Dispersion relation in the limit of high frequency for a hyperbolic system with multiple eigenvalues

• Hyperbolic systems with multiple eigenvalues are studied.
• Plane-wave solutions to linearized hyperbolic systems are considered.
• The dispersion relation in the high-frequency limit is analyzed.
• Recurrence equations for determining the dispersion relation are derived.
• It is shown that linear stability implies stability of weak-discontinuity waves.

The results of a previous paper (Muracchini et al., 1992) are generalized by considering a hyperbolic system in one space dimension with multiple eigenvalues. The dispersion relation for linear plane waves in the high-frequency limit is analyzed and the recurrence formulas for the phase velocity and the attenuation factor are derived in terms of the coefficients of a formal series expansion in powers of the reciprocal of frequency. In the case of multiple eigenvalues, it is also verified that linear stability implies λλ-stability for the waves of weak discontinuity. Moreover, for the linearized system, the relationship between entropy and stability is studied. When the nonzero eigenvalue is simple, the results of the paper mentioned above are recovered. In order to illustrate the procedure, an example of the linear hyperbolic system is presented in which, depending on the values of parameters, the multiplicity of nonzero eigenvalues is either one or two. This example describes the dynamics of a mixture of two interacting phonon gases.