Dictums for problem solving and approximation in mathematical acoustics: Examples involving low-frequency vibration and radiation
Sketch illustrating the problem of acoustic radiation caused by the vibrations of a long rigid cylinder oscillating in a compliant baffle.
Sketch illustrating terminology used in the statement of the acoustic energy corollary in terms of volume and surface integrals.
Sketch illustrating the use of the energy corollary for transient radiation from the moving cylinder. The outer radius is selected to be so large that the disturbance has not yet reached that radius.
Sketch illustrating the problem of the radiation of surface waves and other pressure disturbances by an oscillating horizontal cylinder on the surface of a fluid bounded by a compliant surface.
General sketch of the complex plane for the complex angular velocity ω, indicating general location of singularities for the Fourier transform of a function of time that is zero before some initial time.
Sketch illustrating the concepts of inner and outer regions for problems involving an oscillating cylinder at the edge of a half-space.
Sketch illustrating the concept of an outgoing cylindrical wave as being a superposition of waves from point sources spaced along the symmetry axis.
Deformed contour used as an initial step for the determination of an approximate expression of the outer solution for small arguments.
Sketch showing location of hypothetical line source and image source used in the mathematical construction of an outer solution for the general problem discussed in the present paper. The vibrating interface serves as a third source in the construction.
Selection of branch cuts and location of poles for the contour integration when the Hankel function is represented as a Fourier transform involving an integration over the horizontal wave number.
Resulting path of integration in the complex horizontal wavenumber plane in the limit when the artificial damping parameter goes to zero. (The poles appear when the possibility of surface waves is taken into account.)
Deformed contour for the outer solution of the general problem discussed in the present paper, the deformation being such that a term corresponding to an outwardly propagating surface wave is evident.
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