563 solve the following boundary value problem for

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Unformatted text preview: · ∇p = c2 ∇ · ∇ρ . 132 Thus ρ must satisfy the three-dimensional wave equation ∂2ρ = c2 ∇ · ∇ρ = c2 ∂ t2 ∂2ρ ∂2ρ ∂2ρ + + 2 2 ∂x ∂y ∂z2 . (5.20) If the sound wave ρ is independent of z , (5.20) reduces to ∂2ρ = c2 ∂ t2 ∂2ρ ∂2ρ + 2 ∂x ∂ 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 circumstances. 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: Force...
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