By Bertelle R., Russo M.R.
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Extra info for An approach to the Gummel map by vector extrapolation methods
In a laboratory frame (z, t), the eﬀective velocities take the values (V ∓ a1 )(V ∓ a2 ) a1 a2 ∓ V a +V = . ±¯ a−V ±¯ a−V Particularly, for V = 0 we obtain vst = ± a−1 whereas for V = ∞ −1 , vtemp = ± a . 70) for vst and vtemp when we apply them to the case γ1 = γ2 . 1) governing the wave propagation through an immovable elastic bar represents an Euler equation generated by the action density 2 2 ∂u ∂u 1 1 − k . 74) ∂z ∂ ∂u ∂t ∂z ∂t Wtt = Wtz Wzt Wzz = ∂u ∂Λ 1 −Λ=− ρ ∂z ∂ ∂u 2 ∂z ∂u ∂t 2 1 − k 2 ∂u ∂z 2 − the momentum flux density.
75) we have the rate of increase DW of the Dt energy of a unit segment of the bar; this rate is calculated as the sum of the ∂Wtz tt that is brought into a unit segment local change ∂W ∂t and the energy ∂z tt is equal to the through its endpoints per unit time. 77) − − 2 ∂t ∂t ∂t ∂z produced, per unit time, by an external agent against the variable property pattern. 76). In this section, we shall see in detail how the energy-momentum balance manifests itself through homogenization. 76). 31). 76) to the following form (Wtt )t + (Wtz )z − V δ −1 (Wtt )ξ + δ −1 (Wtz )ξ 1 = V δ −1 ρξ (ut − V δ −1 uξ )2 − kξ (uz + δ −1 uξ )2 , 2 (Wzt )t + (Wzz )z − V δ −1 (Wzt )ξ + δ −1 (Wzz )ξ 1 = − δ −1 [ρξ (ut − V δ −1 uξ )2 − kξ (uz + δ −1 uξ )2 ].
Eﬀective parameters K versus P with variable V (case ρ 1 ρ −k the signs of v1 and v2 are the same if V 2 is taken within the interval 1 , a21 for the slow range, ¯ ρ¯ k1 1 k ≤ 0). 67) and within the interval a22 , k¯ ¯ 1 ρ for the fast range. 59) that such intervals may exist in the irregular case. For each of them, the homogenized waves propagate in the same direction relative to a laboratory frame; this direction may be switched to opposite as we go from V to −V . We thus arrive at what will be termed coordinated wave propagation.