

Landau treats liquid helium hy an approach similar in lhat ol the Debye theory of solids. The longitudinal and transverse sound waves, which are the elementary cxcitalions of that theory of solids. corresponU in the case... 

These purely imaginary relaxation times reflect the oscillatory behavior of the undamped sound waves. Thus, Uj can be identified with the set of thermodynamic derivatives which appear in the nondissipative part of the theory. This implies that the diffusive process must arise from Uij, which now can be identified with the transport coefficients appearing in the dissipative part of the theory. 

The first result of this calculation is that the inertial motion causes almost no dephasing. This result is a direct contrast to models like the IBC theory, which attribute all the dephasing to collisional, i.e., inertial, dynamics. The difference between these theories lies in their assumptions about correlations in the solvent motion. The IBC explicitly assumes that the collisions are independent, i.e., the solvent motion has no correlations. As a result, the collisions are an effective sink for phase memory from the vibration. On the other hand, within the VE model the solvent motions appear as sound waves. Their effect on the vibrational frequency decays as they propagate away from the vibrator, but they remain fully coherent at all times. Because they remain coherent, they cannot destroy the phase... 

Newton immediately attacked the wave theory. Using some complex calculations, he showed that particles, too, would obey the laws of reflection and refraction. He also pointed out that, if truly a wave form, light should be able to bend around corners, just as sound does instead it cast a sharp shadow, further supporting the corpuscular theory. 

Ultrasonic velocity has been almost exclusively measured in ultrasonic studies of fat crystallization, but the attenuation coefficient also can reveal interesting information. As the sound wave passes, the fluid is alternately compressed and rarefied which results in the formation of rapidly varying temperature gradients. Heat energy is lost because the conduction mechanisms are inefficient (thermal losses) and together with molecular friction (viscous losses) cause an attenuation of the sound given by classical scattering theory (5) ... 

A theory for spinning detonation has been put forth by Fay (6). He shows that spin is a self-excited transverse vibratory motion in the burned gas, akin to a st inding sound wave, but with helical symmetry. The possible modes of vibration can be calculated from the properties of the gas. The fundamental mode always has a pitch equal to about three tube diameters higher modes have an apparent pitch smaller than this. Several different modes have been observed and good correlations are found with Fay s theory. 

Acoustic (sound) waves are also transmitted by atomic vibrations. Atomic vibrations are often called phonons, i.e., "particles" of sound. Theories of heat transfer [13] in insulators usually attempt to relate X to other physical properties which are mainly determined by atomic vibrations, such as the velocity of sound and the heat capacity. 

Rayleigh (1894) laid out the foundation for the scattering theory of sound wave propagation in fluids that contain suspended solids. He discussed the plane-wave disturbance produced by small obstacles and observed that (a) the zero-order term in the partial wave expansion of the disturbed field is a manifestation of the compressibility difference between the particles and the suspending fluid and (b) the first-order term is determined from the density difference as well as from the relative motion of particles (viscous drag losses). 

Another process that attenuates sound waves is wave scattering. A theory that describes wave propagation in solid suspension (Ishimaru, 1978) has been well established for a medium in which uniform particles are homogeneously suspended. The attenuation can be determined from the coefficients of the reflected compression wave, viz.,... 

The light-scattering spectrum which is related to 7 (q, /) by Eq. (3.3.3) consequently probes how a density fluctuation <5/ (q) spontaneously arises and decays due to the thermal motion of the molecules. Density disturbances in macroscopic systems can propagate in the form of sound waves. It follows that light scattering in pure fluids and mixtures will eventually require the use of thermodynamic and hydrodynamic models. In this chapter we do not deal with these complicated theories (see Chapters 9-13) but rather with the simplest possible systems that do not require these theories. Examples of such systems are dilute macromolecular solutions, ideal gases, and bacterial dispersions. ... 

---