Joseph Zemanek

An Experimental and Theoretical Investigation of Elastic Wave Propagation in a Cylinder

A high‐speed computer was used to investigate the problem of wave propagation in an isotropic elastic cylinder. Dispersion curves corresponding to real, imaginary, and complex propagation constants for the symmetric and the first four antisymmetric modes of propagation are given. The radial distributions of axial and radial displacements and of shear and normal stresses are given for the symmetric mode. By using a finite number of modes of propagation, an approximate solution is found for the problem of the L(0,1) mode impinging on a traction‐free interface. The reflection coefficient is determined in this way and the accompanying generation of higher order modes at the interface is shown to cause a high‐amplitude end resonance. Experimental results obtained by using the resonance method in conjunction with a long rod are presented to substantiate the calculated reflection coefficient and the frequency of end resonance. Phase velocities, based on measurements of the wavelength of standing waves and resona...

Attenuation and Dispersion of Elastic Waves in a Cylindrical Bar

The resonance method has been used to study the attenuation and dispersion of the first longitudinal mode of propagation and the dispersion of the first flexural mode of propagation of elastic waves in a cylindrical, aluminum alloy (24ST) rod. Q was found to decrease monotonically from 2.5×105 to 1.2×105 as the frequency increased from 0.84 to 100 kc. Longitudinal and flexural phase velocities are compared to Pochhammer‐Chree theory dispersion curves. Agreement of experimental and theoretical curves is within 0.3%. Similar agreement is obtained when normal flexural modes computed by a modified Timoshenko theory are compared to the experimental resonant frequencies. Measurements of torsional mode frequencies indicate dispersion does not exceed approximately 0.01% in the frequency range of approximately 0.5 to 100 kc.

Determination of formation permeability from a long-spaced acoustic log

Disclosed is a method for determining the permeability of a formation which includes the steps of traversing a bore-hole with a tool having a means for transmitting low frequency acoustic energy. The transmitting means is pulsed and the acoustic energy is detected by a first wide band receiver having a frequency response of between at least 0.1k hertz and 30k hertz. The receiver is coupled to the transmitting means only by means of a cable which has a length exceeding about 5 feet and preferably a length of about 15 feet. The amplitude of tube waves detected by the receiver is determined at a plurality of locations in the bore-hole. The change of amplitude in tube waves at various of these locations provides a measure of permeability. The receiver employed preferably comprises a cylinder of piezoelectric material having end plates at the ends of the cylinder and a passageway through one of the end plates which transmits ambient but not dynamic pressure changes.

Experimental Investigation of Elastic Wave Propagation in an Aluminum Cylinder

The resonant‐bar technique was used to measure the dispersion of waves corresponding to symmetric and first and second antisymmetric modes of propagation. Wavelength information was obtained by measuring the spacing of the nodes along the length of the cylinder. The results of these measurements are compared to curves computed from the Pochhammer‐Chree theory. In general, deviation between experimental and theoretical dispersion curves is less than 0.2%. The reflection coefficient for the L(0,1) mode obtained in this manner is compared to an approximation based on the Pochhammer‐Chree theory. Standing‐wave patterns at resonant frequencies in the vicinity of the end resonance show the build‐up of high‐amplitude displacements near the end of the cylinder. The measurements extend into the range of two propagating modes. Standing‐wave patterns obtained in this range show a spatial‐beating phenomenon due to interference between the two modes of propagation.

Theoretical Study of Symmetric Wave Propagation in a Semi‐Infinite Elastic Isotropic Cylinder

The boundary conditions at the traction‐free interface of a semi‐infinite elastic cylinder can, with one exception, be satisfied exactly only if an infinite number of modes of propagation is considered. For the low‐frequency range (diameter<wavelength), there exists only a single mode of propagation with a real propagation constant [the L(0,1) mode]; however, there is also an infinite number of modes of propagation with complex propagation constants. All of these modes are required to satisfy the boundary conditions. An approximate solution is found for the problem of the L(0,1) mode impinging on a traction‐free interface by using only a finite number of modes of propagation. The reflection coefficient for the L(0,1) mode is given. The accompanying generation of higher‐order modes with complex propagation constants is shown to cause high‐amplitude end resonances at isolated frequencies. Results are given for approximations with 3, 5, 7, and 9 modes.