Pressure Release Boundary Condition Research Articles

The 1952 Ph.D. dissertation of T.F.W. Embleton on nonlinear acoustic propagation and reflection

In his doctoral research Tony Embleton studied nonlinear acoustics experimentally using techniques available in the Physics Department of Imperial College, University of London. Originally developed by V. Timbrell, the interferometric equipment was substantially improved by Tony Embleton, enabling his Ph.D. findings. Studying propagation in a tube and reflection by both closed and open ends, Tony Embleton was able to substantially confirm the theory of the time, developed by Stokes, Earnshaw, Rankine, Rayleigh, and Taylor, many pieces confirmed for the first time. He also found discrepancies regarding the attenuation coefficients regarding the first and second half cycle of each pulse as well as a shortening of a pulse at reflection from a closed end. These findings are still worthy of further investigation. The results of reflection of pulses from the open end of a tube, forming a pressure release boundary condition, still are very interesting to the present author. A pulse propagating toward the open end steepens, but the reflected wave “unsteepens” which is a unique nonlinear acoustic phenomenon. [Work supported by the Penn State College of Engineering and its United Technologies Corporation Professorship.]

Reconstruction of the Acoustic Field Radiated Directly from a Vibrating Structure Located Near a Pressure-Release Boundary

Near-field acoustical holography is a powerful tool for reconstructing the three-dimensional acoustic field radiated from a vibrating structure located in free space, but it is not applicable when the source is in a half space bounded by a reflecting boundary. This paper develops a method based on half-space spherical wave function expansion for reconstructing the acoustic field radiated directly from a source located near a pressure-release boundary. First, the series of half-space spherical wave basis functions satisfying the pressure-release boundary condition is formulated. Then the acoustic field in a half space is modeled using an expansion in this basis. The expansion coef?cients are determined by solving an overdetermined linear system of equations, obtained by matching this expansion to the measured half-space acoustic pressures. The pressures radiated directly from the source can finally be reconstructed using the free-space spherical wave function expansion with the obtained expansion coefficients. Numerical simulation examples of a vibrating plate located in water near a pressure-release boundary are demonstrated to validate the proposed method. The effects of various parameters, such as the acoustic frequency, the distance between the source and boundary, and the orientation of the source surface, on the reconstruction accuracy are examined.

A new solution for vibroacoustic analysis of two-dimensional cylindrical shells partially liquid-filled or partially submerged in fluid

In this study, a semi-analytical method is proposed to study the vibration and acoustic radiation of a two-dimensional cylindrical shell partially coupled with internal fluid. Different from the conventional vibroacoustic model of cylindrical shells, two different coordinates are established to describe the motion of the shell and the internal fluid respectively. One coordinate system is established on the free surface, where the sound pressure release boundary condition is satisfied automatically by constructing a sine series expression of sound pressure. Meanwhile, with the center of the shell’s cross section as the origin, the other is established to describe the motion of the shell. Then the Galerkin method and the coordinate transformation technique are adopted to deal with the velocity continuous condition on the partially coupled interface. Finally, the vibroacoustic response can be solved theoretically. The applicability and reliability of the present method are validated by comparison with the finite element method via the free vibration, forced vibration and sound radiation cases. The present method is also extended to deal with the vibration and sound radiation of shell coupled with external fluid successfully. Generally speaking, this study proposes a new method to the solution of the vibroacoustic behaviours of an elastic cylindrical shell partially coupled with fluids theoretically, with a wide application scope and good accuracy.

Direct estimation and experimental validation of the acoustic source characteristics of turbocharged diesel engine exhaust system

The present study aims at application of the novel direct method developed by Kumar and Munjal for naturally aspirated engines [Applied Acoustics 135, 70–84 (2018)] to the turbocharged engine’s exhaust system for estimation of acoustic source characteristics. In this paper, transfer matrices of the turbine and compressor of an automotive turbocharger are derived. Mass continuity across the turbine and compressor along with the respective static pressure ratios are used for deriving their transfer matrices. The acoustic source characteristics of each cylinder for the source-load junction downstream of the exhaust valve/s is computed assuming that the cylinder discharges to the ideal pressure release boundary condition with the constant pressure being product of the turbine pressure ratio and mean pressure in the exhaust pipe downstream of the turbine. Source characteristics of each cylinder are used to estimate the source characteristics downstream of the exhaust manifold. Using the derived transfer matrix of the turbine, the source characteristics are further estimated at the source-load junction downstream of the turbine. Finally, the estimated source characteristics of the engine downstream of the turbine are used to predict the unmuffled sound pressure level (SPL) spectrum which is shown to compare reasonably well with the experimentally measured values. Importance of acoustic modelling of the turbine is highlighted by comparing the SPL spectra predicted using the estimated source characteristics with and without the turbine. The estimated source characteristics and unmuffled approximate SPL spectrum of a turbocharged engine being thus known at the engine’s inception stage, the present study can be used for the integration of design and analysis of the engine and muffler.

Acoustic focusing via guided waves in bianisotropic fluid layers

Willis fluids display a complex, effective dynamic response stemming from microstructural asymmetry, long range order, and/or time-varying material properties. These materials are characterized by constitutive relations that couple the pressure and momentum density to both the particle velocity and the volume strain. This coupling has been shown to be analogous to electromagnetic bianisotropic media, which exhibit coupled electric and magnetic fields [Phys. Rev. B 96, 104303 (2017)]. In a recent study, an acoustic lens composed of an array of Willis fluid layers was demonstrated, where the pressure phase imparted by each array element was tuned by varying the direction of Willis polarization and was determined using finite element analysis. In this work, we report on analytical solutions for guided waves in layers of fluid displaying Willis coupling and derive dispersion relations for layers with rigid and pressure-release boundary conditions. Special cases highlighting the effect of Willis polarization direction are presented via numerical examples. The acoustic lens is revisited with array phases obtained from analytical solutions, eliminating the need for finite element methods. Lenses designed for focusing and steering incident pressure waves are demonstrated using fully analytical models. [Work supported by NSF, ONR, and ARL:UT.]

An Analytical Solution for Free Flexural Vibration of a Thin Cylindrical Shell Submerged in Acoustic Half-Space Bounded by a Free Surface

An analytical solution is proposed for the free flexural vibration of a finite cylindrical shell submerged in half-space bounded by a free surface in the low frequency range. The motion of the shell is described by the Flügge shell theory and the fluid surrounding the shell is assumed to be an acoustic media. The free surface effect is considered by satisfying the pressure release boundary condition. The accuracy of the present method is verified through comparison with the finite element solution. To throw light on the influence mechanism of free surface on the coupled modal frequencies, a modal added mass is introduced and calculated. Numerical results show that when the shell is close to the free surface, the presence of free surface will make a negative contribution to the modal added mass and finally result in the corresponding increase of the coupled modal frequencies. But the free surface effect will decrease when the immersion depth of the cylindrical shell increases. Finally, the free surface effect can be neglected if the immersion depth is higher than four times the shell radius. This conclusion is helpful to select proper test environment for an experiment on the dynamic characteristics of submerged cylindrical shells.

Semi-analytical research on acoustic-structure coupling calculation of partially submerged cylindrical shell

Vibroacoustic analysis of a partially submerged cylindrical shell-liquid coupling system is a typical acoustic-structure interaction problem in an acoustic half-space. Generally, the vibration and acoustic solutions of this problem almost depend on numerical method in previous studies. However, whether from the aspect of verifying the numerical methods or from the aspect of revealing the vibroacoustic mechanism of the acoustic-structure interaction system, the development of an analytical or semi-analytical method is indispensable. In this study, a semi-analytical method is proposed to address the vibroacoustic response of a horizontal cylindrical shell partially immersed in water The acoustic coordinate system is established on the free surface, and the sine series function is introduced to express the sound pressure to meet the pressure release boundary condition on the free surface automatically. Then based on the two-dimensional Flgge shell theory, the motion equation of the shell-liquid coupling system is established with using the center of the shell as the origin of the coordinate system. By using the Galerkin method, the velocity continuous condition on the fluid-structure interface is established. Then the relation matrix between the acoustic pressure amplitude and the shell displacement amplitude is derived after converting the sound pressure from different coordinate systems into that in the same coordinate system by using the geometry relationship of two coordinate systems. Then the vibration and sound radiation of this coupling system are predicted by solving the coupled matrix equations, and the coupled free vibration can also be addressed by solving the characteristic equation after setting the determinant of the coefficient matrix to be zero. To verify the accuracy and reliability of the present method, the coupled natural frequencies, the vibration response and the distribution of the sound pressure obtained from the present method are compared with those obtained from the finite element method, showing that they are in good agreement with each other. Then the shapes of the first four order modes for the coupled system are presented and compared with those of the system submerged in unbounded fluid field, and the couple between different modes, and the couple between the symmetric and antisymmetric modes are observed due to the effect of the free surface. The characteristics of the root mean square velocity of the shell with different immersed depths are discussed, which reveals that the peaks of response curves shift to the lower frequencies with increasing immersed depth. The characteristics of far field sound pressure directivity are presented and explained by the image method in detail. This study provides a novel method to analytically predict the vibroacoustic response of an elastic structure partially coupling with the fluid field when bounded in a sound half space.

Acoustic wave propagation in a sensor port: Experimental measurements and analytical model predictions

In this paper, results from an experimental study on pressure fluctuations in a sensor port are presented. The sensor ports are attached to a large flat plate over which plane acoustic waves propagate. In addition, an analytical model is developed by solving the acoustic wave equation subject to rigid and pressure release boundary conditions. The experimental data is used as input to the model from one end of the sensor port in order to predict the pressure fluctuations at the other end of the sensor port. Two sensor ports are used; a short and a long one. The short sensor port resonates at 1000Hz, whereas the long sensor port resonates at 245Hz. The acoustic source generates harmonic and random plane acoustic waves, which can be modeled easily. Pressure fluctuation measurements are made both on the plate surface and inside the sensor ports for various plane wave inputs; harmonic and random. Power spectra, sound pressure levels and transfer functions are then obtained at the various microphone locations. The experimental results for the short sensor port show an amplification factor of 45 between the inlet and the rigid end of the sensor port at a frequency of 1000Hz. For the long sensor port, the amplification is nearly 50 at a frequency of 245Hz, with lower amplifications at higher frequencies. Using the measured power spectrum at the rigid end of the sensor port for a random plane wave input, the absorption coefficient is calculated using the half-power law. Using this absorption coefficient in our analytical model, predictions were made at both ends of the sensor port using experimental data as input. The model predictions for both sensor ports are in good agreement with experimental data.

Development of a standing wave apparatus for calibrating acoustic vector sensors and hydrophones.

An apparatus was developed to calibrate acoustic hydrophones and vector sensors between 25 and 2000 Hz. A standing wave field is established inside a vertically oriented, water-filled, elastic-walled waveguide by a piston velocity source at the bottom and a pressure-release boundary condition at the air/water interface. A computer-controlled linear positioning system allows a device under test to be precisely located in the water column while the acoustic response is measured. Some of the challenges of calibrating hydrophones and vector sensors in such an apparatus are discussed, including designing the waveguide to mitigate dispersion, understanding the impact of waveguide structural resonances on the acoustic field, and developing algorithms to post-process calibration measurement data performed in a standing wave field. Data from waveguide characterization experiments and calibration measurements are presented and calibration uncertainty is reported.

Numerically exact three-dimensional propagation

Solving propagation in a three-dimensional environment is one of the most challenging problems in computational physics. However, for cases where the environment possesses some kind of symmetry be it rotational or translational, the propagation problem can be simplified considerably. It can be shown that in problems where the environment is translationally invariant, the 3D wave equation can be Fourier transformed along the direction of translational symmetry to reduce it to a 2D equation for each spectral component. The 3D solution can be obtained by solving the 2D wave equation for each spectral component and performing the inverse Fourier transform. In this paper, we use the above technique to compute propagation in an ideal and a penetrable wedge. For the ideal wedge, the pressure-release boundary condition is applied to both boundaries and for the penetrable wedge, the pressure-release boundary condition is applied to the horizontal surface and the continuity of pressure and normal velocity is imposed on the sloped interface. To obtain a numerically exact solution, we use the virtual source technique to solve the 2D problem for each spectral component.

Development of a standing-wave calibration apparatus for acoustic vector sensors

An apparatus was developed to calibrate acoustic hydrophones and vector sensors between 25 and 2000 Hz. A standing-wave field is established inside a vertically oriented, water-filled, elastic-walled waveguide by a piston velocity source at the bottom and a pressure-release boundary condition at the air/water interface. A computer-controlled linear positioning system allows reference hydrophones and/or the device under test to be scanned through the water column while their acoustic response is measured. Some of the challenges of calibrating vector sensors in such an apparatus are discussed, including designing the waveguide to mitigate dispersion, mechanically isolating the apparatus from floor vibrations, understanding the impact of waveguide structural resonances on the acoustic field, and developing processing algorithms to calibrate vector sensors in a standing-wave field. Data from waveguide characterization experiments and calibration measurements will be presented. [Work supported by ARL IR&D.]

Paraxial Coupling of Propagating Modes in Three-Dimensional Waveguides with Random Boundaries

We analyze long range wave propagation in three-dimensional random waveguides. The waves are trapped by top and bottom boundaries, but the medium is unbounded in the two remaining directions. We consider scalar waves, and motivated by applications in underwater acoustics, we take a pressure release boundary condition at the top surface and a rigid bottom boundary. The wave speed in the waveguide is known, but the top boundary has small random fluctuations that cause significant cumulative scattering of the waves over long distances of propagation. To quantify the scattering effects, we study the evolution of the random amplitudes of the waveguide modes. We obtain that in the long range limit they satisfy a system of paraxial equations driven by a Brownian field. We use this system to estimate three important mode-dependent scales: the scattering mean free path, the cross-range decoherence length, and the decoherence frequency. Understanding these scales is important in imaging and communication problems, because they encode the cumulative scattering effects in the wave field measured by remote sensors. As an application of the theory, we analyze time reversal and coherent interferometric imaging in strong cumulative scattering regimes.

Three-dimensional surface scattering using a parabolic equation model

Calculations of acoustic rough surface scattering have previously been performed by numerous researchers using parabolic equation models in two dimensions. Approaches have varied from exact solutions that re-map the rough interface to the pressure release boundary condition through a generalized method of images, approximate methods based on a perturbation approach valid for small slopes, and extended domain approaches that define the rough surface as a boundary between water and air with an additional absorbing boundary above the rough surface. In this work, we examine extensions of these approaches to the three-dimensional propagation. It is shown that the exact solution based on a generalized method of images does not yield realizable expressions for implementation. The other approaches are relatively easily adapted to three-dimensions, and some simple test cases involving Bragg scatter are analyzed for accuracy. Implications for other rough surface scattering model studies are also discussed.

Wave Transmission through Random Layering with Pressure Release Boundary Conditions

This paper considers the statistical properties of the waves generated by a point source in the subsurface and transmitted towards the surface through a randomly layered medium. The problem is analyzed in a regime of separation of scales and with pressure release boundary conditions at the surface. Using a probabilistic representation of the spectral density of the waves received at the surface, the transmitted intensity is analyzed. Conserved quantities specific to the pressure release boundary conditions are evaluated and the power delay spread is computed. In particular a waveguiding effect that is produced by the reflections from the surface and from the random medium is identified and analyzed. This effect is different from the propagation in a standard waveguide from the point of view of the decay of the intensity.

Cross Correlation and Deconvolution of Noise Signals in Randomly Layered Media

It is known that cross correlation of waves generated by noise sources, propagating in an unknown medium and recorded by a sensor array, can provide information about the medium. In this paper the medium is a three-dimensional small-scale randomly layered medium with slow macroscopic variations. The main objective here is to set forth a framework for analysis of cross correlations of waves generated by noise sources and propagating in such a medium and, moreover, to use this framework to design estimators for macroscale medium features. The noise sources are located at the bottom of a random medium slab and generate a random wave field that is scattered by the rapid random fluctuations of the medium and then recorded at the surface. Taking into account the pressure release boundary conditions at the surface, this situation corresponds to the so-called daylight configuration. The analysis is carried out in the asymptotic framework where the typical wavelength is small compared to the scale of the macroscopic variations of the background medium and large compared to the decoherence length of the microscopic random fluctuations of the medium. It is shown that the cross correlation of the waves recorded at the surface contains statistically stable information about the macroscopic background medium.

Numerical difficulties with boundary element solutions of interior acoustic problems

Although boundary element methods have been applied to interior problems for many years, the numerical difficulties that can occur have not been thoroughly explored. Various authors have reported low-frequency breakdowns and artificial damping due to discretization errors. In this paper, it is shown through a simple example problem that the numerical difficulties depend on the solution formulation. When the boundary conditions are imposed directly, the solution suffers from artificial damping, which may potentially lead to erroneous predictions when boundary element methods are used to evaluate the performance of damping materials. This difficulty can be alleviated by first computing an impedance or admittance matrix, and then using its reactive component to derive the solution for the acoustic field. Numerical computations are used to demonstrate that this technique eliminates artificial damping, but does not correct errors in the reactive components of the impedance or admittance matrices, which then causes nonexistence and nonuniqueness difficulties at the interior resonance frequencies for hard-wall and pressure release boundary conditions, respectively. It is shown that the admittance formulation is better suited to boundary element computations for interior problems because the resonance frequencies for pressure release boundary conditions do not begin until the smallest dimension of the boundary surface is at least one half the acoustic wavelength. Aside from producing much more accurate predictions, the admittance matrix is also much easier to interpolate at low frequencies due to the absence of interior resonances. For the example problem considered, only the formulation using the reactive component of the admittance matrix produces accurate solutions as long as the surface element discretization satisfies the standard six-element-per-wavelength rule.

Sound scattering by a compact circular pore

The aim of this paper is to study the three-dimensional scattering of an oblique wave incident on a flanged circular compact pore of finite depth. The multipole structure with the scattering is resolved by the method of matched asymptotic expansion, where we assume smallness of ε = k * a * , the product of the incident wavenumber k * and the pore radius a * . Two distinguished cases are solved: the rigid boundary condition and the pressure-release boundary condition. The study presents by far the most complete solutions to these problems, with the outer solution up to O ( ε 5 ) and the inner solution up to O ( ε 2 ) . In particular, the sophisticated interplay between the pore depth and the incident angle is revealed in the different orders of solution. It is shown that the leading order of the outer wave field for both cases is O ( ε 3 ) . For the rigid boundaries, there is one dipole dependent on the incident angle and one monopole. Interestingly, the monopole arises from the second-order interaction of the pore volume and the small but non-negligible compressibility in the inner field. This is one of the few examples analytically solvable to demonstrate this property. On the other hand, only one dipole is found for the pressure-release boundary. The next order in the outer solutions for both types of the boundary conditions is of O ( ε 5 ) and is shown to contain quadrupoles and octupoles. The multipole structures for both types of boundaries are tabulated, explicitly with the effects of the incident angle and the pore depth.

The design and modeling of a lateral acoustic particle manipulator exhibiting quarter-wave operation

Prior work [Petersson et al., 2004] has demonstrated the operation of a half-wave acoustic particle manipulator, whose forces act in the plane of a silicon substrate. In a half-wave device particles are directed to the centre of a channel. Devices acting in plane have manufacturing advantages, and lend themselves to many microfluidic applications. We demonstrate, for the first time, such a device with a quarter-wave mode that is able to manipulate particles to the side of a channel in addition to a near half wave mode. The design utilises resonant “islands” to create the necessary pressure release boundary condition. The device is conventionally milled in brass, permitting cheaper and quicker fabrication than in silicon. Finite element modelling is presented to elucidate the operation of both half and quarter-wave devices. In contrast to essentially one-dimensional planar devices, the two-dimensional distribution of the velocity and pressure fields result in particles being constrained to a line within the channel.

Numerical difficulties with boundary element solutions of interior acoustic problems

Although boundary element methods have been applied to interior problems for many years, the numerical difficulties that can occur have not been thoroughly explored. Various authors have reported artificial damping due to acoustic radiation and low‐frequency breakdowns. In this presentation, it is shown through a simple example problem that the numerical difficulties depend on the solution formulation. When the boundary conditions are imposed directly, the solution suffers from artificial damping due to sound radiation to the farfield. This difficulty can be alleviated by first computing an impedance matrix or admittance matrix, and then using its reactive component to derive the solution for the acoustic field. Numerical computations are used to demonstrate that this technique eliminates artificial damping, but does not correct errors in the reactive components of the impedance or admittance matrices, which then cause nonexistence/nonuniqueness difficulties at the interior resonance frequencies for hard‐wall or pressure release boundary conditions, respectively. It is shown that the admittance formulation is better suited to boundary element computations for interior problems because the resonance frequencies for pressure release boundary conditions do not begin until the smallest dimension of the boundary surface is at least one half of the acoustic wavelength.

Comparing numerical and analytical solutions for squeeze-film levitation force

In this paper, a two-dimensional numerical study on the levitation force induced by pressure radiation in a gas squeeze film is carried out. In particular, the validity of the pressure release boundary condition and the isothermal assumptions are examined by a CFD scheme. The results are compared to a one-dimensional analytical solution, leading to findings on the role and accuracy of commonly used boundary conditions. It appears that when taking into consideration the energy leakage near the edges of the levitated object, the levitation force due to the squeeze film could be smaller by up to 50% in comparison with the analytical, one-dimensional solution that assumes pressure release conditions.