Displacement Waveforms Research Articles

A model for the propagation of short pulses of ultrasound in a solid

A simple model has been introduced which allows calculations to be made of the particle displacement and velocity waveforms which arise when short pulses of ultrasound are radiated into solid media from transducers typical of those used in nondestructive evaluation. The model makes use of the plane- and edge-wave nature of the pulses radiated and, for simplicity, incorporates an empirical extension of the impulse response method for propagation in a fluid medium. Calculations made using the model are compared with other theoretical results obtained previously by harmonic synthesis, as well as with experimental measurements of pulse waveforms made using a miniature piezoelectric receiving probe. The results obtained are of particular relevance to the characterization of small targets using ultrasound.

Rupture characteristics and tomographic source imaging of ML ∼ 3 earthquakes near Anza, southern California

The rupture characteristics of several ML ∼ 3 earthquakes near Anza, California, are determined from time domain analysis of path‐corrected, displacement pulses. These pulses are corrected for propagation effects (apparent attenuation, scattering, site resonance) using the waveforms of adjacent small events (ML ≤ 2.1) as empirical Green's functions. The earthquakes studied occurred during two swarm sequences near Cahuilla, in an area characterized by swarm activity at shallow depth (< 5 km). The data consist of seismograms recorded by the Anza seismic network, a high dynamic range local array with digital telemetry. The waveforms of adjacent small events (ML ≤ 2.1) are deconvolved from those of the ML ∼ 3 events to yield the source time functions of the larger earthquakes. The ML ∼ 3 earthquakes display a variety of rupture modes, similar to those reported for large shocks but over much shorter time scales and distances. Events with single and multiple ruptures are observed. Relative locations of the subevents are obtained from azimuthal differences in the time separation of pulses in the deconvolved displacement waveforms. Some events (as small as ML 2.4) display unilateral rupture propagation, as manifested by azimuthal variations in the pulse widths of the deconvolved displacement pulses. A tomographic back projection technique is applied to the deconvolved displacement pulses to image the slip velocity on the fault as a function of time after nucleation and distance along the fault, using a one‐dimensional fault model. The tomographic inversion for one of the events (ML 2.4) reveals a unilateral rupture with rupture velocity about equal to the shear wave velocity (Vs) and fault length of 300 m. A similar inversion for an ML 3.7 event resolves two subevents separated by about 200 m and yields an average rupture velocity of 0.8Vs. These rupture velocities are similar to those reported for large earthquakes, indicating that rupture velocity is independent of seismic moment down to 3.1 × 1019 dyn cm, the smallest event studied. There are clear variations in pulse width for neighboring events with similar seismic moments, implying significant differences in static stress drops. Factor of five variations in stress drop (11–61 bars) are calculated for events within 200 m of each other. Thus the stress and/or strength along this fault zone varies considerably over distances of hundreds of meters.

The dynamically shifted oscillator

The dynamically shifted oscillator is described by an equation of motion which includes both a Hookian restoring force and a nonlinear force which depends only upon the sign of the displacement. Exact expressions are obtained for the amplitude-dependent frequency and for the displacement waveform. The theory is applied to the study of the vibrations of the two-point librator and of the spring doorstop.

Radiated seismic energy and strain energy release in laboratory dynamic tensile fracture

Tensile dynamic fractures were propagated under two experimental congifurations for the purpose of assessing the relative amount of strain energy release that is consumed as fracture energy and radiated as seismic waves. The configurations used were (1) application of localized thermal stresses to 2.29 mm-thick plates of soda-lime glass and (2) double cantilever beam (DCB) experiments in 12.7 mm-thick glass plates, in which a fracture is propagated from a notch at one end of the specimen by application of a transverse load. Fracture propagation velocities of 0.35–2 mm/μs were obtained for fractures in the first configuration. A capacitance transducer with a point-like probe was used for measuring the seismic displacement waveforms from propagating fracture sources. This transducer is capable of measuring absolute surface displacements with a resolution of 0.01 nm. It has a flat frequency response in the range 10 kHz to 6 MHz. Measured seismic efficiencies, or the ratio of radiated seismic energy to strain energy released, are in the range 10−5 to 10−3.

The dynamically shifted oscillator

The dynamically shifted oscillator is a nonlinear system described by a differential equation of motion which includes a Hooke's law restoring force plus a stiffness force which depends only upon the sign of the displacement. The natural frequency of the oscillator is a function of the displacement amplitude; for positive stiffness it increases as the amplitude decreases. Because of the special nature of the nonlinearity it is possible to find an exact expression for the displacement waveform as a function of time. In the limits of zero Hookian force or zero stiffness, the waveform becomes, respectively, a cosine wave or a parabola wave. Conservation of energy leads to an integral equation which can be solved to find an exact expression for the amplitude‐dependent frequency. If the model equation is extended to include damping then the model predicts the time dependence of the frequency. The latter is compared with the measured time‐dependent frequencies of two systems to which the dynamically shifted oscillator theory can be applied, the spring doorstop and the two‐point librator.

Ultrasonic measurement of small displacements and deformations of tissue.

A signal processing technique is described for analyzing radio frequency M-mode signals to obtain significantly better depth resolution than previously available. Displacement and deformation waveforms may be measured for displacements less than 0.1 mm and deformations of less than 1 percent. Waveforms of liver dynamics caused by aortic pulsation and vessel diameter variations are presented, and possible tissue characterization applications discussed.

Ultrasonic measurement of small displacements and deformations of tissue

A signal processing technique is described for analyzing radio frequency M-mode signals to obtain significantly better depth resolution than previously available. Displacement and deformation waveforms may be measured for displacements less than 0.1 mm and deformations of less than 1 percent. Waveforms of liver dynamics caused by aortic pulsation and vessel diameter variations are presented, and possible tissue characterization applications discussed.

Laser‐generated ultrasonic waveforms in metals

A Q‐switched, Nd:YAG laser has been used to generate elastic waves within aluminum and mild steel disks. A single 24‐ns pulse from the laser irradiated one face of each disk, and acoustic displacements were detected at the other face following propagation through the substrate. A capacitance transducer was used as the detector, providing undistorted displacement waveforms over a wide bandwidth. The displacement waveform depended markedly on the irradiation conditions. At low laser power densities, such that generation occurred primarily by thermoelastic expansion, a common waveform was produced whose amplitude varied linearly with optical energy. At higher power densities in the presence of material ablation, the longitudinal amplitude increased substantially whereas that of the shear decreased. Wave propagation theory in a plate has been applied to the problem, and theoretical waveforms in good agreement with experiment have been produced; this leads to conclusions concerning the nature of the acoustic source formed by laser irradiation.

Theoretical body wave interactions with upper mantle structure

Long‐period seismograms are generated by a spectral method for body waves interacting with the upper mantle discontinuities that define the low‐velocity and transition zones of the spherically symmetric but radially inhomogeneous earth models, PEM‐C and T7. A complex velocity profile incorporates the SL8 Q model. Models PEM‐C and T7 predict distinct differences in displacement waveforms for P waves that interact with the 400‐ and 670‐km discontinuities. The waveforms demonstrate that later arrivals such as interference head waves can be important constraints upon earth models. Theoretical seismograms were computed by a spectral method that modeled the velocity profile with arbitrarily thick inhomogeneous layers (full wave theory). The seismograms agreed well with those generated by methods that approximate the velocity profile with thin homogeneous layers (the Cagniard and reflectivity methods) provided the displacements were convolved with source‐time functions and a long‐period instrument response. Minor differences among the results of the methods do not affect the interpretation of observed waveform data but do help define the limitations of the methods. For example, the thin‐layer methods suffer from numerical limitations on layer thickness and the number of layers needed to approximate the earth model. The full wave theory may occasionally also require the use of thin layers when the zero‐order asymptotic theory it uses fails to describe wave propagation properly in layers having rapid velocity variation with depth.

A fiber optic lever for noncontact shock and vibration measurements

Frequency response of accelerometers used in shock and vibration measurements is usually limited by linearity considerations to one‐third of the resonant frequency of the accelerometer. In addition, for highly sensitive measurements, the added mass of the accelerometer may alter the characteristics of the surface motion. And finally, to obtain displacement waveform measurements, the signal obtained through an accelerometer must be integrated twice with great loss of frequency response and dynamic range. The use of noncontacting fiber optic levers offers an attractive alternative in some shock and vibration problems, particularly where knowledge of displacement waveform characteristics is desirable. In this paper, displacement, velocity, and acceleration detection limits as well as frequency response and dynamic range of a fiber optic lever developed for shock and vibration measurements are presented and compared with corresponding published accelerometer specifications. Acceleration, velocity, and displacement waveform measurements from an impactively excited plate by a miniature shock accelerometer and a fiber optic lever are also compared.

Thixotropy in human skin

A rotary displacement servo has been used to apply in-plane torques in vivo to an annulus of human skin between a driven disc and fixed guard ring. Using the servo, skin on the forearms of 44 normal healthy individuals was studied with a series of rotational displacements. A trapezoidal displacement vs time waveform was employed to provide separate periods of constant strain rate and stress relaxation. Employing sinusoidal displacements from 0.004 to 20 Hz, the dynamic mechanical characteristics of skin on the forearms of 6 other normal individuals were studied with the same device. Thixotropic behaviour in human skin was indicated by a constant phase of 10° with torque leading displacement in the frequency range of 0.004–1 Hz. Residual torques of 0.64 mN m ( s.d. 0.32) at the end of the stress relaxation portions of the trapezoidal displacements indicated an elastic-gel behaviour in the skin. These two experimental findings add supporting evidence to the hypothesis that skin has significant thixotropic viscous properties.

Theoretical Analysis of Acoustic Signal Generation in Materials Irradiated with Microwave Energy

Stress gradients generated by thermal expansion, electrostriction, and radiation pressure are sources of elastic waves in microwave irradiated materials. A theoretical analysis taking into account induced volume and surface forces due to these interaction mechanisms is presented. Complete solutions of the dynamical equations for the one-dimensional special case are given for different boundary conditions. The closed-form solutions were found to consist of both a stationary part, whose effect is important only in the immediate region of the incident electromagnetic wave, and a traveling part which propagates through the elastic material. Expressions for the Fourier transforms of these solutions are also given. To quantify these results, pressure and displacement waveforms in microwave irradiated physiological saline were computed. Thermal expansion was considerably more effective than either electrostriction or radiation pressure in converting electromagnetic energy to acoustic energy.

Strong ground motion of the San Fernando, California, earthquake: Ground displacements

abstract Two hundred and thirty-four components of ground displacement are the basis of an investigation of long-period strong ground motion in southern California arising from the San Fernando, California, earthquake. The displacement data are obtained from the double integration of strong-motion accelerograms via the base-line adjustment and filtering operations routinely performed in the series “Strong Motion Earthquake Accelerograms”. These procedures can recover long-period data from strong-motion accelerograms with considerable accuracy. Many-station comparisons of displacement data for which the station spacing is small compared to the wavelengths of interest reveal that uncertainties in displacement are less than 1 cm in the period range 5 to 8 sec, 1 to 2 cm at periods near 10 sec, and 2 to 4 cm in the period range 10 to 15 sec, for a data sensitivity of approximately 7.6 cm/g. For limited variations in epicentral distance (R) and source-station azimuth (ϕ), ground displacements show a strong coherence; for wider variations in R and ϕ, many of the observed variations in the displacement wave forms are easily attributable to well-understood seismological phenomena. Seismic moment, source dimension, radiation pattern, rupture propagation, the development of surface waves and their subsequent dispersion, and azimuthal variations in the gross geological structure all appear to have first-order significance in fashioning the gross amplitude and frequency content of the displacement wave forms and in explaining observed variations with R and ϕ. The essential simplicity of these displacement wave forms offers considerable optimism that long-period strong ground motion can be realistically synthesized with advance knowledge of the earthquake source parameters and gross geological structure.

Pitch of noise plus noise delayed

Rosenberg has demonstrated [J. Acoust. Soc. Am. 47, 118 (1970)] that if white noise is added to itself delayed by time T, the composite stimulus gives rise to a perceived pitch equal to the reciprocal of T. Clearly, this experiment raises problems for place theorists, since the spectrum of the composite stimulus is also “white.” Rosenberg's results raise no less of a problem for those who propose periodicity in the stimulus waveform as the determinant of pitch. This stimulus has neither periodicity nor concentrations of spectral energy of any kind. We have applied the theory of pitch perception presented by Ravindran and Thomas at this meeting to this stimulus as follows. The stimulus waveform was generated with the aid of a computer for delay times of 1 and 2 msec. This signal was made available through D/A conversion for monitoring and recording and for input to an analog device whose output represents the waveform of basilar membrane displacement for various membrane locations and for two possible middle-ear transfer functions. The resulting signal was then further analyzed using the steps called for in the Ravindran and Thomas model. The result for any given delay is a “distribution of perceived pitches” which has a distinct maximum at the appropriate pitch (1/T). Changes in membrane location and in middle-ear characteristics have minor effects on the distribution and do not alter the frequency of its maximum. These results lend support to the proposed model. [Work supported by NIH.]

Model for pitch perception

The three important stages in pitch perception are (1) acoustic to mechanical transformation, (2) mechanical to neural changes, and (3) neural firings to perception. The developed model incorporates all three of these stages for explaining pitch perception. Flanagan's transfer function (based on Békésy's experiments) relating the basilar membrane displacement to stapes movement is used for obtaining the waveform of displacement at the attended region of the basilar membrane for a given stimulus. The axis crossing intervals of this band-pass-filtered stimulus (not those of the stimulus external to the ear) are encoded in terms of intense and sparse firings, since only unidirectional movement causes appreciable neural activity. According to the model, these individual axis crossing intervals are stored and compared successively. Whenever adjacent intervals differ by 10 or more μsec, it is presumed that a second-order neuron is activated at a higher level to mark the time of occurrence. The model proposes that various combinations of time intervals between these higher-order firings give the reciprocal of different pitches associated with the stimulus. This model predicts the pitch of the so-called “difference” tones, residue, and cubic difference tones without calling for any nonlinear mechanism. [Work supported by NIH.]

青森県東方沖地震の発震機構の再調査

A particular type of phase designated as the P2-phase is found in the P wave train around 9 seconds after the first arrival of the earthquake off east coast of Aomori prefecture of March 29, 1965. In the previous paper, the P2-phase has been interpreted as the stopping phase from the analysis of the motion direction of this phase. In the present paper, the P2-phase is re-examined in the wave form of the theoretical displacement of P wave from a fault near the earth's surface, because the unreasonable interpretation for the fault length and the rupture velocity has been given. The examination shows that the P2-phase concerned does not consist of the stopping phase, but of reflected sP phase, and that the wave form of direct P wave from a fault with a fault length greater than its depth may be disturbed by the reflected waves.The seismic moment and the apparent stress for the present earthquake are estimated to be 3.20×1025dyne-cm and 5 bars, respectively.

Detection of Interaural Time Differences in Short-Duration Low-Frequency Tones

The detectability of interaural time differences (Δt) was investigated as a function of stimulus duration, level, and frequency. Temporally dichotic tone bursts of 250, 500, and 1000 Hz were presented in a two-interval forced choice, in such a way that the signal to one ear led by Δt μsec during either the first or second interval. The tone bursts were 10, 25, 50, 100, and 205 msec in duration and were gated with a rise-decay time of 2.5 msec. The levels used were 50 and 70 dB SPL. Threshold values of Δt were maximal for the shortest durations, then fell rapidly to asymptote in the region from 50 to 100 msec. In addition, thresholds decreased at higher frequencies. Finally, all conditions showed lower thresholds for 70 dB than for 50 dB. The data are fit by a model that treats increased duration as an increase in the number of samples of the interaural difference and describes the improvements found with higher frequencies and levels as a result of the steeper slopes of the displacement waveforms.

Elastodynamic near field of a finite propagating transverse shear fault

A quantitative model of the earthquake focal mechanism incorporating finite source dimensions and finite nonzero rupture velocity has been developed. There are three linearly independent cases: longitudinal shear fault, tensile fault, and transverse shear fault. This paper completes the theory by developing the case of transverse shear faulting, which was not considered earlier. Near-field displacement, particle velocity, and acceleration wave forms were computed by numerically integrating the Green function integrals for an infinite medium. The results are displayed as graphical wave forms at an extensive selection of close-in field positions and as contour plots of amplitude on particular planes.

Pitch of Periodic Pulses without Fundamental Component

The pitch of a periodic pulse train is matched to that of another train whose fundamental component of frequency is rejected. Three modes of pitch perception are found. The first obtains for pulse rates less than 100 pps. Here, in a manner similar to that found previously for unfiltered stimuli, the pitches are matched to equate pulse rate, regardless of the polarity patterns of the trains. The second mode occurs for fundamental frequencies in the approximate range 200–500 cps. Here there is a decided tendency to equate fundamental frequencies. The third mode is manifested for fundamental frequencies of the order of 1000 cps and higher. In this range subjects tend to equate the fundamental of the matching tone to the lowest spectral component present in the matched stimulus. Between the modes, transition regions exist in which the pitch assignment generally is difficult and uncertain. The stimuli are studied with an electrical analog of the basilar membrane. Waveforms of membrane displacement and first spatial derivative of displacement are obtained from the analog. The three (psychophysical) pitch modes are found to be manifested in these mechanical functions of the basilar membrane.

On the Pitch of Periodic Pulses

Subjects adjusted the frequency of one periodic pulse train to match the pitch of another train fixed in frequency. Two modes of pitch perception are found. In the first mode, for pulse rates less than 100 pps, the pulse trains are ascribed a pitch equal to the number of pulses per second, regardless of the polarity pattern of the pulses. In the second mode, for fundamental frequencies in excess of 200 cps, the sounds are assigned a pitch equal to the fundamental frequency. Between these frequency regions a mode transition occurs in which the pitch judgments generally fall between the pulse-rate and fundamental-frequency values. Amplitude and phase spectra are computed for the stimuli. The stimuli are studied on an electrical analog of the basilar membrane. Waveforms of membrane displacement and first spatial derivative of displacement are obtained from the analog. An effort is made to correlate the psychophysical results with the displacement and derivative patterns observed on the analog membrane. The two pitch modes are found to be manifested in the mechanical operation of the cochlea.