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
define the wave front; thus
define its unit normal
and the speed
V
. An easy manipulation provides
Since in a weak wave the fields
have no jump across the front, the jumps in the gradients must
have the direction of
and we may write
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 (10) the matrix
and the productions are equal on both sides of the wave, since
both only depend on
and since
is continuous. Thus, if we take the difference of the equations
on the two sides and use (13
) and (11
), we obtain
Non-trivial solutions for
require that this linear homogeneous system have a vanishing
determinant
Insertion of (11) into (15
) 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 (15
) is called the
characteristic equation
of the system (10
) of field equations. By (11
) it may be written in the form
![]() |
Speeds of Propagation in Classical and Relativistic
Extended Thermodynamics
Ingo Müller http://www.livingreviews.org/lrr-1999-1 © Max-Planck-Gesellschaft. ISSN 1433-8351 Problems/Comments to livrev@aei-potsdam.mpg.de |