Sound speed, bulk

Shock velocity The velocity of the shock wave as it passes through the material. In the limit of an infinitesimally small shock wave it is equal to the bulk sound speed of the material. 

C° = bulk sound speed in the compressed (shocked state) driver. 

The Lagrangian sound speed is obtained in the following heuristic way. We consider small departures from the shock-compressed state, where the bulk and shear moduli are K and G. The Eulerian sound speed c is then given by... 

Application of Eq. 19 to the /7-HMX isotherm from simulations leads to the Us-Up curve shown in Fig. 11, where negative curvature in the simulation results is clearly evident (filled circles). While such behavior would be anomalous for metals, it is actually expected for pressures below about one GPa in the case of polyatomic molecular crystals, due to complicated molecular packings and intramolecular flexibility, and has in fact been reported for the high explosives pentaerythritol tetranitrate (PETN) where careful studies were performed for low levels of compression [77], By contrast, the experimental results for /3-HMX in the Us-Up plane do not exhibit significant curvature due to lack of data at pressures below about one Gpa [78], Thus, estimates of isothermal sound speeds, and hence isothermal bulk moduli, based on... 

To obtain an adequate impedance match, the desired bulk impedance of the material used can be calculated from equation 9, knowing the speed of sound in the material and the wedge angle. For materials with sound speeds less than that of water the desired bulk impedance is seen to be somewhat less than water. In practice, however, the range of acceptable impedance values is so wide that this... 

The equation for this linear relationship is again U = Cq + su. The constant Co is called the bulk sound speed, but this is misleading. Cq has no real physical meaning other than the fact that it is they-axis intercept on a straight line drawn through the data points. The units in this expression are mm//u,s (or km/s) for the terms U, Cq, and u, and the term 5 is dimensionless. Table 17.1 lists U-u... 

The quantities a, c, f, F, r, and p are the thermal diffusivity, sound speed, heat capacity ratio, bulk viscosity coefficient, shear viscosity coefficient, and density of the sample, respectively and Eo, a, P and Cp are the energy fluence of the laser beam, the optical absorption coefficient, the volume expansion coefficient, and the isobaric heat capacity, respectively, of the fluid. Tlie first and second terms in Eq. 2 describe the time dependences of the thermal and acoustic modes of wave motion, respectively. Since the decays of the acoustic and thermal mode densities back to their ambient values take place on such different time scales (microsecond time scale for acoustic mode and millisecond time scale for thermal mode), they were recorded on the oscilloscope using different time bases. 

In Tables S10.9 and SIO.IO are given these derivatives for metals and ionic crystals MX, which show that elastic moduli change directly proportional to the densities of solids the average coefficients of proportionality for metals and ionic crystals are equal to 5.0 and 5.4, respectively, for compresing, and 4.2 for heating. Knowing these derivatives we can estimate sound speeds, Debye s temperatures, and other properties of substances under high pressures. In Table S10.13 are presented the experimental bulk moduli of elements at different temperatures. 

By making reasonable assumptions about the form of the intermolecular potential, it was possible to calculate bulk modulus as an analytic function of volume (18). The calculation agrees fairly well with experimental data, and with the assumption that volume is the primary factor determining bulk modulus and, by extension, sound speed. 

In applying Rao s rule to solid polymers, it should be kept in mind that the longitudinal sound speed (eq. 1) includes not only a bulk modulus term, but also a shear modulus term that must be taken into account (5). This has been done for both linear and cross-linked polymers (44-47). Van Krevelen has named the additive variable for bulk modulus the Rao function, Ur, and for shear modulus the... 

Group contributions to U-r and [7h for some common molecular groups, taken from Van Krevelan (42), are listed in Table 1. The bulk and shear moduli for various polymers calculated using the values from Table 1 are in good agreement with experimental values (42). Sound speeds can be predicted from the calculated elastic moduli and densities. 

Results have also been obtained for the density and bulk modulus of cross-linked epoxies (44). Agreement is about the same as for linear polymers. Finally, shear modulus has been related to an additive property (47). These results, along with the additive results for density and bulk modulus, show how both longitudinal and shear sound speeds are related to molecular components. 

Thus, the speed of sound is related to the bulk modulus by... 

For our purpose, it is convenient to classify the measurements according to the format of the data produced. Sensors provide scalar valued quantities of the bulk fluid i. e. density p(t), refractive index n(t), viscosity dielectric constant e(t) and speed of sound Vj(t). Spectrometers provide vector valued quantities of the bulk fluid. Good examples include absorption spectra A t) associated with (1) far-, mid- and near-infrared FIR, MIR, NIR, (2) ultraviolet and visible UV-VIS, (3) nuclear magnetic resonance NMR, (4) electron paramagnetic resonance EPR, (5) vibrational circular dichroism VCD and (6) electronic circular dichroism ECD. Vector valued quantities are also obtained from fluorescence I t) and the Raman effect /(t). Some spectrometers produce matrix valued quantities M(t) of the bulk fluid. Here 2D-NMR spectra, 2D-EPR and 2D-flourescence spectra are noteworthy. A schematic representation of a very general experimental configuration is shown in Figure 4.1 where r is the recycle time for the system. 

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