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T ∂x d. Linearized Boussinesq equation for water and elastic waves 2 4 ∂ 2φ 2∂ φ 2 ∂ φ − a − μ = 0. ∂t 2 ∂x2 ∂x2 ∂t 2 Because these equations are linear with constant coefficients, they can be readily solved by Fourier-Laplace transform. The Fourier-Laplace transforms of these equations are (a) (ω2 − c2 α 2 − μ2 )φ˜ = H1 (α, ω). (b) (ω2 − μ2 α 4 )φ˜ = H2 (α, ω). (c) (ω + μα 3 )φ˜ = H3 (α). (d) (ω2 − a2 α 2 + μ2 α 2 ω2 )φ˜ = H4 (α, ω). The right-hand side of each of these equations represents some arbitrary initial conditions.
Note that the entire solution propagates to the right at a dimensionless speed of unity. Discretize the x derivative by the 7-point optimized finite difference stencil. Solve the initial value problem computationally. You may use any time marching scheme with a small time step t. Demonstrate that the numerical solution separates into two pulses. One pulse propagates to the right at a speed nearly equal to unity. The other pulse, composed of high wave numbers, propagates to the left. Explain why the solution separates into two pulses.
This is an extremely important point and should be clearly understood by all CAA investigators and users. 1 Dispersive Waves of Physical Systems Many physical systems support dispersive waves. Examples of commonly encountered dispersive waves are small-amplitude water waves, waves in stratified fluid, elastic waves, and magnetohydrodynamic waves. A common feature of these dispersive waves is that they are governed by linear partial differential equations with constant coefficients. The following are some of these equations.