A steerable non-paraxial Gaussian beam expansion for a steerable parametric array loudspeaker.

A steerable parametric array loudspeaker (PAL) aims to steer a highly directional audio beam without the need to mechanically rotate the source. The Gaussian beam expansion (GBE) method is often used to model PALs because it is a computationally efficient approach, however the method relies on a paraxial approximation that can result in significant inaccuracies at large steering angles. To address this limitation, a steerable non-paraxial GBE is proposed in this article, where the mainlobe of the steered ultrasonic beam is included in the calculation by rotating the coordinate system. A non-paraxial approximation is then used to improve the accuracy of the method when integrating the virtual audio sources. The numerical results obtained using the proposed method are compared against those using the conventional GBE, as well as an exact solution. For a typical configuration, it is shown that for a conventional GBE the prediction error can be more than 30 dB at large angles, whereas the proposed method reduces this to less than 1 dB. The advantage of the proposed method is more significant at large steering angles, low audio frequencies, and those locations outside of the paraxial region. This improvement in performance is achieved with a computational cost that remains the same as the conventional GBE.

A spherical wave expansion for a steerable parametric array loudspeaker using Zernike polynomials.

A steerable parametric array loudspeaker (PAL) can electronically steer highly directional audio beams in the desired direction. The challenge of modelling a steerable PAL is to obtain the audio sound pressure in both near and far fields with a low computational load. To address this issue, an extension of the spherical wave expansion is proposed in this paper. The steerable velocity profile on the radiation surface is expanded as Zernike polynomials which are an orthogonal and form a complete set over a unit circle. An expression for the radiated audio sound is then obtained using a superposition of Zernike modes. Compared to the existing methods, the proposed expansion is computationally efficient and provides a rigorous transformation of the quasilinear solution of the Westervelt equation without paraxial approximations. The proposed expansion is further extended to accommodate local effects by using an algebraic correction to the Westervelt equation. Numerical results for steering single and dual beams are presented and discussed. It is shown that the single beam can be steered in the desired direction in both near and far fields. However, dual beams cannot be well separated in the near field, which cannot be predicted by the existing far field models.

Scattering by a rigid sphere of audio sound generated by a parametric array loudspeaker.

This work investigates the scattering by a rigid sphere of audio sound generated by a parametric array loudspeaker (pal). A computationally efficient method utilizing a spherical harmonic expansion is developed to calculate the quasilinear solution of audio sound fields based on both Kuznetsov and Westervelt equations. The accuracy of using the Westervelt equation is examined, and the rigid sphere scattering effects are simulated with the proposed method. It is found the results obtained using the Westervelt equation are inaccurate near the sphere at low frequencies. Contrary to conventional loudspeakers, the directivity of the audio sound generated by a pal severely deteriorates behind a sphere, as the ultrasounds maintaining the directivity of the audio sound are almost completely blocked by the sphere. Instead, the ultrasounds are reflected and generate audio sound on the front side of the sphere. It means that a listener in front of the pal will hear the audio sound scattered back after introducing the sphere as if it is reflected by the sphere. The experiment results are also presented to validate the numerical results.

Quiet zone generation in an acoustic free field using multiple parametric array loudspeakers.

This paper investigates the feasibility of remotely generating a quiet zone in an acoustic free field using multiple parametric array loudspeakers (PALs). A primary sound field is simulated using point monopoles located randomly in a two-dimensional plane, or three-dimensional (3D) space, whereas the secondary sound field is generated by multiple PALs uniformly distributed around the circumference of a circle sitting on the same plane as the primary sources, or on the surface of a sphere for 3D space. A quiet zone size is defined as the diameter of the maximal circular zone within which the noise reduction is greater than 10 dB. The size of this quiet zone is found to be proportional to 0.19λN for N secondary sources with a wavelength λ when the primary and secondary sources are in the same plane, whereas it is found to be 0.55λN1/2 for the 3D case. The size of the quiet zones generated by PALs is similar to that observed with traditional omnidirectional loudspeakers; however, the effects of using PALs on the sound field outside the target zone is much smaller due to their sharp radiation directivity and slow decay rate along the propagation distance. Experimental results are also presented to validate these numerical simulations.

A cylindrical expansion of the audio sound for a steerable parametric array loudspeaker.

In this work, a cylindrical expansion for the audio sound generated by a steerable baffled parametric array loudspeaker (PAL) based on the phased array technique is derived from the Westervelt equation. The expansion is a series of twofold summations with uncoupled angular and radial components in the cylindrical coordinate system. The angular component is determined by the trigonometric functions, and the radial component is an integral containing the Bessel functions and an arbitrary excitation velocity profile. The numerical results for a typical steerable PAL are presented and compared to those obtained using the convolution model. It is found that the prediction of the audio sound using the proposed cylindrical expansion improves the agreement with the experimental results when compared to the existing models. This is because no further approximations are required in the cylindrical expansion of the quasilinear solution of the Westervelt equation, whereas the complex near field nonlinear interactions between the ultrasonic waves cannot be correctly captured in a convolution model. The proposed cylindrical expansion does, therefore, provide an alternative approach to modeling a phased array PAL and high accuracy with a relatively low computational cost.

The near field, Westervelt far field, and inverse-law far field of the audio sound generated by parametric array loudspeakers.

The near and far fields of traditional loudspeakers are differentiated by whether the sound pressure amplitude is inversely proportional to the propagating distance. However, the audio sound field generated by a parametric array loudspeaker (PAL) is more complicated, and in this article it is proposed to be divided into three regions: near field, Westervelt far field, and inverse-law far field. In the near field, the audio sound experiences strong local effects and an efficient quasilinear solution is presented. In the Westervelt far field, local effects are negligible so that the Westervelt equation is used, and in the inverse-law far field, a simpler solution is adopted. It is found that the boundary between the near and Westervelt far fields for audio sound lies at approximately a2/λ - λ/4, where a is transducer radius and λ is ultrasonic wavelength. At large transducer radii and high ultrasonic frequencies, the boundary moves close to the PAL and can be estimated by a closed-form formula. The inverse-law holds for audio sound in the inverse-law far field and is more than 10 meters away from the PAL in most cases. With the proposed classification, it is convenient to apply appropriate prediction models to different regions.

Atmospheric sound propagation in a moving fluid above an impedance plane: Application of the semi-analytic finite element method.

This article uses a normal mode approach to predict atmospheric sound propagation over a locally reacting impedance plane. The semi-analytic finite element method is used to compute the normal modes, which enables the exact governing wave equation for a moving fluid to be solved in two dimensions. A locally reacting surface is added using the general Ingard-Myers boundary condition, and the transmission loss is obtained for cylindrical and spherical spreading for range independent problems. The approach developed in this article will, in principle, converge toward the exact solution and so has the potential to provide benchmark predictions for complex, range independent, outdoor sound propagation problems. Predictions are shown to provide good agreement with benchmark solutions available in the literature, including those with a logarithmic wind velocity profile. Results are also reported for a combination of a logarithmic wind velocity profile and a temperature inversion for ranges of up to 5 km. Finally, transmission loss predictions are reported for a relatively wide frequency range, and it is concluded that finite elements can provide an alternative approach for computing range independent outdoor sound propagation that converges to the exact solution.

Atmospheric sound propagation in a stratified moving media: Application of the semi analytic finite element method.

This article presents a finite element based solution of the exact governing wave equation for a stratified inhomogeneous moving media. The model is applied to a two dimensional range independent problem in outdoor sound propagation in which the ground is treated as perfectly reflecting. The sound pressure field is expanded as a sum over eigenmodes propagating in the range direction, and the semi analytic finite element method is used to solve the governing eigenequation. This delivers faster solution times when compared to traditional finite element based methods while simultaneously accommodating continuous variations in fluid properties in the vertical direction. In principle, the method converges toward the exact solution and so delivers a benchmark method for range independent problems. The method is shown to provide excellent agreement with analytic solutions, and good convergence is demonstrated for more complex problems, including temperature inversions and logarithmic profiles for wind velocity. Finally, qualitative comparisons are made against infrasound predictions, including those obtained using wide angle parabolic equations. The method is shown to provide a focussed image of the sound pressure field over large distances, as well as to reproduce multiple turning points and ground interactions.

Reflection of audio sounds generated by a parametric array loudspeaker.

The reflection of audio sounds generated by a parametric array loudspeaker (PAL) is investigated in this paper. The image source method and the non-paraxial PAL radiation model under the quasilinear approximation are used to calculate the reflected audio sound from an infinitely large surface with an arbitrary incident angle. The effects of the surface absorption in the ultrasound frequency range are studied, and the simulation and experiment results show that the reflection behavior of audio sounds generated by a PAL is different from those generated by traditional audio sources. The reason is that the reflected sound generated by the PAL consists of the reflection of audio sounds generated by incident ultrasounds and the audio sounds generated by the reflected ultrasound, and it is the latter that determines the directivity of the reflected audio sound.

Insertion loss of a thin partition for audio sounds generated by a parametric array loudspeaker.

Unlike the audio sound generated by traditional sources, the directivity of that generated by a parametric array loudspeaker (pal) deteriorates significantly after passing through a thin partition. To study this phenomenon, the pal radiation model based on the Westervelt equation, and the plane wave expansion method are used to calculate the sound fields behind a sheet of aluminum foil and a porous material blanket under the quasi-linear assumption, where the paraxial approximation is assumed only for ultrasonic waves. The audio sounds generated by a point monopole and a traditional directional source are presented for comparison. Both simulation and experiment results show that the transmitted sound from a pal behind the thin partition is small and less focused on the radiation axis because most of the ultrasounds forming the directivity of the pal is blocked by the thin partition which has little effect on the traditional audio sources.

Mode matching in axisymmetric fluid-filled pipes: Scattering by a flange.

Long range ultrasonic testing of pipelines sends an ultrasonic wave along a pipe wall and then detects scattering from defects present. It is well known that scattering by pipe fixtures and fittings, such as a flange, can cause distortion and interfere with the ability to identify defects. This article develops a theoretical model to investigate scattering from a flange in a fluid-filled pipe with elastic walls. Mode matching is used as this is a computationally efficient way to examine long lengths of pipe and for enforcing the appropriate axial continuity conditions over area discontinuities. A recent article presented a mode matching approach for a similar problem, and it is demonstrated here that a re-casting of the equations is necessary to ensure all of the appropriate matching conditions are enforced. Mode matching predictions are also compared with an alternative point collocation approach in order to provide an independent benchmark. Excellent agreement between mode matching and point collocation is demonstrated, and reflection and transmission coefficients are generated in order to show the resonant behaviour of a flange and illustrate that its influence is significant and strongly frequency dependent.

A spherical expansion for audio sounds generated by a circular parametric array loudspeaker.

The existing non-paraxial expression of audio sounds generated by a parametric array loudspeaker (pal) is hard to calculate due to the fivefold integral in it. A rigorous solution of the Westervelt equation under the quasilinear approximation is developed in this paper for circular PALs by using the spherical harmonics expansion, which simplifies the expression into a series of threefold summations with uncoupled angular and radial components. The angular component is determined by Legendre polynomials and the radial one is an integral involving spherical Bessel functions, which converge rapidly. Compared to the direct integration over the whole space, the spherical expansion is rigorous, exact, and can be calculated efficiently. The simulations show the proposed expression can obtain the same accurate results with a speed of at least 15 times faster than the existing one.

A non-paraxial model for the audio sound behind a non-baffled parametric array loudspeaker.

It has been reported that audible sounds can be heard behind a parametric array loudspeaker in free field, which cannot be predicted by existing models. A non-paraxial model is developed in this paper for the finite size and disk-shaped parametric source based on quasilinear approximation and disk scattering theory. The sounds on both front and back sides are calculated numerically and compared with the existing non-paraxial model for the parametric source installed in an infinitely large baffle. Both simulation and experiment results show that audible sound exists on the back side. The mechanism of the phenomenon is explored.

On the scattering of torsional waves from axisymmetric defects in buried pipelines.

This article develops a numerical model suitable for analysing elastic wave scattering in buried pipelines. The model is based on a previous so-called hybrid approach, where a nominally infinite length of pipe is split up into uniform and non-uniform regions. The key challenge for buried structures is in enforcing the appropriate boundary conditions in both the axial and radial directions, which must encompass the entire length of the structure, as well as the surrounding material. Accordingly, the focus of this article is on developing a model suitable for accurately applying these boundary conditions, and so the analysis is restricted here to the study of axisymmetric defects and to an incident sound field that consists of the fundamental torsional mode only. It is shown that this problem may be addressed in a numerically efficient way provided one carefully choses a perfectly matched layer for the surrounding material, and then integrates over this layer using a complex co-ordinate stretching function. This enables the use of mode matching to deliver a convergent system of equations that enforce the appropriate axial and radial boundary conditions.

On the scattering of elastic waves from a non-axisymmetric defect in a coated pipe.

Viscoelastic coatings are often used to protect pipelines in the oil and gas industry. However, over time defects and areas of corrosion often form in these pipelines and so it is desirable to monitor the structural integrity of these coated pipes using techniques similar to those used on uncoated pipelines. A common approach is to use ultrasonic guided waves that work on the pulse-echo principle; however, the energy in the guided waves can be heavily attenuated by the coating and so significantly reduce the effective range of these techniques. Accordingly, it is desirable to develop a better understanding of how these waves propagate in coated pipes with a view to optimising test methodologies, and so this article uses a hybrid SAFE-finite element approach to model scattering from non-axisymmetric defects in coated pipes. Predictions are generated in the time and frequency domain and it is shown that the longitudinal family of modes is likely to have a longer range in coated pipes when compared to torsional modes. Moreover, it is observed that the energy velocity of modes in a coated pipe is very similar to the group velocity of equivalent modes in uncoated pipes. It is also observed that the coating does not induce any additional mode conversion over and above that seen for an uncoated pipe when an incident wave is scattered by a defect. Accordingly, it is shown that when studying coated pipes one need account only for the attenuation imparted by the coating so that one may normally neglect the effect of coating on modal dispersion and scattering.

A three dimensional investigation into the acoustic performance of dissipative splitter silencers.

Splitter silencers are found in ventilation and gas turbine systems and consist of parallel baffles of porous material placed within a duct so that they split the mean gas flow. Theoretical investigations into dissipative splitter silencers have generally been limited to two dimensions and this limits the analysis to finding the silencer eigenmodes or, for a finite length silencer, to rectangular baffles only. In this article a numerical point collocation approach is used to extend theoretical predictions to three dimensions. This facilitates the analysis of more complex silencer designs such as "bar" silencers and theoretical predictions are validated by comparison with experimental measurements. The insertion loss of different silencer designs is evaluated and the performance of a bar silencer is compared to traditional designs for rectangular and circular ducts. It is shown that a bar silencer with a volume of material identical to an equivalent parallel baffle design delivers a significant improvement in insertion loss at higher frequencies, although this is at the expense of a small reduction in performance at low frequencies. It is also shown that under most circumstances it is possible to get good agreement between prediction and experiment even for relatively large Helmholtz numbers.

Measurement of complex acoustic intensity in an acoustic waveguide.

Acoustic intensity is normally treated as a real quantity, but in recent years, many articles have appeared in which intensity is treated as a complex quantity where the real (active) part is related to local mean energy flow and the imaginary (reactive) part to local oscillatory transport of energy. This offers the potential to recover additional information about a sound field and then to relate this to the properties of the sound source and the environment that surrounds it. However, this approach is applicable only to multi-modal sound fields, which places significant demands on the accuracy of the intensity measurements. Accordingly, this article investigates the accuracy of complex intensity measurements obtained using a tri-axial Microflown intensity probe by comparing measurement and prediction for sound propagation in an open flanged pipe. Under plane wave conditions, comparison between prediction and experiment reveals good agreement, but when a higher order mode is present, the reactive intensity field becomes complicated and agreement is less successful. It is concluded that the potential application of complex intensity as a diagnostic tool is limited by difficulties in measuring reactive intensity in complex sound fields when using current state of the art acoustic instrumentation.

A hybrid finite element approach to modeling sound radiation from circular and rectangular ducts.

A numerical model based on a hybrid finite element method is developed that seeks to join sound pressure fields in interior and exterior regions. The hybrid method is applied to the analysis of sound radiation from open pipes, or ducts, and uses mode matching to couple a finite element discretization of the region surrounding the open end of the duct to wave based modal expansions for adjoining interior and exterior regions. The hybrid method facilitates the analysis of ducts of arbitrary but uniform cross section as well the study of conical flanges and here a modal expansion based on spherical harmonics is applied. Predictions are benchmarked against analytic solutions for the limiting cases of flanged and unflanged circular ducts and excellent agreement between the two methods is observed. Predictions are also presented for flanged and unflanged rectangular ducts, and because the hybrid method retains the sparse banded and symmetric matrices of the traditional finite element method, it is shown that predictions can be obtained within an acceptable time frame even for a three dimensional problem.

Modeling sound propagation in acoustic waveguides using a hybrid numerical method.

Sound propagation in an acoustic waveguide is examined using a hybrid numerical technique. Here, the waveguide is assumed to be infinite in length with an arbitrary but uniform cross section. Placed centrally within the guide is a short component section with an irregular nonuniform shape. The hybrid method utilizes a wave based modal solution for a uniform section of the guide and, using either a mode matching or point collocation approach, matches this to a standard finite element based solution for the component section. Thus, one needs only to generate a transverse finite element mesh in uniform sections of the waveguide and this significantly reduces the number of degrees of freedom required. Moreover, utilizing a wave based solution removes the need to numerically enforce a nonreflecting boundary condition at infinity using a necessarily finite mesh, which is often encountered in studies that use only the standard finite element method. Accordingly, the component transmission loss may readily be computed and predictions are presented here for three examples: an expansion chamber, a converging-diverging duct, and a circular cylinder. Good agreement with analytic models is observed, and transmission loss predictions are also presented for multimode incident and transmitted sound fields.

Analytic mode matching for a circular dissipative silencer containing mean flow and a perforated pipe.

An analytic mode matching scheme that includes higher order modes is developed for a straight-through circular dissipative silencer. Uniform mean flow is added to the central airway and a concentric perforated screen separates the mean flow from a bulk reacting porous material. Transmission loss predictions are compared with experimental measurements and good agreement is demonstrated for three different silencers. Furthermore, it is demonstrated that, when mean flow is present, the axial kinematic matching condition should equate to that chosen for the radial kinematic boundary condition over the interface between the airway and the material. Accordingly, if the radial matching conditions are continuity of pressure and displacement, then the axial matching conditions should also be continuity of pressure and displacement, rather than pressure and velocity as previously thought. When a perforated screen is present the radial pressure condition changes, but the radial kinematic condition should always remain equivalent to that chosen for the axial kinematic matching condition; here, results indicate that continuity of displacement should be retained when a perforated screen is present.