^{1,a)}and Amadou G. Thiam

^{1}

### 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.

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.

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

### Key Topics

- Boundary value problems
- 29.0
- Surface waves
- 13.0
- Real functions
- 12.0
- Philosophy of science
- 11.0
- Fourier transforms
- 10.0

## Figures

Sketch illustrating the problem of acoustic radiation caused by the vibrations of a long rigid cylinder oscillating in a compliant baffle.

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 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 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.

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.

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 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.

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.

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.

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.

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.)

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.

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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