Volume 22, Issue 5, September 1950

On the Non‐Specular Reflection of Plane Waves of Sound
View Description Hide DescriptionThe non‐specular or scattered reflection of plane waves of sound by various rigid, non‐absorbent non‐porous surfaces composed of either semicylindrical or hemispherical bosses (protuberances) on an infinite plane is analyzed.Exact solutions for the problem of the single boss and a plane wave at an arbitrary angle of incidence are derived through consideration of a cylinder or sphere and two simultaneously incident “image waves.” Finite patterned distributions of such bosses are then treated and the far field solution obtained subject to the restriction that the secondary excitations of the various bosses be neglected. (The equivalent problems for cylinders and spheres are also considered as well as the second‐order solution, for the cylindrical case, which takes into account the interaction of neighboring elements.) These solutions are found to contain the characteristic Fraunhofer terms for a grating or lattice. The asymptotic solutions for the single bosses (Kr≫1, Ka<1) are then extended to consider both finite and infinite uniform random distributions. The solutions for the finite distributions are found to contain the characteristic Fraunhofer terms for similarly shaped apertures. The solutions for the infinite distributions (of semicylinders or hemispheres) are found to be remarkably similar when expressed in terms of the volumetric departure from the plane per cm^{2} of distribution. Some extensions and ramifications of the results are also considered.

Short‐Time Autocorrelation Functions and Power Spectra
View Description Hide DescriptionThe reciprocal relations between autocorrelation functions and power spectra, known as Wiener's Theorem, axe extended in a modified form to the case of experimental results obtained by means of filters with finite time constants. If the short‐time autocorrelation function φ_{ t }(τ) and power spectrum G_{t} (ω) are properly defined, it is found that where 1/α is a time constant. These equations may be used to relate the autocorrelation‐function representation of a speech wave to the corresponding spectrographic representation.

Response Peaks in Finite Horns
View Description Hide DescriptionIn this paper the term hyperbolic horn is used to designate those horns whose area law is given by: . Where S(x) is the cross‐sectional area of the horn at distance (x) from the throat, (a) is the flare constant; (T) is the shape parameter and (R _{0}) is the throat radius.
The pressure on the axis due to a circular mouth, unbaffled horn loudspeaker is derived. In this calculation use is made of the recent results of Levine and Schwinger [Phys. Rev. 73, 383 (1948)] and some useful additional functions are computed from their data and presented here in graphical form.
It is shown how to calculate the frequencies at which peaks occur in hyperbolic horn type loudspeaker frequency response curves.
It is shown how to calculate the parameters of hyperbolic and exponential horn type loudspeakers having pre‐determined peaks in their frequency response curves.
Some experimental confirmation of the theory is presented.

Resonance Characteristics of a Finite Catenoidal Horn
View Description Hide DescriptionExpressions for the impedance components of a finite catenoidal horn are derived and a comparison with similar exponential and conical horns made. The impedance of a section of a catenoidal horn is also calculated and it is shown how, for the finite as well as the infinite horn, this approaches that of the exponential as more length is trimmed from the throat end.
The assumption that the resonancecharacteristics of a horn are the same as that of a uniform tube, provided the higher velocity of sound for the horn is used, seems to be borne out for the catenoidal horn but for the exponential horn the agreement is not very good except at higher frequencies.

On the Propagation of Sound Waves in a Cylindrical Conduit
View Description Hide DescriptionThe characteristic impedance and propagation constant of a cylindrical conduit are calculated on the basis of an equivalent electrical T‐section. Numerical values of the results are plotted for air at 20°C, for a range of values of the independent variable which includes the region of transition from isothermal to adiabatic conditions.

A Method for Measuring Source Impedance and Tube Attenuation
View Description Hide DescriptionIf the active face, or acoustic output terminal, of a sinusoidal sound source moves as a plane piston, then the source can be characterized by a blocked pressure and an acoustic output impedance. If this piston is coupled to a microphone by means of a closed air column, the pressure at the microphone depends on the acoustic impedance of the microphone, on the impedance of the source, and on the air column. An expression for this pressure as a function of the length of the air column is developed, and data are presented which show how source impedance, tube attenuation and other quantities may be obtained.

Propagation of Sound in Rarefied Helium
View Description Hide DescriptionThe velocity and attenuation of sound at 1 mc/sec. were measured in helium at pressures as low as 0.1 mm Hg (sound frequency about twice mean collision rate). The observed dispersions are very large and agree well with those predicted by existing theories except at very low pressures where uncertainties are introduced by lack of gas purity.

On the Acoustics of Coupled Rooms
View Description Hide DescriptionIn many acoustically coupled systems the methods of geometrical acoustics do not apply. Reverberation formulas as ordinarily used would lead to incorrect results. This paper approaches the problem of coupled rooms from the “wave” point of view, treating the coupled rooms as a boundary value problem in obtaining an approximate solution. The results explain some discrepancies noted by earlier researchers between experiment and predictions from geometrical acoustics; for example, the dependence of absorption in a room on the position of the open area which couples the room to an adjacent one. For the case where the window area which couples one room to another is comparable in size with the partition which separates the rooms, the effect of the partition will be least when it is at a particle‐velocity node. For the case where the window area is small compared with the partition which separates the two rooms, the effect of the coupling window depends on the square of the unperturbed pressure at the window. Thus the effect of the window varies with position and is least at a pressure node. Experimental data on isolated modes of vibration of a coupled system are given which check the results predicted by this application of the wave theory.

Piezoelectric Equations of State and Their Application to Thickness‐Vibration Transducers
View Description Hide DescriptionThe electromechanical equations of state are written in a number of forms, extending from Voigt's formulation to the recently introduced formulation with g and h as the piezoelectric constants. Those equations that are most appropriate in the theoretical treatment of the two chief types of crystal transducer are pointed out. A detailed treatment of the thickness‐vibration transducer is then given, leading to expressions for the electrical characteristics and for the acoustic vibrational amplitude and intensity. Some special cases are also considered.

A Derivation and Tabulation of the Piezoelectric Equations of State
View Description Hide DescriptionThe linear piezoelectricequations of state with strain, electric displacement, and entropy, as independent variables, are derived from the principle of conservation of energy. These thirteen tensor equations are then specialized to the adiabatic case and used as the starting point for the development of all possible linear piezoelectricequations of state using in turn as independent variables, stress and electric field, strain and polarization, and stress and polarization. This process gives as a by‐product the set of relations among the elastic, electric, and piezoelectric coefficients for the various pairs of independent parameters and emphasizes the frequently overlooked fact that the piezoelectricequations of state are mutually convertible. M.k.s. units and the I.R.E. notation are used throughout the discussion and a table is given which compares the I.R.E. notation with the former notations of Cady and Mason.

The Sound Field of a Straubel X‐Cut Crystal
View Description Hide DescriptionThe sound beam generated in a liquid by a 7.5 megacycle Straubel X‐cut quartz crystal has been investigated by the phenomenon of streaming. The streaming patterns in carbon tetrachloride have been determined by measuring the velocities of sugar particles which have very nearly the same density as the liquid. The velocity distributions produced by a cylindrical sound beam in liquid contained a concentric cylindrical tank are found to be in good agreement with those predicted by Eckart's solution of the hydro‐dynamical flow equations. Comparison measurements with square X‐cut quartz crystals of the same frequency give poor agreement with theory. Since Eckart's analysis assumes uniform sound intensity across the beam, these experiments indicate that the sound beam radiated by a Straubel cut crystal is very uniform.

A Second‐Order Gradient Noise Canceling Microphone Using a Single Diaphragm
View Description Hide DescriptionA close talking, noise canceling microphone has been developed which responds to the second order of the pressure gradient and which has only one diaphragm. Since there are four sound pressures involved in a second‐order gradient microphone it has been deemed necessary in the past to have four surfaces for the four pressures to act upon. This microphone has sound entrances to the two surfaces of a single diaphragm spaced and oriented in such a manner as to produce the second‐order effect, thereby increasing the signal‐to‐noise ratio over that obtained in a first‐order gradient microphone. Mathematical analyses are made of the microphone first as a purely theoretical microphone with infinitesimal spacing of the sound entrances, then as a microphone with dimensions between sound entrances which are practical for use in a microphone of this type.

American Standard Method for the Coupler Calibration of Earphones Z24.9—1949 (Abridged)
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American Standard Method for the Pressure Calibration of Laboratory Standard Microphones Z24.4—1949 (Abridged)
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Uniform Speech‐Peak Clipping in a Uniform Signal‐to‐Noise Spectrum Ratio
View Description Hide DescriptionA graphical function W(r, c) has been determined experimentally, in which W is word articulation, r is the relative level of unclipped speech and the noise, and c is the amount of uniform, symmetrical, speech‐peak clipping. Pre‐emphasis of the speech signal gave an approximately uniform speech spectrum prior to clipping. Uniform, random noise was mixed with the clipped speech before postequalization, making the final noise spectrum similar in shape to the speech spectrum. The real‐ear response of the earphones was compensated electrically to yield a uniform orthotelephonic response for the communication system, in the frequency range contributing significantly to articulation index. For constant clipping c_{n} the function W(r, c_{n} ) approaches W(r, 0) as a limit for sufficiently large values of r. For , W≒W(r, 0). For the case of no clipping, W(r) when transformed to W(A) (A being articulation index) resembles closely the curve by Pollack.

On the Masking Pattern of a Simple Auditory Stimulus
View Description Hide DescriptionThe masking audiogram of a pure tone is complicated by phenomena that arise from the interaction of the test tone with the masking stimulus. The production of beats and of difference tones results in a masking audiogram that does not represent the pattern of activity in the cochlea or nerve due to a simple masking stimulus. In the present experiments a narrow band of noise was used to mask pure tones. The “beat” heard in the immediate vicinity of the masking noise is not prominent, and a test tone higher in frequency than the band of noise is detected in terms of the characteristic pitch of that tone rather than by means of a difference tone.
With a band of noise slightly wider than a critical band (Fletcher), the amount by which the test signal at the center of the band exceeds the level/cycle of the noise is less than that obtained with a considerably wider band of noise. This smaller signal‐to‐noise ratio obtained with the narrow band of noise is probably due to “beats,” since the test tone is heard as a “buzz” or “rattle.” For purposes of deriving excitation or loudnesspatterns, a better estimate of the maximum amount of masking is obtained by using a wide band of noise whose pressure spectrum level is the same as that of the narrow band.

The Masking of Clicks by Pure Tones and Bands of Noise
View Description Hide DescriptionPure tones and various bands of noise were used to mask impulsive acoustic stimuli (clicks) that were produced by electrical square pulses through a moving‐coil type earphone. Results indicate that it is difficult to generalize about the masking of clicks by pure tones from observations on only a few observers because of sizeable individual differences. The masking of clicks by noise, however, is related unambiguously to the intensity and the spectral distribution of the noise. Both pure tones and narrow bands of noise provide maximum masking in the region between 1000 and 3000 c.p.s. Generally speaking, noise is a more efficient masker of clicks than are single pure tones. Although the present measurements were made with masking frequencies below 6500 c.p.s., there is some preliminary evidence to suggest that the audibility of certain clicks in noise is markedly dependent on frequencies above 6500 c.p.s. The linear (slope = 1) relation between the masking of speech or pure tones by noise and the intensity of the noise does not apply here. Rather a trimodal function indicates that the masking of unfiltered clicks by “pinkish‐white” noise does not change over a 30‐db range of moderate intensities. Below and above this range masking increases with the intensity of noise, sometimes at a slope greater than one.

The Trill Threshold
View Description Hide DescriptionTwo tones of different frequencies alternated successively five times per second. When the difference in frequency was small, the alternation sounded like a continuous up‐and‐down movement of the pitch. When the difference in frequency was large, the alternation sounded like two unrelated, interrupted tones. The transition point between these two perceptual organizations is called the trill threshold. The trill threshold was measured as a function of frequency for 14 subjects; the results are summarized in Fig. 1.
 LETTERS TO THE EDITOR


Attenuation of (n,0) Transverse Modes in a Rectangular Tube
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Some Effects of Side‐Tone Delay
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