Past exam of the mathematics course of the University of Cambridge / 2019 / ii / Paper 1 / 39A / b / iii

Here

X_{p} (s) = - \frac{s ^{1 + α}}{1 + α}, \dot{X}_{p} (s) = - s^{α} .

Write

A = (γ + 1) /2

. Substituting

x = (c_{0} - a) t

into the

C_{+}

characteristic equation gives

a t = c_{0} s + A t s^{α} - (A - \frac{1}{1 + α}) s^{1 + α} .

For

0 < a < A

and

0 < α ≪ 1

, the relevant emission time is exponentially small. The terms containing an explicit factor

s

are therefore beyond all algebraic orders, leaving

s^{α} \sim \frac{a}{A} .

s (x, t) \sim (\frac{a}{A})^{1/ α} = (\frac{2 a}{γ + 1})^{1/ α} .

Although

s \to 0

, the limiting state is nontrivial:

u \sim - \frac{2 a}{γ + 1}, c \sim c_{0} - \frac{γ - 1}{γ + 1} a,

which is the centered-rarefaction state on the ray

x / t = c_{0} - a

when

α = 0

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