 Solution

ID: past-exam-of-the-fluids-course-of-the-university-of-oxford/2021/2/b/solution

\partial_{t} ρ + \partial_{x} (ρ u) = 0, \partial_{t} u + u \partial_{x} u = - ρ^{- 1} \partial_{x} p .

Substitute

u = U_{0} + v_{1}

and the stated background-plus-perturbation expressions. Dropping products of perturbations gives the linearization

D_{0} ρ_{1} + ρ_{0} \partial_{x} v_{1} = 0, ρ_{0} D_{0} v_{1} = - \partial_{x} p_{1}, D_{0} = \partial_{t} + U_{0} \partial_{x} .

p_{0} (1 + \frac{ρ _{1}}{ρ _{0}})^{γ} = p_{0} + \frac{γ p _{0}}{ρ _{0}} ρ_{1} + O (ρ_{1}^{2}), p_{1} = c_{0}^{2} ρ_{1}, c_{0}^{2} = \frac{d p}{d ρ}_{ρ_{0}} = \frac{γ p _{0}}{ρ _{0}} .

Since the background coefficients are constant,

D_{0}

commutes with

\partial_{x}

. Applying

D_{0}

to the linearized continuity equation and using the linearized Euler equation therefore yields

D_{0}^{2} ρ_{1} = - ρ_{0} \partial_{x} D_{0} v_{1} = \partial_{x}^{2} p_{1}, (\partial_{t} + U_{0} \partial_{x})^{2} ρ_{1} = c_{0}^{2} \partial_{x}^{2} ρ_{1} .

x - U_{0} t

, it becomes the ordinary one-dimensional wave equation. Positive

c_{0}^{2}

is required for propagating sound waves.

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