Richard K. Cook

Variation of Elastic Constants and Static Strains with Hydrostatic Pressure: A Method for Calculation from Ultrasonic Measurements

The elastic constants and static strains of a solid subjected to large hydrostatic pressures can be deduced from measurements of resonant frequencies (or transit times) for ultrasonic waves in specimens of suitable crystallographic orientations. The pressure changes the specimens size, shape, and density as well as the elastic constants, and all of the effects influence the resonant frequencies. An algorithm for separating out the effects due to variations in elastic constants from the effects due to static strains is presented and applied to cubic crystals and hexagonal crystals, these structures being of immediate interest to investigators concerned with the properties of metals. The results apply also to isotropic and transversely isotropic solids. The only measurements needed while the specimen is under hydrostatic pressure are resonant frequencies (or transit times). Also required are the size and density at zero pressure, or the elastic constants at zero pressure.

Measurement of Correlation Coefficients in Reverberant Sound Fields

Reverberation chambers used for acoustical measurements should have completely random sound fields. We denote by R the cross‐correlation coefficient for the sound pressures at two points a distance r apart. Ru2009=u2009〈p1p2〉Av/(〈p12〉Av〈p22〉Av)12, where p1 is the sound pressure at one point, p2 that at the other, and the angular brackets denote long time averages. In a random sound field, R = (sinkr)/kr, where k = 2π/(the wavelength of the sound). An instrument for measuring and recording R as a function of time is described. A feature of this instrument is the use of a recorders servomechanism to measure the ratio of two dc voltages. The results of correlation measurements in reverberant sound fields are given.

Absolute Pressure Calibration of Microphones

A tourmaline crystal disk was used both as a microphone (direct piezoelectric effect) and as a sound source (converse piezoelectric effect). Application of a principle of reciprocity to the acoustic measurements gave an absolute determination of the piezoelectric modulus d33u2009+u20092d31 of tourmaline under hydrostatic pressure. A condenser microphone was calibrated by the tourmaline disk. The same principle was applied to data obtained by using a condenser microphone as both source and microphone to secure an absolute calibration of another condenser microphone. It was proved experimentally that the tourmaline disk and the condenser microphones satisfied the principle of reciprocity. The absolute acoustic determination of the piezoelectric modulus gave d33u2009+u20092d31u2009=u20092.22u2009×u200910−17u2009coulomb/dyne. The “reciprocity” calibrations agreed with the results of electrostatic actuator, pistonphone, and “smoke particle” calibrations, but disagreed with thermophone calibrations of the condenser microphones.

Acoustic Impedance of a Right Circular Cylindrical Enclosure

At low frequencies, the acoustic impedance of a right circular cylindrical enclosure (containing air, or other gases) is affected by the cooling effects of the walls. Analytical expressions for the temperature distribution have been obtained, and computations of the effect on the impedance are given in the form of plotted correction factors. These corrections are used in making absolute pressure calibrations of condenser microphones at low frequencies.The solution presented for the average temperature distribution applies also to the heat‐conductivity problem of a uniform volume distribution of sinusoidal heat sources, which are considered to be in phase, inside a cylindrical enclosure having isothermal walls.

Absorption and Scattering by Sound Absorbent Cylinders

The absorption and scattering of a plane wave of sound by an infinitely long circular cylinder, whose axis is perpendicular to the direction of propagation of the wave, are calculated. The surface of the cylinder is assumed to have a known normal acoustic impedance. The calculations take account of diffraction effects. Absorption measurements were made on long cylinders placed in a reverberation room, where the incident wave directions are at all angles to the axes of the cylinders, and were compared with the calculated values. In order to make the comparison, The reverberation room statistics appropriate for cylinders (which are different from the statistics for flat patches of absorbent material) are developed and applied. The theory predicts, and measurements confirm, that absorbent cylinders can have coefficients of absorption greater than unity. Fairly good agreement between the calculated and measured coefficients is found. The reverberation room statistics appropriate for spherical absorbers are al...

Free‐Molecule Propagation of Sound Through Gases

Measurements have been made of the attenuation and phase shift of sound propagated through gases at very low pressures. The 11 Mc/sec double‐crystal interferometer described earlier was used. The range of pressures studied extends from about 1.5 mm Hg to less than 0.1 mm. The amplitude and phase shift of the received sound were measured at distances of less than the mean free path. A simple theory, based on the assumption of no collisions, was developed. This theory shows that the propagation approaches (ix)12u2009exp[−3(ix/2)23] asymptotically for large x, xu2009=u2009(γ/2)12ωh/c0 (c0 = the ordinary velocity of sound in the gas, γ = ratio of specific heats, ω = 2π times frequency, h = distance from source to receiving crystal). The theory agrees with the main features of the measured propagation. For example, with increasing h, the phase velocity increases roughly like h13 and is independent of pressure.

Absorption by Sound‐Absorbent Spheres

The absorption of a plane wave of sound by a sphere is computed. The calculations are based on the assumption that the complex ratio of sound pressure at a point on the spheres surface to the normal component of particle velocity is a constant independent of the direction of incidence (“normal impedance assumption”). Absorption measurements were made on hair felt‐covered spheres placed in a reverberation room, and were compared with the computed absorption by means of the reverberation room statistics appropriate for spheres. The theory and measurements both show that absorbent spheres can have absorption coefficients greater than unity. The discrepancies between theoretical and experimental coefficients seem to indicate that the normal impedance assumption is not valid for the hair felt used in the experiments.

Sound Wave Fields Within Cavities

Cylindrical cavities of circular cross section are extensively used in acoustics in the calibration of microphones and receivers. Such a cavity having plane ends is assumed to be driven by motion of one of the plane ends. Solutions of the wave equation in the form of Fourier‐Bessel expansions for the distribution of pressure and particle velocity (as a function of frequency) within the cavity are obtained. The theoretical results are compared with experimental measurements of amplitude and phase of sound pressure within cylindrical cavities.

Measurements on Sound Absorbers for Jet‐Engine Test Cells

The space absorbers were long cylindrical shells of perforated metal, containing glass fiber. The absorbers were to be used in an aero‐engine test cell, and these conditions were simulated in the measurements by mounting a few absorbers in a long concrete pipe, so that the cross‐sectional ratio (absorber/enclosure) was the same in each case. The direction of propagation of sound was parallel to the axis of the cylinders so that absorption occurred at grazing incidence. The concrete pipe was terminated by a rigid reflector, and the attenuation figures were deduced from the resulting interference pattern using impedance tube theory. The theory was extended to the case of octave bands of noise, when the interference pattern for the mean squared pressure is (p2)u2009=u2009[cosh2axu2009sinh(2ax/3)]/(2ax/3)u2009+u2009[cos(2kx)u2009sin(2kx/3)]/(2kx/3), where a is the sound attenuation constant, k is the wave number at the center frequency of the band, and x the distance along the pipe measured from the reflecting end. Experimental resu...

New Method of Recording the Sound Transmission Loss of Walls as a Continuous Function of Frequency

An apparatus has been developed which records as a continuous function of frequency the difference in decibels between the sound levels in two rooms separated by the partition under test. The quantity recorded differs from the transmission loss only by a small correction for the Quiet‐room absorption and partition area. Experimental results are presented.An electronic potentiometer recorder is used as a self‐balancing device. The voltages produced by microphones in the Loud and Quiet rooms are applied to the recorder, which drives a logarithmic attenuator and continuously reduces the Loud voltage to the same value as the Quiet voltage. The voltages are balanced at an attenuator setting which is determined by the sound transmission loss and this is plotted linearly in db on an 11 inch scale, which may be chosen as 0–60 db or less.