Smith, Malcolm Gordon
(2004)
Sound radiation from a vibrating surface under a boundary layer.
*University of Southampton, Doctoral Thesis*.

## Abstract

This thesis investigates the sound field generated by a plane vibrating surface radiating into a moving fluid. The mean flow considered consists of a semi-infinite fluid flowing parallel to a surface, with a boundary layer next to the surface.

The usual governing equations for sound propagation in a shear flow, using pressure and acoustic particle velocity, have singularities if either the mean velocity gradient becomes infinite or at a critical layer where the Doppler shifted frequency of wall vibration becomes zero. It is shown that the first of these singularities may be avoided by using the particle displacement as a wave variable in place of the particle velocity. Both singularities may be avoided by using the displacement impedance as a variable. The new formulation has enabled a list of properties of solutions at a critical layer to be compiled.

The sound radiation problem is solved by numerically integrating the governing equations for sound waves radiating away from the surface at each horizontal wave number, applying a boundary condition corresponding to the wave spectrum of the wall vibration, and then using an inverse spatial Fourier transform to determine the radiated sound field.

The effect of the flow Mach number and boundary layer thickness on the radiation efficiency of individual wave number components, and the power and sound field radiated by a uniform piston vibrating in an otherwise rigid wall are considered. The model is validated by replicating analytic results for a uniform flow over a compact piston, an increase in sound power output and a convective amplification effect that increases the pressure upstream of the source and reduces the level downstream. It is shown that, as the boundary layer thickness is increased, the power output and convective amplification are reduced, and the sound field is further modified by refraction of upstream propagating sound away from the surface and by downstream channelling of sound in the boundary layer.

The model is used to investigate the performance of a flush mounted device in the wall that uses speed of sound propagation between a source and one or more receivers to determine properties of the flow. It is shown that by measuring propagation speed at two appropriate frequencies it is possible in principle to measure the free-stream Mach number, the boundary layer displacement thickness and the direction of the flow.

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