Non-linear wave propagation in a relaxing gas

We consider the propagation of waves of small finite amplitude ε in a gas whose internal energy is characterized by two temperatures T (translational) and T i (internal) in the form e = C vf T + C vf T i , and T i is governed by a rate equation dT i / dt = ( T − T i )/τ. By means of approximations appropriate for a wave advancing into an undisturbed region x > 0, we show that to order εδ, the equation satisfied by velocity takes the non-linear form \[ (τ∂/∂ t+1)\{∂ u/∂ t+(a_1+γ + 1/2u)∂ u/∂ x-{1/2}λ∂^2u/∂ x^2\}=(a_1-a_0)∂ u/∂ x, \] where a 1 , a 0 are the frozen and equilibrium speeds of sound in the undisturbed region, δ = ½(1 − ( a 2 0 / a 2 1 )), and λ is the diffusivity of sound due to viscosity and heat conduction (λ may be neglected except when discussing the fine structure of a discontinuity). Some numerical solutions of this model equation are given. When ε is small compared with δ, it is also possible to construct a solution for the flow produced by a piston moving with a constant velocity by means of a sequence of matched asymptotic expansions. The limit reached for large times for either compressive or expansive pistons is the expected non-linear solution of the exact equations. For a certain range of advancing piston speeds, this is a fully dispersed wave with velocity U in the range a 0 < U < a 1 . If U > a 1 the solution is discontinuous, and indeterminate in the absence of viscosity; a singular perturbation technique based on λ is then used to determine the structure of the wave head.