As the velocity decreases from its leftmost entrance value, u ( x → −∞ ) = u$_{s}$ , to its rightmost exit value, u ( x → + ∞ ) = u$_{s}$ − u$_{p}$ , the stationary nature of the wave requires that the fluxes of mass, momentum, and energy remain constant throughout: ( ρu )$_{x}$ = ( ρu )$_{cold}$ = ( ρu )$_{hot}$ ; ( P$_{xx}$ + ρu $^{2}$)$_{x}$ = ( P + ρu $^{2}$)$_{cold}$ = ( P + ρu $^{2}$)$_{hot}$ ; ( ρu )[( e + ( P$_{xx}$/ρ ) + ( u $^{2}$/ 2)]$_{x}$ + Q$_{x}$ = ( ρu )[ e + ( P/ρ ) + ( u $^{2}$/ 2)]$_{cold}$ = ( ρu )[ e + ( P/ρ ) + ( u $^{2}$/ 2)]$_{hot}$ . The notation here is conventional, with the pressure tensor P and heat flux vector Q assumed to be calculable from the density ρ , velocity u , energy e , and their gradients. Temperature 11,12,15–17 is our special interest in this work. Temperature is most simply and usefully defined as a velocity fluctuation, the “kinetic temperature”: kT$_{xx}$ ≡ m 〈 ( ˙ x − 〈 ˙ x 〉 ) $^{2}$〉 ; kT$_{yy}$ ≡ m 〈 ( ˙ y − 〈 ˙ y 〉 ) $^{2}$〉 . ρk ( T$_{xx}$ − T$_{yy}$ ) / (2 m ) = ( P$_{xx}$ − P$_{yy}$ ) / 2 [Dilute Gas] , h$_{hot}$ − h$_{cold}$ = [ e + ( P/ρ )]$_{hot}$ − [ e + ( P/ρ )]$_{cold}$ = [ ρ$_{cold}$u 2 $_{s}$/ 2] − [ ρ$_{hot}$ ( u$_{s}$ − u$_{p}$ ) $^{2}$/ 2] . The angular brackets imply a local average. The velocities here are individual particle velocities, whose local average would be the hydrodynamic flow velocity u . Temperature is just the fluctuation about this average. It is evident that T$_{xx}$ and T$_{yy}$ can differ. In dilute-gas kinetic theory, the difference corresponds to a shear stress: where k is Boltzmann’s constant and m is the particle mass, which we choose equal to unity in what follows. In dense fluids there is no simple relationship between the two tensors so that special evolution equations for T$_{xx}$ and T$_{yy}$ need to be developed, as we do in Section III. The cold fluid, initially moving to the right at the entrance velocity, or “shock velocity” u$_{s}$ , is slowed by its encounter with the wave until it reaches its exit velocity u$_{s}$ − u$_{p}$ , where u$_{p}$ is the “piston velocity” or “particle velocity”. In this irreversible deceleration the kinetic energy lost by the decelerating fluid is converted into additional hot fluid enthalpy ( H = E + P V ↔ h = e + P v ):