Perturbations sound wave

The previous subsection described single-experiment perturbations by J-jumps or P-jumps. By contrast, sound and ultrasound may be used to induce small periodic perturbations of an equilibrium system that are equivalent to periodic pressure and temperature changes. A temperature amplitude 0.002 K and a pressure amplitude 5 P ss 30 mbar are typical in experiments with high-frequency ultrasound. Fignre B2.5.4 illustrates the situation for different rates of chemical relaxation with the angular frequency of the sound wave... 

In ultrasonic relaxation measurements perturbation of an equilibrium is achieved by passing a sound wave through a solution, resulting in periodic variations in pressure and temperature.40,41 If a system in chemical equilibrium has a non-zero value of AH° or AV° then it can be cyclically perturbed by the sound wave. The system cannot react to a sound wave with a frequency that is faster than the rates of equilibration of the system, and in this case only classical sound absorption due to frictional effects occurs. When the rate for the host-guest equilibration is faster than the frequency of the sound wave the system re-equilibrates during the cyclic variation of the sound wave with the net result of an absorption of energy from the sound wave to supply heat to the reaction (Fig. 4). 

Sound waves provide a periodic oscillation of pressure and temperature. In water, the pressure perturbation is most important in non-aqueous solution, the temperature effect is paramount. If cu (= 2 nf, where/is the sound frequency in cps) is very much larger than t (t, relaxation time of the chemical system), then the chemical system will have no opportunity to respond to the very high frequency of the sound waves, and will remain sensibly unaffected. 

Molecules of a liquid which can exist in two or more states of different energy may be perturbed by the passage of a sound wave. The disturbance may result from pressure or temperature changes accompanying the sound wave, and the experiment may be repeated in a wide range of frequencies. The... 

Relaxation techniques based on ultrasonics have been widely used to obtain rate constants for fast reactions. In this method a system at equilibrium is perturbed by the passage of a sound wave, which induces pressure and temperature variations. Ah... 

The expectation value of the property A at the space-time point (r, t) depends in general on the perturbing force F at all earlier times t — t and at all other points r in the system. This dependence springs from the fact that it takes the system a certain time to respond to the perturbation that is, there can be a time lag between the imposition of the perturbation and the response of the system. The spatial dependence arises from the fact that if a force is applied at one point of the system it will induce certain properties at this point which will perturb other parts of the system. For example, when a molecule is excited by a weak field its dipole moment may change, thereby changing the electrical polarization at other points in the system. Another simple example of these nonlocal changes is that of a neutron which when introduced into a system produces a density fluctuation. This density fluctuation propagates to other points in the medium in the form of sound waves. 

Relaxation methods can be classified as either transient or stationary (Bernasconi, 1986). The former include pressure and temperature jump (p-jump and t-jump, respectively), and electric field pulse. With these methods, the equilibrium is perturbed and the relaxation time is monitored using some physical measurement such as conductivity. Examples of stationary relaxation methods are ultrasonic and certain electric field methods. Here, the reaction system is perturbed using a sound wave, which creates temperature and pressure changes or an oscillating electric field. Chemical relaxation can then be determined by analyzing absorbed energy (acous-... 

Before recombination, the radiation pressure is so great that the Jeans length is greater than the horizon size, and so no perturbations within the horizon can grow they can only oscillate as sound waves. Conversely, after recombination, the pressure drops precipitously and all of a sudden, perturbations within the horizon can grow. 

Stationary relaxation methods include sound absorption und dlNpcrRlon and dielectric dispersion. A sound wave is used to perturb thc system (hat causes temperature and pressure alterations on an oscillating electric field. Then, chemical relaxation is measured by determining adsorbed energy (acoustical absorption or dielectric loss), or a phase lag that is dependent on the frequency of a forcing function (Bernasconi, 1986 Sparks, 1989). In this chapter, only transient relaxation methods will be discussed. 

Molecular dynamics with periodic boundary conditions is presently the most widely used approach for studying the equilibrium and dynamic properties of pure bulk solvent,97 as well as solvated systems. However, periodic boundary conditions have their limitations. They introduce errors in the time development of equilibrium properties for times greater than that required for a sound wave to traverse the central cell. This is because the periodicity of information flow across the boundaries interferes with the time development of other processes. The velocity of sound through water at a density of 1 g/cm3 and 300 K is 15 A/ps for a cubic cell with a dimension of 45 A, the cycle time is only 3 ps and the time development of all properties beyond this time may be affected. Also, conventional periodic boundary methods are of less use for studies of chemical reactions involving enzyme and substrate molecules because there is no means for such a system to relax back to thermal equilibrium. This is not the case when alternative ensembles of the constant-temperature variety are employed. However, in these models it is not clear that the somewhat arbitrary coupling to a constant temperature heat bath does not influence the rate of reequilibration from a thermally perturbed... 

If a sound wave is passed through the solution it will perturb the equilibrium slightly and a wave of new equilibrium positions is called for. If the concentrations can alter rapidly enough to enable... 

Electro-optic and magneto-optic phenomena contain terms of nonlinear optics effects (see Eqs. (4.32), (4.33), (4.36), and (4.39)). On the other hand, acousto-optic effects which arise from a periodical density fluctuation of the medium, analogous to the Brillouin scattering phenomenon, do not contain terms of nonlinear optics, as a general rule.5) The perturbation of light propagation by sonic waves differs from that induced by electric and magnetic fields. As the electric susceptibility, xe, is a function of the density of the medium, it will be influenced by the periodical density fluctuation induced in a medium by sound waves. 

The variation of dT with position may be neglected, since the sound wavelength is much larger than the range of correlations at controllable temperature intervals from the critical point. Only a first-order perturbation theory will be attempted, so when Eq. (102) is used in Eq. (101) the equilibrium G may be used where it multiplies 8T. If is the perturbation in G induced by the sound wave, we have... 

Local fluctuations in temperature, density and pressure are always present, even in homogeneous, premixed systems. In systems whose sensitivity to perturbations is high, these fluctuations can have surprising consequences and alter the behavior from conventional expectations. We saw this above when we investigated the sensitivity of the hydrogen-oxygen mixture to the presence of sound waves. It has also been shown that the same systems are just as sensitive to entropy perturbations (12). Below we describe a system in which the combined effects of sound wave and entropy perturbations cause non-ideal behavior behind incident shocks. 

Figure 2. Calculated temperature vs. time at three locations in reactive mixtures that are perturbed by a sound wave.

Thus we have seen another case where the interaction of sound waves and entropy perturbations with chemical reactions has altered the timescales and even the location of the physical processes. This is a much more complicated and less idealized example than the sound wave study described in the previous section. The sound wave calculations, however, did isolate the interactions and looked at what was required to quantify the effect. Here we have not only seen the direct effect of variations in the induction times due to the presence of perturbations, but also the indirect effect of an alteration of the background physical conditions,... 

Thus we see that the factors that determine the detonation cell size are complicated interactions of fluid dynamics and chemical kinetics. The fluid dynamics here involves a number of interacting shock waves, pressure waves, sound waves, and perturbations due to energy release. The chemical reactions are occurring in an environment which is always subjected to fluctuations and pressure perturbations. Thus we have gone up in level of complexity in the flow properties. However, this problem encompasses all of the issues we discussed in the previous section. 

In the examples given above we have tried to describe some of the phenomena which arise as a result of chemical kinetic-fluid dynamic coupling. First, we described studies of the isolated effects of chemical-acoustic coupling, emphasizing the effects on the chemical kinetics. The major conclusion is that sound waves and entropy perturbations can alter chemical timescales, and that this effect can be quantified. We then described a system in which sound waves and entropy perturbations behind a shock wave caused early ignition at unpredictable locations and at reduced ignition times. A series of reaction centers formed and one of these close to the shock front eventually ignited. 

Sound is, by nature, a concept that is defined in continuous media. Typical treatments begin with the perturbation of a continuous fluid with some density and pressure [ 1,2], These lead to a wave equation for pressure variations in which the speed of propagation (speed of sound) can be related to fluid density and other parameters. From these results and thermodynamic arguments other important quantities, such as the energy density of a sound wave, can be determined. 

Typical characteristic scales for sound waves are T = 0.1 s and Z = 10 m. If a particle displacement scales as L, local accelerations scale as Zr 1000 m/s. Clearly, gravity is negligible for such strong accelerations and the Coriolis force can likewise be neglected on snch short timescales (/t 1). For a basic state, a nniform flow u in any direction and locally uniform density p, pressnre p, and temperature T are assumed. Under these conditions the linearized perturbation equations become... 

Acousto-optic modulator (AOM) A light modulator that consists of an acoustic medium, such as glass, to which a piezoelectric transducer is bonded. When an electrical signal is applied to the transducer, a sound wave propagates through the acoustic medium, causing perturbations in the index of refraction proportional to the electrical excitation, which in turn modulates a laser beam traversing the acoustic medium. 

Most porous materials absorb incident and airborne sound waves well. A small change in pressure perturbations can generate loud noise, which dissipates into heat as it travels through the tortuous path in these materials. On the other hand. 

We will establish the basic principles that govern the behavior of aU acousto-optic devices whether of bulk or waveguide (SAW) construction. An acousto-optic modulator is composed of an acoustic medium (such as water, glass, lithium niobate, rutile, etc.) and a transducer. The transducer converts electrical signals into sound waves propagating in the acoustic medium with an acoustic frequency spectrum that is limited by the bandwidth of the transducer that matches the electrical excitation. The sound wave causes a perturbation in the index of refraction of the material, setting up a refractive index grating of the form... 

The ultrasonic relaxation observed in CTAB/Pentanol/water-systems has previously been interpreted in terms of a perturbation of the equilibrium between alcohol molecules in the water phase and micelle phase respectively. Generally speaking the sound wave detects the alcohol molecules jumping in and out of the CTAB-micel-les. The interesting question is whether or not this process is affected by additives that make the system viscoelastic. It is important to notice that the ultrasonic absorption data at different frequencies for CTAB in pure water are pretty close to those obtained from CTAB made viscoelastic by adding Sodium Salicylate. Thus the ultrasonic wave hardly detects the change that causes the viscoelasticity. 

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