Nonlinear Geometrical Optics
Using asymptotic methods, one can reduce complicated systems of equations to simpler model equations. The model equation for a single, genuinely nonlinear, hyperbolic wave is Burgers equation. Reducing the gas dynamics equations to a Burgers equation, leads to a theory of nonlinear geometrical acoustics. When diffractive effects are included, the model equation is the ZK or unsteady transonic small disturbance equation. We describe some properties of this equation, and use it to formulate asymptotic equations that describe the transition from regular to Mach reflection for weak shocks. Interacting hyperbolic waves are described by a system of Burgers or ZK equations coupled by integral terms. We use these equations to study the transverse stability of interacting sound waves in gas dynamics.
KeywordsSound Wave Burger Equation Weak Shock Eikonal Equation Shock Strength
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