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Unformatted text preview: ewrite this equation as
∂v
1
+ (v · ∇)v = − ∇p.
∂t
ρ (5.18) Note that this equation is nonlinear because of the term (v · ∇)v.
To ﬁnish the Euler equations, we need an equation of state , which relates
pressure and density. The equation of state could be determined by experiment,
the simplest equation of state being
p = a2 ργ , (5.19) where a2 and γ are constants. (An ideal monatomic gas has this equation of
state with γ = 5/3.)
The Euler equations (5.16), (5.18), and (5.19) are nonlinear, and hence quite
diﬃcult to solve. However, one explicit solution is the case where the ﬂuid is
motionless,
ρ = ρ0 , p = p0 , v = 0,
where ρ0 and p0 satisfy p0 = a2 ργ .
0 Linearizing Euler’s equations near this explicit solution gives rise to the linear
diﬀerential equation which governs propagation of sound waves.
Let us write
ρ = ρ0 + ρ , p = p 0 + p , v = v ,
where ρ , p and v are so small that their squares can be ignored.
Substitution into Euler’s equations yields
∂ρ
+ ρ0 ∇ · (v ) = 0,
∂t
∂v
1
= − ∇p ,
∂t
ρ0
and p = [a2 γ (ρ0 )(γ −1) ]ρ = c2 ρ , where c2 is a new constant. It follows from these three equations that
∂2ρ
∂v
= −ρ0 ∇ ·
∂ t2
∂t = ∇...
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 Winter '14
 Equations

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