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Unformatted text preview: · ∇p = c2 ∇ · ∇ρ .
132 Thus ρ must satisfy the three-dimensional wave equation
= c2 ∇ · ∇ρ = c2
∂ t2 ∂2ρ
∂z2 . (5.20) If the sound wave ρ is independent of z , (5.20) reduces to
∂ t2 ∂2ρ
∂ y2 , exactly the same equation that we obtained for the vibrating membrane.
Remark: The notion of linearization is extremely powerful because it enables
us to derive information on the behavior of solutions to the nonlinear Euler
equations, which are extremely diﬃcult to solve except for under very special
The Euler equations for a perfect gas and the closely related Navier-Stokes
equations for an incompressible ﬂuid such as water form basic models for ﬂuid
mechanics. In the case of incompressible ﬂuids, the density is constant, so no
equation of state is assumed. To allow for viscosity, one adds an additional term
to the expression (5.17) for the force acting on a ﬂuid element:
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- Winter '14