Circular Plane Piston Research Articles

Analysis and Optimization of Micro Speaker-Box Using a Passive Radiator in Portable Device

Abstract With the rapid development of multimedia devices such as smart phones and tablet PCs, microspeakers have been recently increasingly used for audio equipment. Improving the acoustic performance of a microspeaker is always a main concern, especially in the low frequency range. To avoid sound cancelation, a microspeaker unit is usually inserted into a speaker box. A passive radiator is also used in speaker boxes to improve the sound performance in the loudspeaker system. However, passive radiators have not been applied into microspeaker system. In this study, a speaker box with a passive radiator was analyzed and optimized to achieve a higher Sound Pressure Level (SPL) in a microspeaker system. The Finite Element Method (FEM), two-degree-of-freedom (DOF) vibration theory, and a plane circular piston sound source were used to study the electromagnetic, vibration, and acoustic characteristics, respectively. Optimization was conducted by changing the mass, stiffness, and size of the passive radiator. Based on the optimized parameters, a new sample was manufactured. The experiment results show that the SPL of the optimized speaker box with a passive radiator is improved by 5 dB at 200 Hz compared with the one without a radiator. The analysis results also matched the experiment results.

System and method for calculating the directivity index of a passive acoustic array

A software system and method is presented that calculates the sensor directivity at all spatial angles, and stores these values in a two dimensional matrix. These values are then used as additional “weighting” coefficients in the equation for the pressure detected by an array of directive sensors. The azimuthal and polar angles of a particular sensor normal vector are used to “rotate” the sensor directivity matrix to account for the angular orientation of each sensor within the array. The software system receives as input sensor and array geometry, shading functions, sensor shape, array structure baffling, steering, and shading. A sensor can have the shape of a point, line, plane, volume, baffled ring, or circular plane piston. The sensors within the array can either be baffled by the array structure or retain their free field directivity response, and array as a whole can have steering or no steering.

Experimental determination of end reflection loss (ERL) over a range of common types and configurations of duct terminations

The two-microphone impedance tube method ASTM E1050 was applied to directly measure the termination impedance of rectangular duct configurations of different aspect ratios and end conditions over a 25- to 500-Hz frequency band. Measured values of the acoustic impedance, and hence the reflection coefficient, of the duct termination permits computation of end reflection loss. ERL is essentially the insertion loss of duct termination. The measurement program is challenging, as the precision required for accurate ERL computation is much higher than that for accurate determination of the reflection coefficient. The measurement program is part of a project to assess the available published data on duct ERL. The ASHRAE Handbook contains a tabulation of ERL data derived from predictions of the radiation impedance of a circular plane piston, and there have been some validation measurements made on a limited set of circular duct configurations and orifice plate terminations. Results to date closely conform to published ERL values. [Work supported by ASHRAE.]

A theoretical model for the evaluation of measurement uncertainty of a sound level meter calibration by comparison method in an anechoic room

A theoretical model for the evaluation of measurement uncertainty of a sound level meter (hereafter as `SLM') calibration by comparison method in an anechoic room was developed. Through this model, the uncertainties in the semi-automatic calibration and that in the full-automatic calibration were estimated for the recently developed SLM calibration system. In order to estimate the standard uncertainty against the SLM positioning, which is a significant uncertainty component, the sound field curve-fitting formulae were adopted. The validity of the curve-fitting method was proven by the similarity of the spatial distributions of radiation sound field produced by the plane circular piston source and that by the cone shape source. A linear equation was used to fit the measurements of the sound field distribution along the radiation axis. A quadratic equation was used to fit the measurements along the radial axis normal to the radiation axis. The fitting parameters gave us the sensitivity coefficients of the propagation of the uncertainty. In addition, one of the quadratic fitting parameters was found to be a positional uncertainty itself. Using this model, the expanded uncertainties were evaluated for the semi-automatic and full-automatic calibration of SLM.

Spatial averaging in the beam of a piston transducer

This paper presents a theoretical analysis of the spatially averaged free-field responses of phase-sensitive and phase-insensitive receivers centered in the beam of a harmonically excited piston transmitter. The responses of unfocused circular plane piston receivers are analyzed, and both unfocused and spherically focused piston transmitters are considered. A set of closed-form expressions figures prominently in the analysis. The expressions are based on the Lommel diffraction formulation which is, in turn, based on the Fresnel approximation. Although approximate, the expressions allow for quick and easy estimation of phase-sensitive or phase-insensitive unfocused piston receiver responses. It is shown that the spatial averaging effects associated with phase-sensitive and phase-insensitive receivers are virtually identical when γ ≤ 0.1, where γ = b/a is the ratio of receiver radius b to transmitter radius a. In addition, numerical results obtained from the closed-form expressions are compared with previously reported results. The comparisons indicate that the approximate results are valid from the m = 3 maxima forward under the assumption of linear propagation when ka > 58, where k is the circular wave number. Finally, it is pointed out that the closed-form expressions may prove useful in the estimation of the potential for bioeffects associated with diagnostic ultrasound. (E-mail: narpci@rit.edu)

Diffraction effects in hydrophone measurements

Theoretical and experimental methodology are presented for accurately determining the effective radii of unfocused, circular plane piston transducers as well as using tone burst hydrophone measurements to verify steady-state theoretical calculations. Experiments using two specially fabricated unfocused, composite piezoelectric transducers demonstrate the validity of the methodology. For spherically focused circular transducers, a simple model is used to estimate the transient diffraction encountered in co-axial flat hydrophone measurements.

An experimental investigation of the nonlinear pressure field produced by a plane circular piston

An experimental investigation of the nonlinearly generated harmonics in the sound beam of a plane circular piston in water has been carried out. Special attention was paid to the higher harmonic sidelobes called fingers. All experiments were performed using a high precision, computer-controlled tank facility where movement resolution is better than 50 μm. Harmonic beam patterns generated by a 1-MHz, 12-mm radius plane circular piston driven with a face sound pressure level of 223 dB (re: 1 μPa) were measured. Beam patterns of the first ten harmonics (out to the first sidelobe of the fundamental beam pattern) are shown. The first four harmonic beam pattern data are compared with results produced by a numerical solution of the Khokhlov–Zabolotskaya–Kuznetzov (KZK) equation. Agreement between experiment and theory is excellent. Finally, careful measurements of the decay of the fingers were made. The decay of the first finger in the second harmonic beam pattern generated by a 2.2-MHz, 3-mm radius circular piston [217 dB (re: 1 μPa)] out to 30 Rayleigh distances was measured. As predicted by Naze Tjo/tta and Tjo/tta [J. Acoust. Soc. Am. 83, 487–495 (1988)], the finger decays as 1/z (neglecting absorption).

Distortion and high-frequency generation due to nonlinear propagation of short ultrasonic pulses from a plane circular piston

A numerical finite-difference model for the prediction of nonlinear propagation has been adapted to examine the high-frequency components generated by short ultrasonic pulses. Comparisons are made in both the time and frequency domains between the model predictions and experimental measurements. The experimental measurements were obtained, in water, using a 1-mm-diam membrane hydrophone to sample the pressure field generated by a plane transducer (38 mm in diameter) that was driven by a diagnostic medical ultrasound system. The pulses generated were a few cycles long with a zero-crossing frequency of about 2 MHz and a maximum peak positive pressure of about 300 kPa. It is shown that the theoretical model can be used to provide accurate predictions (typically better than 10%) both on and off the acoustic axis for pulsed pressure fields in water. An important factor in obtaining good agreement is the initial characterization of the transducer and pulse.

Effect of wave diffraction on the measurement of ultrasonic power output using the radiation force method with a plane reflecting target angled at 45 degrees (biomedical equipment)

The radiation force method utilizing a reflecting target is widely used for measuring the total power outputs from medical ultrasonic equipment. The investigation described shows that, by varying the distance between transducer and reflector, measured values of power output from a 0.75 MHz plane circular piston transducer (diameter=2.5 cm) differ by up to 17% within the near field. The possibility that the distance dependence arises because of wave diffraction is considered. It is recommended that researchers should quote transducer-reflector distance when reporting values of spatial average intensity obtained using this method. The validity of the equation which is used to determine power output of transducers from the measured radiation force is questioned, and it is shown that the weight of the reflector Mg (after correcting for buoyancy) should be much greater than the radiation force but this requirement may not be necessary where conical or symmetrical wedge-shaped reflectors are used.

The nonlinear pressure field of a plane circular piston: Theory and experiment

The measured nearfield pressure levels of a plane circular piston are compared with a numerical solution of the parabolic approximation to the nonlinear wave equation under conditions of high nonlinearity (100 kPa at the piston face). The solution allows for nonlinearity, diffraction, and absorption in continuous wave pressure fields. The measurements were made in water, using a transducer (38-mm diameter) driven at 2.25 MHz and a 1-mm-diam membrane hydrophone. Comparisons are made along and across the acoustic axis for the amplitudes of the fundamental, second, and third harmonics and the phases of the second and third harmonics. Good agreement is shown between experiment and theory within the known limitations of both.

A further study into the Gauss Laguerre formulation of the nearfield of a circular plane piston transducer

The advantages of interpretation using Gauss Laguerre/Hermite formulation was adopted to acoustics from optics by Cook and Arnoult [J. Acoust. Soc. Am. 59, 9–11 (1976)]. Recently Cavanagh and Cook [J. Acoust. Soc. Am. 62, S22(A) (1977)] have shown this formulation has possibilities in acousto‐optic projection analysis of ultrasound waves. The question remains of the number of modes required to adequately describe the field from plane piston transducers. This paper describes the ongoing research into that problem for a circular transducer.

Integrated acousto-optical effect produced by pulsed sound fields

Precision acousto-optic transducer calibration requires an understanding of the influence of the nearfield of a transducer on the integrated optical effect. This influence has previously been studied theoretically for continuous waves from circular and rectangular plane piston transducers [F. Ingenito and B. Cook, J. Acoust. Soc. Am. 45, 572–577 (1969)]. For pulsed ultrasonic transducers, the nearfield can be considerably different than for continuous waves. The influence of the differences on the integrated acousto-optic effect is analyzed by linear superposition. For a circular transducer with piston motion, it is possible to reduce the formulation with appropriate approximations and restrictions to a universal form. For long pulses, i.e., greater than ten wavelengths, the integrated acousto-optic effect is the same as for continuous waves. For shorter pulses, the nearfield can influence the pulse shape, but not substantially. This influence is dependent on field position and can be minimized.

Data requirements for the mapping of ultrasonic fields with conventional light diffraction

A procedure has been proposed by Cook [J. Acoust. Soc. Am. 53, 330(A) (1972)] through which a cross-sectional mapping of a progressive sound beam could be obtained by conventional light diffraction. This technique requires that a series of optical diffraction measurements be taken laterally across the field, for a large number of rotations about the sound field’s major axis. Since the number of measurements could become excessive, and investigation of the data requirements of this technique has been performed. Computer-generated data were calculated of the integrated optical effect (both magnitude and phase) produced by the pressure field of a circular plane piston in an infinite rigid baffle. From these data a cross-sectional map was obtained using an effective two-dimensional Fourier transform, as well as the two-dimensional fast Fourier transform algorithm. The cross-sectional mapping was evaluated at different axial positions, using one set of data obtained by varying the data set and comparing the results to those obtained from the diffraction integral. The data requirements were found not to be excessive. Subject Classification: 35.65.

Data requirements for the mapping of ultrasonic fields with conventional light diffraction

A procedure has been proposed by Cook [J. Acoust. Soc. Am. 53, 330(A)(1972)] through which a cross-sectional mapping of a progressive sound beam could be obtained by conventional light diffraction. This technique requires that a series of optical diffraction measurements be taken using a parallel beam of light passing through and perpendicular to the sound beam. These measurements are taken laterally across the field, for a large number of rotations about the sound field's major axis. Since the number of measurements could become excessive, an investigation of the data requirements of this technique has been performed. Computer-generated data were calculated of the integrated optical effect (both magnitude and phase) produced by the pressure field of a circular plane piston in an infinite rigid baffle. From these data a cross-sectional map was obtained using an effective two-dimensional Fourier transform, as well as the two-dimensional fast-Fourier-transform algorithm. The cross-sectional mapping was evaluated at different axial positions, using one set of data generated at a single axial distance. Data requirements were obtained by varying the data set and comparing the results to those obtained from the diffraction integral. The data requirements were found to be not excessive.

Parametric detection of low-frequency acoustic waves in the nearfield of an arbitrary directional pump transducer

We analyze by first-order perturbation theory the manner in which a low-frequency acoustic wave and a high-frequency pump beam interact nonlinearly to produce a sum- and a difference-frequency component. Provided that its acoustic beam is well collimated, the pump source can be of arbitrary shape, with arbitrary acoustic distribution over its surface. For the case where the incident low-frequency wave is plane, we derive the following result: To a close approximation, the acoustic particle velocity at the difference frequency is proportional to the product of (a) the velocity potential of the pump beam, calculated at the difference frequency, and (b) the response function, at the low frequency, of an endfire array stretching from the pump source to the plane of a receiver. The result also holds for the sum frequency. We also prove that the result holds—with a modified endfire response function—when the low-frequency source is at a finite distance. For a circular plane piston as pump source, we discuss the achievable directivity in detail; also, we show that focusing the pump beam to obtain a stronger signal can succeed only at the expense of good directivity.

Parametric Detection of a Low-Frequency Plane Wave by a Circular Piston Beam

Westervelt's prediction [J. Acoust. Soc. Amer. 35, 535–537 (1963)] that an acoustic signal of low frequency can be detected by a parametric endfire array—i.e., by nonlinear interaction with a “pump” beam of high frequency—is analyzed in the case of a plane wave of low frequency and a highly directional beam from a circular plane piston used as a pump. To first-order perturbation theory, at field point (r,z), the z component of particle velocity for the interaction component at the difference frequency is proportional to the product of the piston-beam function at (r,z) (but calculated at the difference frequency) and the response function of a truncated endfire array of length z (calculated at the low frequency). The receiving directivity of the system is thus determined by the endfire response function alone, regardless of the piston's properties. The piston-beam function is, however, a factor in determined the system's receiving sensitivity. Work continues on generalizing the theorem to arbitrary directional pump transducers.

Journal of the Acoustical Society of America : Acoustic field of circular plane piston in limits of short wavelength or large radius Rogers, P.H., Williams Jr, A.O., 52 (September 1972) 865

Journal of the Acoustical Society of America : Acoustic field of circular plane piston in limits of short wavelength or large radius Rogers, P.H., Williams Jr, A.O., 52 (September 1972) 865

Acoustic Field of Circular Plane Piston in Limits of Short Wavelength or Large Radius

The acoustic beam of a circular plane piston (radius a) is analyzed as it approaches either the short-wavelength limit (SWL) or the large-radius limit (LRL). Using a new expression for the beam, we show that with no absorption the SWL would be a cylindrically collimated beam, with radius a, of plane waves. If absorptive effects are minor, the chief deviations during approach to the SWL occur in two decreasing volumes, a horn-shaped paraxial region and a cylindrical shell of mean radius a. One consequence is that any second harmonic, self-generated in the beam, approaches its own SWL more rapidly than does the first harmonic. Approach to the LRL yields a known paradox: Even the piston's nearfield need not approach a uniform plane wave. We show that the presence of absorption does make the LRL a plane wave and that in the absence of absorption there is a valid reason for the paradox: The piston beam must have a unique axis of cylindrical symmetry; an unbounded plane wave cannot.

Calculation of Second-Harmonic Generation in a Piston Beam

We discuss the nonlinear generation of a second-harmonic component in the first-harmonic beam from a circular plane piston, radiating into a fluid. First-order perturbation theory is used; higher harmonics, absorption, and depletion of each harmonic are ignored, although a partial correction for the last two effects is applied later. The ratio of piston diameter to first-harmonic acoustic wavelength is assumed to be about 30 or greater. In particular, the axial behavior of the second harmonic, which has marked structure, is found to agree well with experimental results in water. Also, the average value of the second-harmonic field is calculated over a circle coaxial with the source, parallel to it, and of the same size. This average differs considerably from the result of an oversimplification that treats the first-harmonic beam as cylindrically collimated plane waves. When compared with measurements in water, the average agrees well at a source pressure of 3 atm but progressively less well at 5 and at 6 atm. It is suggested that both theory and experiment contribute to the discrepancies.

Measured Structure of Harmonics Self-Generated in an Acoustic Beam

The acoustic beam from a circular plane piston 1.01 cm in radius, driven sinusoidally to finite amplitudes in water, at 2.58 Mc/sec, has been mapped with two quite different small probes about 2 mm in diameter. Equivalent acoustic pressures at the source ranged from 0.2 to 6.0 atm. The fundamental component and the self-generated second and third harmonics were filtered separately, and beam patterns were photographed on a CRO screen. Fine structure down to the size of the probes was recorded. Reliability was demonstrated both by agreement of the two different probes and by measurements on theoretically known patterns at very low amplitude; maxima and minima were, respectively, lowered and raised in the records, but approximate analytical corrections are available. Rough estimates (±30%) of sensitivity showed that both probes were substantially linear and stable. The second-harmonic component of the beam was found to be more nearly cylindrically collimated than was the fundamental. Fine structure in second and third harmonics was rather less complex than in the fundamental, and different in detail, yet the higher harmonics were not much nearer than the fundamental to being collimated plane waves.