2.4 Characteristic speeds

The system of field equations (1View Equation), (2View Equation) may be written as a quasilinear system of n equations in the form
A ∂F---u = π. (10 ) ∂u ,A
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 ϕ(xD ) = 0 define the wave front; thus

∂-ϕ- -∂ϕ- V- ∂xa = |grad ϕ|na and ∂ct = − |grad ϕ| c (11 )
define its unit normal n and the speed V. An easy manipulation provides
V2 gABϕ,A ϕ,B -2-= 1 + --------2-. (12 ) c |grad ϕ|

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

V [ ∂u ] [u,a] = δuna, [u,0] = −--δu, where δu = na---- . (13 ) c ∂xa
δu 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 (10View Equation) the matrix A ∂F--- ∂u and the productions are equal on both sides of the wave, since both only depend on u and since u is continuous. Thus, if we take the difference of the equations on the two sides and use (13View Equation) and (11View Equation), we obtain

∂F A ϕ,A ----δu = 0. (14 ) ∂u
Non-trivial solutions for δu require that this linear homogeneous system have a vanishing determinant
( A) ∂F--- det ϕ,A ∂u = 0. (15 )
Insertion of (11View Equation) into (15View 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 (15View Equation) is called the characteristic equation of the system (10View Equation) of field equations. By (11View Equation) it may be written in the form
( ) ∂F a V ∂F 0 det -----na − ------- = 0. (16 ) ∂u c ∂u

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