On the diffraction of acoustic waves by a quarter-plane

This paper follows the work of A.V. Shanin on diffraction by an ideal quarter-plane. Shanin's theory, based on embedding formulae, the acoustic uniqueness theorem and spherical edge Green's functions, leads to three modified Smyshlyaev formulae, which partially solve the far-field problem of scattering of an incident plane wave by a quarter-plane in the Dirichlet case. In this paper, we present similar formulae in the Neumann case, and describe a numerical method allowing a fast computation of the diffraction coefficient using Shanin's third modified Smyshlyaev formula. The method requires knowledge of the eigenvalues of the Laplace-Beltrami operator on the unit sphere with a cut, and we also describe a way of computing these eigenvalues. Numerical results are given for different directions of incident plane wave in the Dirichlet and the Neumann cases, emphasising the superiority of the third modified Smyshlyaev formula over the other two. © 2011 Elsevier B.V.

Precise description of the different far fields encountered in the problem of diffraction of acoustic waves by a quarter-plane

This paper provides a review of important results concerning the Geometrical Theory of Diffraction and Geometrical Optics. It also reviews the properties of the existing solution for the problem of diffraction of a time harmonic plane wave by a half-plane. New mathematical expressions are derived for the wave fields involved in the problem of diffraction of a time harmonic plane wave by a quarter-plane, including the secondary radiated waves. This leads to a precise representation of the diffraction coefficient describing the diffraction occurring at the corner of the quarter-plane. Our results for the secondary radiated waves are an important step towards finding a formula giving the corner diffraction coefficient everywhere. © 2012 The authors.

Modal scattering at an impedance transition in a lined flow duct

An explicit Wiener-Hopf solution is derived to describe the scattering of duct modes at a hard-soft wall impedance transition in a circular duct with uniform mean flow. Specifically, we have a circular duct r = 1, - ∞ < x < ∞ with mean flow Mach number M > 0 and a hard wall along x < 0 and a wall of impedance Z along x > 0. A minimum edge condition at x = 0 requires a continuous wall streamline r = 1 + h(x, t), no more singular than h = Ο(x1/2) for x ↓ 0. A mode, incident from x < 0, scatters at x = 0 into a series of reflected modes and a series of transmitted modes. Of particular interest is the role of a possible instability along the lined wall in combination with the edge singularity. If one of the "upstream" running modes is to be interpreted as a downstream-running instability, we have an extra degree of freedom in the Wiener-Hopf analysis that can be resolved by application of some form of Kutta condition at x = 0, for example a more stringent edge condition where h = Ο(x3/2) at the downstream side. The question of the instability requires an investigation of the modes in the complex frequency plane and therefore depends on the chosen impedance model, since Z = Z (ω) is essentially frequency dependent. The usual causality condition by Briggs and Bers appears to be not applicable here because it requires a temporal growth rate bounded for all real axial wave numbers. The alternative Crighton-Leppington criterion, however, is applicable and confirms that the suspected mode is usually unstable. In general, the effect of this Kutta condition is significant, but it is particularly large for the plane wave at low frequencies and should therefore be easily measurable. For ω → 0, the modulus fends to |R001| → (1 + M)/(1 -M) without and to 1 with Kutta condition, while the end correction tends to ∞ without and to a finite value with Kutta condition. This is exactly the same behaviour as found for reflection at a pipe exit with flow, irrespective if this is uniform or jet flow.

Flexoelectro-optic properties of chiral nematic liquid crystals in the uniform standing helix configuration.

The flexoelectro-optic effect describes the rotation of the optic axis of a short-pitch chiral nematic liquid crystal under the application of an electric field. We investigate the effect in the uniform standing helix, or "Grandjean" configuration. An in-plane electric field is applied. The director profile is determined numerically using a static one-dimensional continuum model with strong surface anchoring. The Berreman method is used to solve for plane-wave solutions to Maxwell's equations, and predict the optical properties of the resulting structure in general cases. By using a chiral nematic with short pitch between crossed polarizers an optical switch may be generated. With no applied field the configuration is nontransmissive at normal incidence, but becomes transmissive with an applied field. For this case, numerical results using the Berreman method are supplemented with an analytic theory and found to be in good agreement. The transmitted intensity as a function of tilt, the contrast ratio, and the tilt required for full intensity modulation are presented. The angular dependence of the transmission is calculated and the isocontrast curves are plotted. For typical material and cell parameters a switching speed of 0.017 ms and contrast ratio of 1500:1 at normal incidence are predicted, at a switch-on tilt of 41.5 degrees. Experimental verification of the analytic and numerical models is provided.

A six sensor method for measuring acoustic properties in ducts.

An accurate description of sound propagation in a duct is important to obtain the sound power radiating from a source in both near and far fields. A technique has been developed and applied to decompose higher-order modes of sound emitted into a duct. Traditional experiments and theory based on two-sensor methods are limited to the plane-wave contribution to the sound field at low frequency. Due to the increase in independent measurements required, a computational method has been developed to simulate sensitivities of real measurements (e.g., noise) and optimize the set-up. An experimental rig has been constructed to decompose the first two modes using six independent measurements from surface, flush-mounted microphones. Experiments were initially performed using a loudspeaker as the source for validation. Subsequently, the sound emitted by a mixed-flow fan has been investigated and compared to measurements made in accordance with the internationally standardized in-duct fan measurement method. This method utilizes large anechoic terminations and a procedure involving averaging over measurements in space and time to account for the contribution from higher-order modes. The new method does not require either of these added complications and gives detail about the underlying modal content of the emitted sound.

Aeroacoustic source measurement methods for characterizing the sound generated by ducted flow devices with higher-order modes.

The International Organization for Standardization (ISO) method 5136 is widely used in industry and academia to determine the sound power radiated into a duct by fans and other flow devices. The method involves placing the device at the center of a long cylindrical duct with anechoic terminations at each end to eliminate reflections. A single off-axis microphone is used on the inlet and outlet sides that can theoretically capture the plane-wave mode amplitudes but this does not provide enough information to fully account for higher-order modes. In this study, the "two-port" source model is formulated to include higher-order modes and applied for the first three modes. This requires six independent surface pressure measurements on each side or "port." The resulting experimental set-up is much shorter than the ISO rig and does not require anechoic terminations. An array of six external loudspeaker sources is used to characterize the passive part of the two-port model and the set-up provides a framework to account for transmission of higher-order modes through a fan. The relative importance of the higher-order modes has been considered and their effect on inaccuracies when using the ISO method to find source sound power has been analyzed.

Dual-mode thin film bulk acoustic wave resonators for parallel sensing of temperature and mass loading.

Thin film bulk acoustic wave resonator (FBAR) devices supporting simultaneously multiple resonance modes have been designed for gravimetric sensing. The mechanism for dual-mode generation within a single device has been discussed, and theoretical calculations based on finite element analysis allowed the fabrication of FBARs whose resonance modes have opposite reactions to temperature changes; one of the modes exhibiting a positive frequency shift for a rise of temperature whilst the other mode exhibits a negative shift. Both modes exhibit negative frequency shift for a mass load and hence by monitoring simultaneously both modes it is possible to distinguish whether a change in the resonance frequency is due to a mass load or temperature variation (or a combination of both), avoiding false positive/negative responses in gravimetric sensing without the need of additional reference devices or complex electronics.

Dual-mode thin film bulk acoustic wave resonators for parallel sensing of temperature and mass loading

Thin film bulk acoustic wave resonator (FBAR) devices supporting simultaneously multiple resonance modes have been designed for gravimetric sensing. The mechanism for dual-mode generation within a single device has been discussed, and theoretical calculations based on finite element analysis allowed the fabrication of FBARs whose resonance modes have opposite reactions to temperature changes; one of the modes exhibiting a positive frequency shift for a rise of temperature whilst the other mode exhibits a negative shift. Both modes exhibit negative frequency shift for a mass load and hence by monitoring simultaneously both modes it is possible to distinguish whether a change in the resonance frequency is due to a mass load or temperature variation (or a combination of both), avoiding false positive/negative responses in gravimetric sensing without the need of additional reference devices or complex electronics. © 2012 Elsevier B.V.