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adiabatic invariant airfoil amplitude approximation arbitrary assume asymptotic expansion behavior boundary condition boundary layer canonical transformation characteristic coefficients consider const constant corresponding curves damping defined depend derive dimensionless discussed in Section exact solution example expansion procedure expressed Figure fixed flow frequency function given gives Hamiltonian infinity initial conditions initial value problem inner expansion integral intermediate limit Jacobi integral Laplace's equation leading term limit process expansion linear matching method motion multiple variable expansion Navier-Stokes equations obtain orbit ordinary differential equations outer expansion outer limit outer solution partial differential equations periodic solutions perturbation phase plane plane potential Reference resonance result right-hand side satisfy shock side of Equation singular slow variable slowly varying solution of Equation solution to 0(1 solve subcharacteristic surface terms of order theory transonic uniformly valid velocity wave weakly nonlinear Zeipel