Non-linearity, relaxation and diffusion in acoustics and ultrasonics

As they propagate through a gas, fluctuating pressure signals of moderate amplitude and of ultrasonic frequency are affected by amplitude dispersion, by relaxation damping and, particularly in ‘shock layers’, by diffusive damping. We derive a ‘high frequency’ theory including all these effects, for disturbances of arbitrary wave form excited by a wide variety of boundary conditions. By introducing a phase variable α, and taking account of non-linearity, we show how the signal propagates along the rays of linear acoustics theory, with constantly changing wave profile. Relaxation dampens the signal, as for linear acoustics, and also diminishes amplitude dispersion. A criterion for shock formation is given, and the importance of non-linearity for signal attenuation exhibited. As shocks form, α surfaces coalesce and diffusive mechanisms are accentuated. Whitham's area rule is shown to be relevant for unsteady three-dimensional flows in relaxing gases, and is used to compute the attentuation of an ultrasonic beam. Supersonic relaxing flow over a wavy wall is also analyzed, and focusing effects are discussed.