On Green’s function for Laplace’s equation in a rigid tube

2021 ◽  
Vol 80 (1) ◽  
pp. 87-98
Author(s):  
P. Martin
Keyword(s):  
Point Source ◽  
Green’S Function ◽  
Potential Theory ◽  
Green's Function ◽  
Classical Problem ◽  
Extensive Literature ◽  
Laplace’S Equation ◽  
Laplace's Equation ◽  
Rigid Tube

A classical problem from potential theory (a point source inside a long rigid tube) is revisited. It has an extensive literature but its resolution is not straightforward: standard approaches lead to divergent integrals or require the discarding of infinite constants. We show that the problem can be solved rigorously using classical methods.

Etude numérique des modes et fréquences propres d'une cavité à l'aide de la fonction de Green

1979 ◽  
Vol 57 (2) ◽  
pp. 208-217
Author(s):  
Jacques A. Imbeau ◽  
Byron T. Darling
Keyword(s):  
Point Source ◽  
Green’S Function ◽  
Green's Function ◽  
Normal Modes ◽  
Preceding Paper ◽  
Double Precision ◽  
Prolate Spheroidal ◽  
Normal Functions ◽  
Normal Frequency ◽  
High Degree

We apply the methods developed in our preceding paper (J. A. Imbeau and B. T. Darling. Can. J. Phys. 57, 190(1979)) for calculating the Green's function of a cavity to obtain the normal modes and normal frequencies of the cavity. As the frequency of the driving point source approaches that of a normal frequency the response (Green's function) of the cavity becomes infinite, and the form of the Green's function is dominated by the normal mode. There is also a 180° reversal of phase in passing through a resonance. In this way, we are able to calculate the normal frequencies of prolate spheroidal cavities to the full precision employed in the calculations (16 significant digits for double precision of the IBM-370). The Green's functions and the normal functions are also obtainable to a high degree of precision, except in the immediate vicinity of the surface of the cavity where they suffer a well-known discontinuity.

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