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 Incoming links: Convected acoustic wave equation

\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.

Use a plane wave proportional to

e^{i (k x - ω t)}

(ω - U_{0} k)^{2} = c_{0}^{2} k^{2}, ω = (U_{0} \pm c_{0}) k .

The two branches have signed phase velocities and group velocities

U_{0} \pm c_{0}

. On a branch with

ω - U_{0} k = s c_{0} k

s = \pm 1

, the linearized Euler equation gives

p_{1} = c_{0}^{2} ρ_{1}, v_{1} = s \frac{c _{0}}{ρ _{0}} ρ_{1} .

0 \leq U_{0} < c_{0}

, the absence of wave reflection selects the outgoing branch on each side:

ρ_{1} (x, t) = ρ_{0} Re [A e^{i (k_{\pm} x - ω t)}], k_{+} = \frac{ω}{U _{0} + c _{0}} (x > 0), k_{-} = \frac{ω}{U _{0} - c _{0}} (x < 0) .

For

U_{0} = 0

, take

A

real and positive by choosing the source phase, and put

k_{0} = ω / c_{0}

. At

ω t = 2 nπ

, the requested spatial profiles are

ρ_{1} = ρ_{0} A cos (k_{0} x), p_{1} = ρ_{0} c_{0}^{2} A cos (k_{0} x), v_{1} = c_{0} A sgn (x) cos (k_{0} x) (x \neq = 0) .

For complex

A

, retain

Re [A e^{i k_{0} ∣ x ∣}]

instead. The normalized sound pressure and acoustic density perturbation lie on the same even cosine curve. The acoustic particle velocity follows that curve on the right and its negative on the left, as in the figure in part 2(a). Compressed fluid moves away from the source on both sides. These are two outgoing travelling waves, not a standing wave. The ideal point loudspeaker supplies the discontinuity of velocity at

x = 0

; the source-free equations apply on either side of it.