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Dictums for problem solving and approximation in mathematical acoustics: Examples involving low-frequency vibration and radiation

### Abstract

A sequence of dictums for mathematical acoustics is given representing opinions intended to be regarded as authoritative, but not necessarily universally agreed upon. The dictums are presented in the context of the detailed solution for a class of problems involving the forced vibration of a long cylinder protruding half-way into a half-space bounded by a compliant surface (impedance boundary) characterized by a spring constant. One limiting case corresponds to a cylinder vibrating within an infinite rigid baffle, and another limiting case corresponds to a vibrating cylinder on the compliant surface of an incompressible fluid. The second limiting case is identified as analogous to that of a floating half-submerged cylinder whose vibrations cause water waves to propagate over the surface. Attention is focused on vibrations at very low frequencies. Difficulties with insuring a causal solution are pointed out and dictums are given as to how one overcomes such difficulties. Various approximation techniques are described. The derivations involve application of the theory of complex variables and the method of matched asymptotic expansions, and the results include the apparent entrained mass in the near field of the cylinder and the radiation resistance per unit length experienced by the vibrating cylinder.

© 2012 Acoustical Society of America

Received 15 March 2011
Revised 07 January 2012
Accepted 10 January 2012
Published online 15 March 2012

Acknowledgments:
The authors have discussed the substance of this paper with several of their colleagues. At the risk of omitting some relevant names, they would like to especially thank James G. McDaniel, William M. Carey, William L. Siegmann, and Richard B. Evans. A.G.T. would like to thank the General Electric Company for its support of his efforts associated with the work reported here.

Article outline:

I. INTRODUCTION
II. POSING THE PROBLEM
A. Vibrating cylinder in a wall
B. Notation
C. Governing equations
1. The artifice of fictitious damping
2. Boundary conditions
3. Causality requirement for unbounded media
4. Energy corollary
5. Proof of uniqueness
D. Free-surface, with gravity; an alternate interpretation
III. FORMULATION FOR FIXED FREQUENCY
A. Introduction of complex amplitudes
B. Causality and Fourier transforms
C. Closing the contour at infinity
D. Parameter regimes
E. Use of dimensionless variables
IV. INNER SOLUTION
A. First approximation to inner solution
B. Hypothesized inner solution
C. Identification of Fourier coefficients
D. Outer expansion of inner solution
V. OUTER SOLUTION, RIGID-BAFFLE LIMIT
A. Use of superposition
B. The Hankel function
C. Approximation for small argument
D. Contour deformation
E. Neglect of contour segment at infinity
F. Expansion in terms of exponential integrals
G. Ordering system
H. Matching of solutions
I. Summary of inner and outer solutions
VI. OUTER SOLUTION, GENERAL CASE
A. Hypothesized form of outer solution
B. Partial differential equation for the Hankel function
C. Hankel function expressed as a Fourier integral
D. Causality requirements on the integrand factor
E. Definition of functions in the complex plane
F. Introduction of branch cuts
G. Satisfying the compliant-surface boundary condition
H. Residue theorem and branch-line integrals
I. Inner expansion of outer solution
J. Matching of inner and outer solutions
VII. ENTRAINED MASS AND RADIATION RESISTANCE
A. Comparison with Ursell’s “results”
B. Further note regarding the Euler–Mascheroni constant
C. Radiation resistance derived from inner solution
VIII. FAR FIELD SOLUTIONS AND RADIATION RESISTANCE
A. Far field for the rigid-surface case
1. Radiation resistance derived from outer solution
B. Far field for the incompressible-fluid case
1. Radiation resistance derived from outer solution
IX. CONCLUDING REMARKS

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2012-03-15

2016-05-30

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