Wave Speed Measurements Research Articles

Applicability of the Biot theory. III. Wave speeds versus depth in marine sediments

Empirical and theoretical methods are used to estimate the elastic moduli of the skeletal frame in sedimentary materials. The Biot model is then used to estimate compressional wave speeds as a function of depth. These estimates are compared to laboratory data and to field measurements. Using the empirical method to estimate the frame moduli, the measured wave speeds are matched within 3% for Ottawa sand pack and within 10% for glass bead pack at pressures corresponding to depths of about 500 m or less. A new method for estimating the frame moduli of partially consolidated materials is used to quantify the effects of grain packing and cementation. Velocity gradients are estimated for shear and compressional waves in terrigenous sand and calcareous clay. Predicted gradients for compressional waves, when corrected for the effect of temperature, lie within the range of published values. Predicted gradients for shear waves fall within a range of shear-wave gradients determined from in situ measurements of saturated terrestrial and marine materials. Using a constant porosity of 0.40 to simulate conditions in pure sands, predicted compressional wave gradients agree with laboratory results.

Applicability of the Biot theory. I. Low-porosity materials

The Biot theory of wave propagation in porous, saturated materials contains 13 parameters. We discuss empirical and theoretical ways to predict values of these parameters for natural materials. The self-consistent theory of composites is used to predict the elastic moduli of the skeletal frame assuming that the inclusions (pore fluid, sediment grains) are needle-shaped at low concentrations. The Biot theory is then used to predict compressional and shear wave speeds in consolidated materials. In a man-made material (porous sintered glass), the predictions agree with experimental data to within 3% (experimental uncertainty is also 3%); these samples were distilled-water-saturated. The predictions agree with data to within 1% for Berea sandstone, and to within 5% for Bedford limestone; all of these samples were brine-saturated. The predictions agree with data to within 8% for distilled-water-saturated Bedford limestone. In the natural sedimentary rocks, the experimental uncertainty in measured wave speeds is estimated at 5%, and the comparisons apply at high effective pressures (>0.3 kbar), where the experimentally determined wave speeds approach ‘‘limiting velocities.’’ In distilled-water-saturated Massilon sandstone, predicted shear wave speeds are about 25% to 35% higher than measured speeds; this may indicate a reduction of the frame shear modulus due to interaction of the pore fluid with the minerals of the rock.

Simulation of Taylor-Couette flow. Part 1. Numerical methods and comparison with experiment

We present a numerical method that allows us to solve the Navier-Stokes equation with boundary conditions for the viscous flow between two concentrically rotating cylinders as an initial-value problem. We use a pseudospectral code in which all of the time-splitting errors are removed by using a set of Green functions (capacitance matrix) that allows us to satisfy the inviscid boundary conditions exactly. For this geometry we find that a small time-splitting error can produce large errors in the computed velocity field. We test the code by comparing our numerically determined growth rates and wave speeds with linear theory and by comparing our computed torques with experimentally measured values and with the values that appear in other published numerical simulations. We find good agreement in all of our tests of the numerical calculation of wavy vortex flows. A test that is more sensitive than the comparison of torques is the comparison of the numerically computed wave speed with the experimentally observed wave speed. The agreements between the simulated and measured wave speeds are within the experimental uncertainties; the best-measured speeds have fractional uncertainties of less than 0.2%.

Interpretation of sample wave speed measured in an impedance tube

The basic measurement in an impedance tube is a complex reflection coefficient, from which the propagation speed and attenuation are computed. This speed is often identified with the dilatational wave speed in the sample material. This is only approximately correct; it depends on the ratio of sample radius to inner radius of the tube. Thus a correction to the measured wave speed is in order. Moreover, by varying the ratio of the radii one may derive two elastic moduli for the sample material from the measurements, as is shown by an experimental example for a Lucite sample.

Ultrasonic wave velocity in double-braided nylon rope

Ultrasonic wave speeds are measured in new Samson double-braided 2-in-1 nylon ropes of diameters ranging from 0.535 cm ( 1 4 in) to 1.588 cm ( 5 8 in). Two resonant frequency transducers with resonant frequencies at 30 k Hz are mounted on the loaded rope and spaced at various distances and relative circumferential locations. An impulse voltage is introduced into the input transducer, and the wave speed is evaluated from the response of the output transducer. The output signal appears as a complicated wave packet, with the maximum amplitude peak arriving much later than the first detectable peak. The wave speed corresponding to the arrival of the first peak increases with increasing rope tension, ranging from 1700 m s −1 to 4500 m s −1. Wave speed at a given rope tension increases with cyclic loading, whereas wave speed at a given rope strain decreases with cyclic loading. This loading history dependence becomes less pronounced both at higher rope tensions and also after three loading cycles on new rope. At any given strain, the wave speed corresponding to the arrival of the first peak is nearly independent of rope diameter. The wave speed corresponding to the arrival of the first peak is also a function of the path angle of propagation through the rope interior. The wave speed measured along a generator on the rope surface is 2.33 times as large as the wave speed measured across the rope diameter. Values of longitudinal wave speed are predicted by measuring the modulus in the rope, and assuming the rope to act as a uniform elastic rod. Predicted wave speeds have an average percentage deviation from wave speeds corresponding to the arrival of the first peak of 13.2%, with the percentage deviation decreasing with increasing rope tensions to 10% or less. The wave speed corresponding to the arrival of the first peak is much faster than the apparent wave speed corresponding to the arrival of the maximum amplitude peak, with an average ratio of the two speeds of 4.1. The wave speeds predicted by assuming the rope to act as a uniform taut string in transverse vibration have an average percentage deviation from the apparent wave speeds corresponding to the arrival of the maximum amplitude peak of 47.9%, with the percentage deviation decreasing at the higher rope tensions to about 30%. Thus, ultrasonic wave speeds in double-braided nylon ropes are significantly affected by tension, strain and loading history. These effects should be taken into account when conducting nondestructive evaluation using ultrasonics and acoustic emission. These effects also suggest the possibility of monitoring the tension and/or strain in a rope via wave speed measurements.

Measurements of radar wave speeds in polar glaciers using a down-hole radar target technique

A new technique for measuring the speed of radar waves in polar ice sheets was developed to investigate a previously reported disagreement between the permittivities of laboratory and glacier ice. The technique involves lowering a cylindrical radar target to several carefully measured depths in a borehole and measuring the travel time of a radar wave transmitted from a surface radar unit to the target in the borehole. The experiment was performed at Dome C, East Antarctica, and Dye-3, Greenland, and useable data were collected for target depths between 200 and 800 m. After computing the range to the target along a straight ray path and after correcting the travel time for delays in the radar receiver, the velocities determined from these experiments were found to be in good agreement with the velocities predicted by Robin's empirical formula. The apparent discrepancy between the permittivity of glacier ice, as measured using the radar wide-angle reflection method, and laboratory ice now seems to be due in large part to signal delay in the radar receiver that was ignored in earlier experiments.

Energy saturation and phase speeds measured on a natural beach

Field measurements of wave height and speed from 7‐m depth shoreward are described. The experiment plan consisted of a shore‐normal transect of closely spaced (compared to a dominant wave length) velocity, pressure, and elevation sensors on an almost plane profile having an inshore slope of 1:50. As the waves shoal and begin to break, the dominant dissipative mechanism is due to turbulence generated at the crest, and wave heights become increasingly depth controlled as they progress across the surf zone. Wave heights in the inner surf zone are strongly depth dependent; the envelope of the wave heights is described by Hrms = 0.42 h. The depth dependence of the breaking wave height is shown to be related to the kinematic instability criterion. Celerity spectra were measured by using phase spectra calculated between pairs of adjacent sensors. Inshore of 4‐m depth, the celerity was found distant over the energetic region of the spectrum. A ‘mean’ celerity was compared with linear theory and was within +20% and −10%, showing good agreement for such a nonlinear, dissipative region.

Experiments on the propagation of composite internal wave trains

Experiments have been performed in a two-layer stably stratified flow where the mean velocity surpasses the phase speed of all internal waves of mode one. Results show the existence of two wave trains propagating in the direction of the mean flow (excited by a two-dimensional wave maker), which are identified by the modulation of the amplitude of the linearly superposed wave trains. Measurements of the composite wave speed and dispersion relation are presented. Wave speeds are shown to be very near the mean. The qualitative behavior of composite wave trains is shown to appear similar to that of unstable Kelvin–Helmholtz waves where spatial growth is dominant, leading to possible misinterpretation.

Elastic vibrations and waves in a superconducting torus

Experiments on a small 16 coil superconducting torus, modeled after a tokamak fusion reactor design, demonstrate a number of dynamic phenomena resulting from mangetoelastic coupling in the torus. Almost all of the vibratory natural frequencies are observed to decrease with current, resulting in eventual buckling of one of the modes. Further,lateral bending wave propagation around the periodic structure was observed. The measured wave speed increases with current. The wave speeds are related theoretically to the group velocity derived from frequency dispersion relations. Induced vibrations in the toroidal field coils due to pulsed poloidal coils are also reported. Finally buckling induced quench in the superconducting coils show large dynamic overshoot and vibrations following the quench of the torus.

On the determination of phase and group velocities of dispersive waves in solids

A new technique is developed to determine the dispersion relation and the propagational speeds of waves in dispersive solids. The phase spectrum of a broadband pulse is linearly related to the dispersion relation of the dispersive medium. The method is simpler than the continuous-wave phase comparison technique. Application is made to measure the phase and group velocities of waves in fiber-reinforced composite materials and in thin wires. This technique is expected to be applicable to measurements of acoustic or electromagnetic wave speeds in other dispersive media.

Effect of electrical state on mechanical and electrical response of a ferroelectric ceramic PZT 95/5 to impact loading

Ferroelectric ceramic specimens 2–10 mm thick were impact loaded in uniaxial strain to a stress of 1.6 GPa. Material velocity-time wave profiles were measured using a laser interferometer system. Specimen states studied were unpoled, poled and shorted, and poled with a high resistive load (E?2 kV/mm). The results show clearly the effect of the specimen electrical state on the mechanical response. Evolution of the mechanical wave was studied by measuring profiles for several different thicknesses with the same electrical state. For poled specimens, simultaneous measurements were made of the depolarization current during the impact loading. Measured wave speed of the first detectable disturbance was sonic in some but not all cases. Measurements of Hugoniot states for unpoled material were also obtained.

A method-of-characteristics calculation of coronary blood flow

The one-dimensional unsteady flow equations for flow in an elastic tube have been solved by employing the method of characteristics and used to predict the development of the flow and pressure wave forms in the left coronary artery system of the horse. Input data to this model include in vivo measurements of wave speed and aortic-root pressure, both of which were carried out in horse experiments. In addition, estimates of vessel taper and fluid losses due to branching were established from plastic casts of the horse coronary arteries and from in vivo flowmeter data. The calculated results have confirmed earlier experiments in revealing a relatively large systolic component of flow in the major epicardial vessels. However, the calculations indicate that, once within the myocardium, the systolic flow component quickly diminishes and the diastolic flow component becomes increasingly important. The pressure pulse does not peak as observed in the aorta, but rather rounds out with systolic pressures decreasing slowly and diastolic pressures decreasing more rapidly with distance from the coronary ostium. The presence of relatively large amplitude, low frequency waves (of the order of 5–10 Hz), which were observed mainly during diastole in horse experiments, has also been confirmed by the computer calculations. Similar calculations carried out for conditions simulating the dog and human coronary systems indicate that such oscillations become higher in frequency and lower in amplitude with decreasing animal size.

Shock attenuation by a single transverse slit

Attenuation of a plane shock due to the interaction with a single slit is examined via spark schlieren photography for shock Mach numbers between 1·17 and 2·44 and slit widths between 0·173 and 3·175 cm. Wave speed measurements by piezoelectric transducers indicate that the attenuation is weak and adequately predicted by Whitham's ray-shock theory. The slit width is observed to produce only a secondary effect on the shock attenuation rate. Stability of the attenuating shock is demonstrated from a wave diagram constructed by the ray-shock technique.

Measurement of release wave speed in shock-compressed polycrystalline alumina and aluminum

Values of release wave speed in shock-compressed polycrystalline alumina and aluminum (2024) have been measured to 400 and 260 kbar, respectively. These values, as well as previous data for aluminum and iron, have been compared with fourth-order finite strain extrapolations of the longitudinal sound speed V_p. For alumina in the pressure range 200–300 kbar and for α iron and ∈ iron the observed release wave speeds fall below any reasonable extrapolation of V_p. In the range 360–400 kbar the data for alumina indicate that M(∂^2M/∂ P^2) ≃ −50 (where M and P are longitudinal modulus and pressure, respectively); this value is in approximate agreement with Graham's ∼200-kbar data on elastic shock compression of single-crystal aluminum oxide. For aluminum the data indicate a value for M(∂^2M/∂ P^2) of 2.7±5.7.

Direct measurement of transverse wave speed on a stretched string

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Measurement of subsonic laser absorption wave propagation characteristics at 10.6 μm

An experimental study of subsonic laser absorption waves arising from the interaction of 10.6-μm radiation with metal and insulator target materials is described. A shock-tube-driven gas-dynamic laser operating up to 400 kW for 4 msec was used as the radiation source. Instrumentation included streak photography to determine wave speed, holographic interferometry to determine the gas density field, time-integrated visible spectroscopy to determine the principal radiating species, and CO2 laser attenuation measurements to determine the plasma absorption coefficient. The results reported here emphasize the characteristics of waves in free and clean air, far from the influence of the target. The measured wave speeds in air at 1 atm pressure ranged from Mach 0.05 to 0.2 for CO2 laser intensities ranging from 1×105 to 5×105 W/cm2. The wave speeds were also found to depend on the laser beam diameter. The spectral emission and CO2 laser absorption measurements showed that the air plasma reaches a temperature of 15 000–20 000°K and is essentially fully ionized. From an analysis of the results, it is concluded that radiation transport in the plasma plays a dominant role as a wave propagation mechanism, and that radial flow ahead of and within the wave is a dominant feature of the wave structure.

Reevaluation of arterial constitutive relations. A finite-deformation approach.

The purpose of this investigation was to use the finite-deformation theory of elasticity to interpret pressure-diameter data for in situ canine aortas and other arterial response data reported in the literature. A meaningful mechanical property for arterial tissue was identified as ∂ W 1 /∂ I , the partial derivative of the strain-energy density function with respect to the first strain invariant. An exponential function was found to characterize the mechanical property ∂W 1 /∂I for all arteries considered. Thin-walled tube stress approximations were found to result in inaccurate values for arterial stresses and incremental elastic mechanical properties. Wave speeds calculated using ∂ W 1 /∂ I for these arterial tissues agreed well with experimental measurements of wave speeds reported in the literature. Elevated values for strain-energy density were found in the inner arterial tissue layers. These high values for strain energy may contribute to atherogenesis in relatively straight arteries (e.g., the abdominal aorta) subjected to hypertension.

Density Gradient Effects on Alfvén Wave Propagation in a Cylindrical Plasma

The propagation of nonaxisymmetric Alfvén waves in a linear discharge tube filled with a nonuniform plasma under a 30-kG axial field has been investigated theoretically and experimentally in the frequency range 1–5 Mc/sec. Typical plasma densities were 3–4 × 1015 ions/cm3 near the center of the tube and fell to low values at the wall. It was found that the density gradient introduces a frequency dependent coupling between the torsional and compressional waves of the uniform plasma. As a result, an axial component of the magnetic field perturbation, which is normally associated only with the compressional wave, is now carried by the torsional wave as well, and was in fact detected at frequencies well below the lowest waveguide cutoff of the compressional wave. At still lower frequencies, this component disappeared, and the remaining transverse field components were found to propagate at a radially varying speed which agreed well with the local Alfvén speed deduced from the measured ion density profile. A theoretical analysis is presented, and its predictions are compared with experimental measurements of signal amplitudes, wave speeds, attenuation rates, and density profiles.

Variable interval between electric and acoustic phenomena in auricular fibrillation

A case of auricular fibrillation in which the interval between QRS and the first heart sound was much longer after a short diastole is described. Two more cases were studied, in one of which the same phenomenon occurred. The length of systole changed in proportion to that of the preceding diastole in the two cases in which the interval between the electric and acoustic phenomena was variable. Measurement of the pulse wave speed by means of the heart sounds does not change the known fact that smaller waves have a slower speed in cases of auricular fibrillation.

A New Method of Measuring Wave Speeds

A New Method of Measuring Wave Speeds