3 Finite Speeds in Non-Relativistic 2 Scope and StructureCharacteristic 2.3 Exploitation of the Entropy

2.4 Characteristic Speeds 

The system of field equations (1Popup Equation), (2Popup Equation) may be written as a quasilinear system of n equations in the form

  equation243

Such a system allows the propagation of weak waves, so-called acceleration waves. There are n such waves and their speeds are called characteristic speeds, which are not necessarily all different. The fastest characteristic speed is the pulse speed . This is the largest speed by which information can propagate.

Let tex2html_wrap_inline3549 define the wave front; thus

  equation253

define its unit normal tex2html_wrap_inline3551 and the speed V . An easy manipulation provides

  equation266

Since in a weak wave the fields tex2html_wrap_inline3369 have no jump across the front, the jumps in the gradients must have the direction of tex2html_wrap_inline3551 and we may write

  equation277

tex2html_wrap_inline3571 is the magnitude of the jump of the gradient of u . The square brackets denote differences between the front side and the back side of the wave.

In the field equations (10Popup Equation) the matrix tex2html_wrap_inline3575 and the productions are equal on both sides of the wave, since both only depend on tex2html_wrap_inline3369 and since tex2html_wrap_inline3369 is continuous. Thus, if we take the difference of the equations on the two sides and use (13Popup Equation) and (11Popup Equation), we obtain

  equation301

Non-trivial solutions for tex2html_wrap_inline3571 require that this linear homogeneous system have a vanishing determinant

  equation309

Insertion of (11Popup Equation) into (15Popup Equation) provides an algebraic equation for V whose solutions - for a prescribed direction n - determine n wave speeds V, of which the largest one is the pulse speed. Equation (15Popup Equation) is called the characteristic equation of the system (10Popup Equation) of field equations. By (11Popup Equation) it may be written in the form

  equation322



3 Finite Speeds in Non-Relativistic 2 Scope and StructureCharacteristic 2.3 Exploitation of the Entropy

image Speeds of Propagation in Classical and Relativistic Extended Thermodynamics
Ingo Müller
http://www.livingreviews.org/lrr-1999-1
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