Volume 37, Issue 4, April 1965

Frequency Equations for the Normal Modes of Vibration for a Plate Bounded by Parabolic Cylinders When Rotary Inertia and Transverse Shear are Considered
View Description Hide DescriptionIn this paper, the partial differential equations governing the motion of plates when transverse shear and rotary inertia are considered are solved for plates bounded by parabolic cylinders. Three cases have to be considered, depending on the shape of the plate. They are: Case I: a plate bounded by two parabolic arcs (v = v _{1}, u = u _{1}, u = −u _{1}); Case II: a plate bounded by three parabolic arcs (v = v _{1}, v = v _{2}, u = u _{1}, u = −u _{1}); Case III: a plate bounded by four parabolic arcs (v = v _{1}, v = v _{2}, u = u _{1}, u = −u _{2}). After converting the partial differential equations and boundary conditions into parabolic coordinates, product solutions are assumed and solutions of the resulting ordinary differential equations appear as definite integrals. The eight different types of boundary conditions entering into the theory, for each case, are satisfied by taking linear combinations of products of these definite integrals. The elimination of the arbitrary constants in each boundary problem leads to the frequency equation for the normal modes of vibration. The frequency equations for Case II and Case III are given by determinants of the 12th order equaled to zero, whereas for Case I the determinant is of the 6th order. In the classical theory, the frequency equations far Case II and Case III are given by determinants of the 8th order equated to zero, whereas for Case I a determinant of the 4th order is required.

Diffraction Constants of Transducers
View Description Hide DescriptionThe concept of a transducerdiffraction constant as the ratio of the blocked diaphragm pressure to the incident free‐field pressure is examined for meaning, definition, and application in the general case. The concept is extended to a transmitting transducer, and a transmitting diffraction constant that is equal in value but reciprocal in concept to the conventional or receiving constant is developed from the acoustical reciprocity theorem. The relationship among the diffraction constant D, the radiation resistanceR, and the directivity factor δ is given as D ^{2} = Rδ/R _{0}, where R _{0} is the radiation resistance of a simple source. The electrical analog of the diffraction constant D is found to be an ideal transformer with the turns ratio D:1. The electrical analog of a complete and reciprocal projector medium hydrophone system is presented.

Scattering of Acoustic Waves by Pressure Fluctuations
View Description Hide DescriptionThe scattering of acoustic radiation by limited regions of pressure fluctuations is considered. The hydro‐dynamic equations are expanded into a hierarchy of acoustic equations of increasing nonlinearity and the first three of these is solved for the particular scattering problem. The scattered energy is given in terms of correlation functions for the pressure of two different points and for the pressure and its time derivative at two different points.

Propagation of Finite‐Amplitude Ultrasound in a Relaxing Liquid
View Description Hide DescriptionPerturbation analysis is used to obtain expressions for the harmonic content of finite‐amplitude waves propagated in a relaxing liquid. Comparison is made with similar expressions for the nonrelaxing case. Experimental measurements were made with 2.57‐MHz rf pulses in a 0.2‐M aqueous solution of to test the theoretical results. Agreement is excellent for the fundamental frequency. The second‐harmonic curves have the correct shape but are 10% 15% smaller in absolute magnitude.

Detection of a Doppler‐Invariant FM Signal by Means of a Tapped Delay Line
View Description Hide DescriptionThe phenomenon of Doppler invariance in an FM signal is discussed first. This is followed by a description of how this signal may be detected by passive sampling, a variable sampling rate being chosen to make each sample correspond to a signal peak. This results in signal folding, which reduces the total number of samples required to detect the signal. Certain characteristics of the signal are described and a formula is derived for the output of the detector when a weighting function designed to reduce the sidelobes is applied to the taps. The output is a discontinuous function because the signal is not always present at all of the taps all of the time, and the nature of this function is illustrated by means of graphs drawn for a number of typical weighting functions.

Acoustic Modes of a Hemispherical Room
View Description Hide DescriptionThe acoustic modes of a hard‐walled hemisphere are derived. Inside the enclosure the standing waves for the velocity potential are described by , where m and l are positive integers, m ⩽l, c is the soundvelocity,P_{t} ^{m} are the associated Legendre functions, j_{t} are the spherical Bessel functions, and ωl_{q} are the ordered radian frequencies of the allowed modes. The eigenvalues are roots of , where a is the radius of the hemisphere. Only the standing waves for which (m+l) is even are shown to satisfy the hemispherical boundary. Implications of the results to acoustic treatment for a hemispherical room are discussed.

Impact‐Noise Characteristics of Female Hard‐Heeled Foot Traffic
View Description Hide DescriptionIt has been said that the ISO tapping machine does not yield data on representative flooring materials that accurately relate to the impact‐noise isolation provided for real, foot‐fall impacts. To understand this problem, we have measured the impact force under both the ISO hammer and a woman's shoe heel. The hammer force is found to be much more intense and generally to contain more high‐frequency components than the heel force. The differences between hammer and heel forces are found to arise from the different mechanical internal admittances and from the different approach velocities. Radiated sound levels for the measured force spectra applied to a concrete slab are calculated and found to agree closely with measured impact sound levels. We conclude that the ISO machine grossly misrepresents the impact isolation provided by some typical floor surfaces.

Shifts in Air‐Conduction Thresholds Produced by Pulsed and Continuous Contralateral Masking
View Description Hide DescriptionIn the present investigations, auditory threshold shifts for either constant or pulsed pure tones were observed while a steady or pulsed narrow band of white noise was delivered to the contralateral ear via an insert receiver. The narrow‐band masker, centered around 4000 cps, was presented at intensity levels of 50, 70, and 90 dB SPL while thresholds were obtained from the test ear at 4000, 1000, and 250 cps. The results demonstrate, first, that larger threshold shifts occur when the test signal and the masker are pulsed simultaneously than when the masker is continuous; second, that a continuous masker may be as effective as the pulsed masker if the test tone is also continuous; third, that more contralateral masking is found when the test tone and masker are close in frequency; and, last, that a small increase in the average threshold shift occurs as the intensity level of the masker increases. Several interpretations are offered in explanation of the threshold shifts for the continuous continuous and pulsed pulsed (simultaneous) conditions.

Use of Noise to Eliminate One Ear from Masking Experiments
View Description Hide DescriptionTo evaluate the possibility that one ear can be eliminated from a masking experiment by use of noise, certain relevant facts were determined. These are: (1) if a tonal signal mixed with noise is received at one ear, the addition of a noise to the other ear slightly reduces the threshold for the tone if the noises are statistically independent; (2) in contrast, the noise added to the nonsignal ear distinctly reduces the threshold for the tone if the noises are perfectly correlated (+1.0); (3) these effects, (1) and (2) above, are observed whether the level of the masking noise at the ear that receives the tonal signal is less than, equal to, or greater than the level of the added noise at the ear that does not receive the tone; (4) if identical tones are presented to the two ears and if the signal‐to‐noise ratio is about 25 dB lower in one ear than in the other, the effect of the signal at the ear with the lower signal to noise ratio is eliminated from the masking experiment.

Distortion Processes in the Cochlear‐Microphonic Response under Normal and Abnormal Physiological Conditions
View Description Hide DescriptionThere is a long history of controversy concerning the locus and mechanism of harmonic‐distortion processes in the cochlea. This controversy has importance for theories of frequency analysis along the basilar membrane and concepts of the transduction process in the hair cells, as well as for the clinical significance of psychophysical tests of aural harmonics. The cochlear‐microphonic response to a pure tone recorded at the round window of the guinea pig was led into three wave analyzers tuned to the 1st‐, 2nd‐, and 3rd‐harmonic frequencies, each component then driving one channel of a pen recorder. The various components exhibit shifts through time, which were studied at various frequencies and intensities, and during recovery from fatiguing and damaging levels of tonal and random‐noise stimulation and anoxia. The data are discussed in terms of mechanical versus transduction concepts of harmonic distortion in the cochlea.

Temporal Effects in Simultaneous Masking by White‐Noise Bursts
View Description Hide DescriptionThe motivation of the research described was to investigate the behavior of a masking transient that indicates that masking of a short signal pulse by a longer white‐noise burst is stronger at the beginning of the masker burst than later. The threshold of signal pulses masked by masker bursts was measured as a function of different variables such as bandwidth and center frequency of the signal, delay between onset of masker and onset of signal, duration of signal and masker, level of masker, and repetition rate. The results reveal very little “overshoot” of the threshold of short pulses as a function of the ON time of the masker if the signal and the masker have the same or similar broad spectra. The overshoot increases up to 13 dB as the bandwidth of the signal decreases down to that of a tone. The size of the “overshoot” and the prior excitation seem to be related to each other. Taking this in account, the thresholds under different conditions can be calculated on the basis of detection models. The measured and the calculated values are in good agreement.

Theoretical Analysis of the Solion Polarized Cathode Acoustic Linear Transducer
View Description Hide DescriptionSolions are electrical components involving charge transport by solution ions as contrasted to electrons, holes, or gas ions. Four general types of solion acoustic transducers may be distinguished on the basis of mechanism underlying the transducer action. One of these, the polarized cathode type, offers a combination of advantages, which includes low threshold pressure, high power gain, stable zero, and direct‐current transmission. Individually constructed models with a nominally linear pressure response have been used successfully in the study of infrasonic disturbances transmitted through the earth and atmosphere as well as under water. Yet, design and construction methods have not previously been perfected to a degree that would ensure wide usage. The relatively sluggish motion of the charge carriers in the electrolyte of the transducer is at the same time a necessary condition for transducer action and a factor limiting the dynamic response. A theoretical analysis is presented to explain the observed behavior of the existing devices of this type and to indicate what may be expected from transducers having the benefit of more‐refined fabrication techniques.

Analysis of the Flute Head Joint
View Description Hide DescriptionThe acoustical nature of the head joint of a transverse flute is examined in mathematical detail. The effective length of the flute is altered in the amount by an embouchure correction Δl_{c} = (c/2πf) arctan , where f is the playing frequency, e is the speed of sound in the contained air, r the (17‐mm) cork‐to‐embouchure distance, and f _{1} a parameter (near 1450 cps which is essentially the frequency of a Helmholtz resonator made by plugging the bore with a second cork a distance r below the embouchure. This correction is roughly constant at 42 mm over the normal playing range. Effects of cork position, lip position, and embouchure‐hole size are discussed in detail, as is the effect of cavity resonances in the player's own mouth. Tuning errors caused by the edgetone‐regeneration mechanism are shown to be correctible by a head joint taper perturbation. Three designs are analyzed, which produces a flattening at low frequencies, zero effect near 600 cps, and a maximum sharpening near 1000 cps. Above, this the correction returns smoothly to zero at the upper playing limit near 2000 cps. In the high register, the correction is produced jointly by the taper and the vent holes. The whole analysis is summarized by means of a detailed comparison of the Boehm design of flute with the older cylindrical‐head, taper‐bore model and with the purely cylindrical flute.

Speech Communications as Limited by Ambient Noise
View Description Hide DescriptionSpeech‐intelligibility scores as a function of noise level are studied for face‐to‐face, sound‐powered‐phone, and amplified speech(earphone and loudspeaker) communication conditions. The speech‐interference level (SIL) for octaves of noise centered at 500, 1000, and 2000 cps (0.5/1/2) is used as the measure of noise level. By using this noise measure, much of the work in this field can be brought together and interpreted. It is noted that “noisy” and “very noisy” spaces are associated with SIL's such that “shouting” or “very loud” voice levels (or 95‐dB speech levels) are required for conversations at 1.5 or 3 ft, and this is the region where telephone conversations are judged to be “difficult” or “unsatisfactory.” All of these adverse noise conditions occur at the region where ear protection will aid intelligibility and at the boundary where ear protection should be used to protect against hearing losses. Where people must converse or communicate via some interior communication device, 0.5/1/2 SIL's above 70 dB should be avoided. At 0.5/1/2 SIL's greater than 90 dB, the wearing of hearing protection should be made mandatory and every noiseproofing technique (except a noise shield for the microphone) should be employed. At 0.5/1/2 SIL's above 100 dB, every noise‐proofing technique should be employed

Sound Propagation in Near‐Stoichiometric Ti‐Ni Alloys
View Description Hide DescriptionExperimental data on sound velocity illustrate the unusual velocity‐temperature relationship of near‐stoichiometric nickel‐titanium alloys. The effect of temperature cycling, annealing, pressure, minimum exposure temperature, and time is also shown.

Revised Grain‐Scattering Formulas and Tables
View Description Hide DescriptionThe current theory of Rayleigh and stochastic scattering in polycrystallinematerials is reviewed and compared with (1) former theory and (2) experiment. Rayleigh scattering giving ultrasonic attenuation equal to STf ^{4} (S is the Rayleigh scattering factor, T the average scattering volume, f the frequency) occurs when λ > 2πD̄ (λ is the wavelength, D̄ the average grain diameter); stochastic scattering yielding ultrasonic attenuation equal to ΣD̄f ^{2} (Σ is the stochastic scattering factor) occurs when λ < 2πD̄. The average scattering volume and average grain diameter must be evaluated by taking their averages over the grain‐size distribution in the metal. When this is done, the current theory accounts rather well for the scattering component of the ultrasonic attenuation in polycrystalline metals. Former theory underestimated the scattering. A tabulation is made of the scattering factors S and Σ in various materials. The computed scattering factors show that polycrystalline samples of the following materials should have low attenuation:aluminum, chromite, chromium,magnesium, magnetite, silicon strontium nitrate, tungsten, vanadium, and YIG.

Ultrasonic Attenuation Caused by Scattering in Polycrystalline Metals
View Description Hide DescriptionUltrasonic‐attenuation measurements have been made on fine‐grained specimens of several metals. The grain‐size distributions and ultrasonic velocities in these metals were also determined. The experimental attenuation is in good quantitative as well as qualitative agreement with current theory. Nickel and three iron alloys, one 30% nickel reported previously, the second 12% chromium (type 416 stainless steel), and the third 17% chromium and 1% carbon (type 440‐C stainless steel), all gave good results. Brass also gave good results, but copper showed much twinning, which as yet is unaccounted for.

Vibrational Relaxation Times of Oxygen in the Temperature Range 100°–200°C
View Description Hide DescriptionVibrational relaxation times in pure oxygen have been measured over the temperature range 100°–200°C. An anomalously rapid decrease in the relaxation time is observed for temperatures above 150°C, confirming earlier conclusions based on a comparison of theory and experiment [J. G. Parker, J. Chem. Phys. 41, 1600–1609 (1964)]. This decrease in the vibrational relaxation time may be interpreted as a transition in oxygen from a low‐temperature state to a high‐temperature state, although the exact nature of the physical mechanism underlying this transition remains obscure.

Numerical Solution for Sound Velocity and Absorption in Cylindrical Tubes
View Description Hide DescriptionA numerical solution of the Kirchhoff equation for the propagation constant of longitudinal sound waves in infinitely long cylindrical tubes has been obtained. The solution, which avoids the wide‐tube approximations, shows that the percentage errors in the von Helmholtz‐Kirchhoff tube velocity correction and tube absorption are both roughly equal to the percentage the velocity correction is of the free‐space velocity. The error in the von Helmholtz‐Kirchhoff equations can be plotted as a function of fD/a, pD/ηa, and γ. (f is the sound frequency, D the tube diameter, a the free‐space velocity,p the gas pressure, η the viscosity, and γ the ratio of specific heats.) Recent absorptionmeasurements in Ar are in agreement with values calculated numerically, but measuredvelocities indicate the need for considering molecular slip at the tube wall. Thermal relaxation is introduced into Kirchhoff's basic equation by using the Eucken relation and considering γ to be the ratio of complex relaxing specific heats. Viscothermal and relaxation effects are found to be additive only if the frequency is near the cutoff frequency for the first unsymmetric mode and the f/p values do not extend to the megacycle/second atmospheres range.

Thermal Stress‐Wave Propagation in Hollow Elastic Spheres
View Description Hide DescriptionIn conventional dynamic‐thermoelastic problems, the heating rates are sufficiently slow so that the inertia terms in the equations of equilibrium are negligible and time enters as a parameter from the transient temperature in the body. In modern technology, however, one has encountered extremely high heating rates and it is necessary to reexamine the role of inertia. To this end, studies of extremely massive bodies and extremely slender bodies under rapid heating conditions have been made by others. In the former, thermal stress‐wave propagation without reflection was examined; in the latter, thermally induced vibrations were considered. It is now of interest to examine the effect of inertia in a rapidly heated body whose dimensions lie between these extremes. As an example, we consider a hollow sphere of arbitrary thickness subjected to a step change of temperature on its inner surface. In the solution, the propagation and reflection of thermal stress waves in the sphere are observed. For comparison, the results of the conventional analysis are obtained as a special case. After passing to appropriate limits, the results are compared to the results of the previous analyses of slender and massive spherical regions.