Are waves with negative spatial damping unstable?

Highlights

• Unconventional harmonic waves have increasing amplitude in direction of propagation.

• Full solutions of models are superpositions of conventional and unconventional waves.

• When conventional waves dominate, the 2nd law of thermodynamics is valid.

• When unconventional waves dominate, the 2nd law of thermodynamics is violated.

Abstract

Conventional plane harmonic waves decay in direction of propagation, but unconventional harmonic waves grow in the direction of propagation. While a single unconventional wave cannot be a solution to a physically meaningful boundary value problem, these waves may have an essential contribution to the overall solution of a problem as long as this is a superposition of unconventional and conventional waves. A fourth order diffusion equation with proper thermodynamic structure, and the Burnett equations of rarefied gas dynamics exhibit conventional and unconventional waves. Steady state oscillating boundary value problems are considered to discuss the interplay of conventional and unconventional waves. Results show that as long as the second law of thermodynamics is valid, unconventional waves may contribute to the overall solution, which, however is dominated by conventional waves, and behaves as these.

Introduction

Any mathematical model is expected to fulfil stability requirements, so that small disturbances do not blow up. The amplitude of a small fluctuation in space should not grow over time (temporal stability), and a small disturbance at a boundary should not induce a large disturbance away from the wall (spatial stability). Hence, new models should be tested for stability of both types. The simplest test is for linear stability, where one considers only equations linearized about a homogeneous equilibrium rest state, for small disturbances. If a model is linearly unstable, also the full non-linear model will suffer from instabilities, which might render the model useless.

A well-known example are the Burnett equations of rarefied gas dynamics [1], [2], [3], which exhibit linear temporal instabilities for disturbances of small wavelengths [4], [5], [6]. Solutions of the fully non-linear Burnett equations for the shock wave structure problem are strongly affected by these linear instabilities. If a numerical time stepping method is used, unavoidable numerical errors of small wavelengths occur on refined grids, which blow up over time, and make the solution impossible [7], [8], or at least introduce artificial oscillations [9].

The (linearized) Burnett equations also exhibit wave solutions with increasing amplitude in direction of propagation, which indicates possible spatial instabilities. The following discussion aims to clarify the relation between such wave modes, and spatial instability.

A plane harmonic wave of frequency $\omega >0$ and wavelength $\lambda =\frac{2\pi }{\left|k\right|}$ is given by the equation [3], [10], [11]

$$u\left(x,t\right)=u_0{e}^{-\zeta x}cos\left[\omega t-kx+\varphi \right],$$

where $u_0$ is the amplitude, $k$ is the (real) wave number, $\varphi$ is the phase offset, and the spatial damping coefficient $\zeta$ describes the change of amplitude in space. The wave travels with the phase speed $v_{ph}=\omega ∕k$ into positive $x$-direction for $k>0$ and into negative $x$ -direction for $k<0$.

On physical grounds, for an individual wave that emerges from a source at $x=0$, one would expect positive spatial damping, such that the amplitude decreases in the direction of travel, i.e., $\zeta >0$ if $k>0$ and $\zeta <0$ if $k<0$ [3], [11]. For instance, the classical heat equation leads to wave solutions with $k=\zeta =\pm \sqrt{\omega ∕2}$, hence behaves as expected. The same is true for the description of damped sound waves by means of the linearized Navier–Stokes equations.

There are, however, models in the literature, where the solution includes waves with negative spatial damping, such that the amplitude grows in the direction of propagation, i.e., $\pm \zeta <0$ for $\pm k>0$. Well-known examples arise in continuum models based on rarefied gas dynamics [2], [3], which extend the Navier–Stokes and Fourier equations by adding higher order terms: The Burnett [1] and super-Burnett [12] equations, and some related models, such as the augmented Burnett equations [13], exhibit wave modes of this unconventional type [3], [6].

In previous work, it was stated that these unconventional waves should not appear in solutions of transport models, so that models that exhibit such “spatially unstable” waves should be discarded [3], [6], [11]. However, these statements were not corroborated by performing actual solutions of the incriminated equations for boundary value problems.

In the following we will show that, in contrast to what was stated earlier [3], [6], [11], unconventional waves contribute to meaningful solutions of proper thermodynamic models. The overall solutions are superpositions of several conventional and unconventional waves, and these behave properly, when unconventional waves contribute, but conventional waves dominate the solution. We consider two test cases, the diffusion equation with an additional fourth order term and the linearized Burnett equations. The results indicate that the superpositions are meaningful when they solve equations that are accompanied by a proper formulation of the second law of thermodynamics.

The fourth order diffusion equation is constructed based on the principles of non-equilibrium thermodynamics [14] such that it is accompanied by a proper statement of the second law of thermodynamics, hence it has an entropy with positive production term. The entropy flux is used to construct entropy generating boundary conditions. The equation is solved for the oscillating steady state with an oscillating boundary. It will be seen that although unconventional waves contribute significantly, the overall result behaves as expected, with a signal leaving the boundary with decreasing amplitude in the direction of propagation.

The Burnett equations [1] are well known to be unstable [4] and violating the second law of thermodynamics [15], at least for large frequencies [16]. We consider the equations for the setting of an acoustic resonator [17], i.e., a boundary value problem. At small frequency the equations describe sound waves leaving the oscillating boundary with decreasing amplitude in the direction of propagation. For larger frequency, however, the unconventional modes are dominant, and the equations produce a nonsensical result, namely waves arriving at the oscillating boundary, with increasing amplitude in the direction of propagation. In this case, the oscillator produces power—a clear violation of the second law.

We are not aware of discussion of unconventional wave modes by others. Hence, this contribution aims to clarify the relevance of such waves. The results indicate that these waves might contribute to reasonable results, as long as the underlying equations do not violate the second law of thermodynamics. Occurrence of these waves does not justify to discard a set of transport equations. However, one has to carefully examine the equations, since unconventional waves, if dominant, lead to nonsensical results.