PDF《gas》

1 ≤ i ≤ 4, is the phase of the i-th wave associated with the characteristic speed λi. Now we use (3.2) in (2.2), expand A and B in Taylor'

s series in powers of ? about U = U0, replace the partial derivatives ? ?X (X being either x or t) by ? ?X + ??1

4 i=1 ?φi ?X ? ?θi , and equate to zero the coe?cients of ?0 and ?1 in the resulting expansions, to obtain O(?0 ) :

4 i=1 I ?φi ?t + A0 ?φi ?x ?U1 ?θi = 0, (3.3) O(?1 ) :

4 i=1 I ?φi ?t + A0 ?φi ?x ?U2 ?θi = ? ?U1 ?t ? A0 ?U1 ?x ? (U1.?U B)0 ?

4 i=1 ?φi ?x (U1.?U A)0 ?U1 ?θi , (3.4) where I is the

4 x

4 unit matrix and ?U is the gradient operator with respect to the dependent variable U. The expressions of (U1.?U B)0 and (U1.?U A)0 are given as (U1.?U B)0 = U12mρ0 x , 0, ?U11(γ ? 1) p0c τρ0 + U12 mγp0 (1 ? bρ0)x + U13 (γ ? 1)c(1 ? bρ0) τ ? U14 (γ ? 1)ρ0 τ , U11 p0c τρ2

0 ? U13 c(1 ? bρ0) τρ0 + U14

1 τ T , Asymptotical Solutions for a Vibrationally Relaxing Gas

427 (U1.?U A)0 = ? ? ? ? ? ? ? U12 U11

0 0

0 U12 ?U11 ρ2

0 0

0 γ b U11 p0 (1?bρ0)2 + γ U13 (1?bρ0) U12

0 0

0 0 U12 ? ? ? ? ? ? ? . Now since the phase functions φi,

1 ≤ i ≤ 4, satisfy the eikonal equation Det I ?φi ?t + A0 ?φi ?x = 0, we choose the simplest phase function of this equation, namely φi(x, t) = x ? λit,

1 ≤ i ≤ 4. It follows from (3.1) that for each phase φi, ?U1 ?θi is parallel to the right eigen- vector R(i) of A0 and thus U1 =

4 i=1 σi(x, t, θi)R(i) , (3.5) where σi = (L(i) ・ U1) is a scalar function called the wave amplitude, that depends only on the i-th fast variable θi. We assume that σi(x, t, θi) has zero mean value with respect to the fast variable θi, that is, lim T →∞

1 2T T ?T σi(x, t, θi) dθi =

0 We then use (3.5) in (3.4) and solve for U2. To begin with we write U2 =

4 j=1 mjR(j) , substitute this value in (3.4), and premultiply the resulting equation by L(i) to obtain the system of decoupled inhomogeneous ?rst order partial di?erential equations:

4 j=1 (λi ? λj) ?mi ?θj = ? ?σi ?t ? λi ?σi ?x ? L(i) (U1 ・ ?B)0 ?

4 j=1 L(i) (U1 ・ ?A)0 ?U1 ?θj ,

1 ≤ i ≤ 4. (3.6) The characteristic ODEs for the i-th equation in (3.6) are given by B θj = λi ? λj for j = i, B θi = 0, B mi = Hi, Math. Model. Anal., 14(4):423C434, 2009.

428 R. Arora where Hi(x, t, θ1, θ2, θ3, θ4) = ? ?σi ?t ? λi ?σi ?x ? L(i) (U1 ・ ?B)0 ?

4 j=1 L(i) (U1 ・ ?A)0 ?U1 ?θj . We asymptotically average (3.6) along the characteristics and appeal to the sub-linearity of U2 in θ, which ensures that the expression (3.2) does not contain secular terms. The constancy of θi along the characteristics and the vanishing asymptotic mean value of B mi along the characteristics implies that the wave amplitudes σi,

1 ≤ i ≤ 4, satisfy the following system of coupled integro-di?erential equations ?σi ?t + λi ?σi ?x + aiσi + Γi iiσi ?σi ?θi (3.7) + i=j=k Γi jk lim T →∞

1 2T T ?T σj(θi + (λi ? λj)s)? σk(θi + (λi ? λk)s)ds = 0, where ? σk = ?σk ?θk and the coe?cients ai and Γi jk are given by ai = L(i) (R(i) ・ ?B)0, Γi jk = L(i) (R(j) ・ ?A)0R(k) . (3.8) The interaction coe?cients Γi jk denote the strength of coupling between the j-th and k-th wave modes (j = k) that can gene........