8.2 Restriction on the velocity

Note that Equation (37View Equation) is, disregarding the difference in notation, identical to the first equation in (35View Equation). However, in the present setting, there is a restriction on va, which did not appear in the polarized case. The essential part of this restriction is the inequality 0 < |va(𝜃)| < 1; the sign of va is not of central importance. The restriction v < 1 a is due to the fact that there is a potential inconsistency with Equation (16View Equation) if va ≥ 1; assuming the derivatives of u and w with respect to time and space to tend to zero, all the terms in Equation (16View Equation) tend to zero except, possibly, e2P− 2τQ2𝜃, which is roughly e2(va(𝜃)−1)τq2𝜃(𝜃 ). In fact, if va(𝜃) ≥ 1 and q𝜃(𝜃 ) ⁄= 0, the expansions are clearly inconsistent. Furthermore, considering Equation (16View Equation), the term e2P− 2τQ2 𝜃 would in that case seem to have the effect of causing P τ to decrease. For a further discussion, see [10Jump To The Next Citation Point].
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