The propagation models in musical wind instruments are generally based on the assumption that a one-dimensional description is sufficient to describe the behavior of sound waves regardless of the geometry of the resonator axis. However, when the resonators are not straight, this description is not exact and modal approaches have shown that the pressure field has no symmetry in curved ducts, both in linear and weakly nonlinear propagation. The aim of the present study is to discuss the dynamics of nonlinear sound propagation in U-shaped ducts with geometries close to some parts of brass instruments resonators, in particular when the pressure level is so high that shock waves appear, as it is the case for brassy sounds. For this purpose, both experiments and numerical simulations in time domain have been performed. The experiments are based on optical measurements requiring a square section of the U-shaped portion of the duct. Numerical simulations have been performed by solving the 2D Euler equations in curvilinear coordinates using a finite-difference time-domain approach. Results reveal the dynamics of shock propagation and show nonlinear phenomena that are not yet taken into account when modeling nonlinear propagation of sound waves in brass instruments resonators.

The irregular reflection of weak acoustic shock waves, known as the von Neumann reflection, has been observed experimentally and numerically for spherically diverging waves generated by an electric spark source. Two optical measurement methods are used: a Mach-Zehnder interferometer for measuring pressure waveforms and a Schlieren system for visualizing shock fronts. Pressure waveforms are reconstructed from the light phase difference measured by the interferometer using the inverse Abel transform. In numerical simulations, the axisymmetric Euler equations are solved using finite-difference time-domain methods and the spark source is modeled as an instantaneous energy injection with a Gaussian shape. Waveforms and reflection patterns obtained from the simulations are in good agreement with those measured by the interferometer and the Schlieren methods. The Mach stem formation is observed close to the surface for incident pressures within the range of 800 to 4000 Pa. Similarly, as for strong shocks generated by blasts, it is found that for spherical weak shocks the Mach stem length increases with distance following a parabolic law. This study confirms the occurrence of irregular reflections at acoustic pressure levels and demonstrates the benefits of the Mach-Zehnder interferometer method when microphone measurements cannot be applied.

Reflection of pressure shock waves on a flat surface can lead to a 3-shock reflection pattern, called Mach reflection, with reference to Ernst Mach, who provided evidence for this non-linear interaction [1]. This phenomenon has been widely studied in cases of supersonic jets and high-amplitude pressure generated by moving supersonic bodies or blast waves. In the case of weak shocks (Mach number < 1.3), 3-shock(s) reflection patterns are also observed, and in particular in the case of acoustic shock waves. If the irregular reflection of shocks on a flat surface is well known, the effects of roughness have been much less studied. In this work we performed numerical simulation of shock propagation over periodic and random surfaces and compared results to experiments. Numerical simulations are based on the temporal integration of axi-symmetric curvilinear Euler equations. Simulations are compared to Schlieren visualizations of the reflection of spark generated N-waves over sandpaper. The results show that near a rough surface, the pressure level is higher than in the case of a flat surface and the pressure decrease above the surface is also changed. The method will also be applied at larger scales, for example for sonic boom or blast waves reflection over the ground.

The authors have recently shown that irregular reflections of spark-generated pressure weak shocks from a smooth rigid surface can be studied using an optical interferometer [Karzova, Lechat, Ollivier, Dragna, Yuldashev, Khokhlova, and Blanc-Benon, J. Acoust. Soc. Am. 145(1), 26-35 (2019)]. The current study extends these results to the reflection from rough surfaces. A Mach-Zehnder interferometer is used to measure pressure waveforms. Simulations are based on the solution of axisymmetric Euler equations. It is shown that roughness causes a decrease of the Mach stem height and the appearance of oscillations in the pressure waveforms. Close to rough surfaces, the pressure was higher compared to the smooth surface.

Impact of a real terrain on the sonic boom signature at the ground has been little studied in the literature. In the current prediction schemes, a flat and perfectly reflecting ground is usually assumed and the reflected boom is obtained by multiplying the incident boom by a constant factor. In this paper, the effects of a non-flat and absorbing ground are investigated. For this, a numerical study is conducted. The 2-D Euler equations are solved in curvilinear coordinates using high-order finite difference schemes. Signatures of the incident boom typical of a classical N-wave and of a low boom are considered. The variability on the waveform at the ground induced by the terrain irregularities is studied for different characteristic length scales of the terrain.