Barrier billiard and random matrices

The barrier billiard is the simplest example of pseudo-integrable models with interesting and intricate classical and quantum properties. Using the Wiener-Hopf method it is demonstrated that quantum mechanics of a rectangular billiard with a barrier in the centre can be reduced to the investigation of a certain unitary matrix. Under heuristic assumptions this matrix is substituted by a special low-complexity random unitary matrix of independent interest. The main results of the paper are (i) spectral statistics of such billiards is insensitive to the barrier height and (ii) it is well described by the semi-Poisson distributions.

Discrete scattering and meta-arrest of locally resonant elastic wave metamaterials with a semi-infinite crack

Previous investigations on wave scattering and crack propagation in the discrete periodic structure are concentrated on the conditional mass–spring model, in which the internal mass is not included. In this work, elastic wave metamaterials with local resonators are studied to show the scattering of elastic waves by a semi-infinite crack and the arrest behaviour. The influences of internal mass–spring structure are analysed and the discrete Wiener–Hopf method is used to obtain the displacement solution. Numerical calculations are performed to show that the dynamic negative effective mass and band gaps can be observed owing to the local resonance of the internal mass. Therefore, the scattering of an elastic wave with a specific frequency by a semi-infinite crack can be avoided by tuning the structural parameters. Moreover, the energy release ratio which characterizes the splitting resistance is presented and the meta-arrest performance is found. It is expected that this study will increase understanding of how to control the scattering characteristics of elastic waves by a semi-infinite crack in locally resonant metamaterials and also help to improve their fracture resistance.

Deep neural networks for waves assisted by the Wiener–Hopf method

In this work, the classical Wiener–Hopf method is incorporated into the emerging deep neural networks for the study of certain wave problems. The essential idea is to use the first-principle-based analytical method to efficiently produce a large volume of datasets that would supervise the learning of data-hungry deep neural networks, and to further explain the working mechanisms on underneath. To demonstrate such a combinational research strategy, a deep feed-forward network is first used to approximate the forward propagation model of a duct acoustic problem, which can find important aerospace applications in aeroengine noise tests. Next, a convolutional type U-net is developed to learn spatial derivatives in wave equations, which could help to promote computational paradigm in mathematical physics and engineering applications. A couple of extensions of the U-net architecture are proposed to further impose possible physical constraints. Finally, after giving the implementation details, the performance of the neural networks are studied by comparing with analytical solutions from the Wiener–Hopf method. Overall, the Wiener–Hopf method is used here from a totally new perspective and such a combinational research strategy shall represent the key achievement of this work.

Analysis of Sound Waves with Semi Perforated Pipe

The paper presents analytical results of radiation phenomena at the far field and solution of the wave equation with adequate boundary condition imposed by the pipe wall. An infinite pipe with perforated part is considered. The solution is obtained by using the Fourier transform technique in conjunction with the Wiener-Hopf Method. Applying the Fourier transform technique, the boundary value problem is described by Wiener Hopf equation and then solved analytically.

Tonal Fan-Noise Radiation From Aero-Engine Bypass With Serrated End Treatments

Chevrons, which are also known as serrations, are initially developed to suppress jet noise radiating from aero-engine nozzles. The associated fluid mechanics are already well known. Compared with jet noise, turbomachinery fan noise has become relatively more important along with the ever-increasing bypass ratio. However, it is still unclear whether the trailing-edge chevrons on the bypass duct would attenuate fan noise and, if the answer is yes, what is the associated mechanism. In this work, we first use a theoretical model based on the Wiener–Hopf method to rapidly conduct parametric studies across a number of different setups. The results from such a theoretical model suggest that the chevrons are also effective in the reduction of fan noise scattering. Next, we perform high-fidelity computational fluid and acoustic simulations for a realistic aero-engine with some representative setups, and the results further confirm the effectiveness of chevrons. Both analytical and numerical results show the associated noise control mechanism, that is, chevrons would induce acoustic mode conversion (especially from low modes to high modes), which shall further result in evanescent waves in the radial direction and the final noise reduction at various radiation angles. The findings may find applications in the next-generation low-noise aero-engine design.