Quarterly of Applied Mathematics

Quarterly of Applied Mathematics

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The potential for a homogeneous cylinder in a cylindrical coordinate system


Author: Wan-xian Wang
Journal: Quart. Appl. Math. 49 (1991), 163-171
MSC: Primary 31B15
DOI: https://doi.org/10.1090/qam/1096238
MathSciNet review: MR1096238
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Abstract: Some special cases of the potential for a homogeneous cylinder in a cylindrical coordinate system may be treated by virtue of simple integrals, for example, the potential for a straight rod or wire segment and that for a homogeneous cylinder at the point on its axis. However, because of the involved mathematical operations, the analytical formula of the potential for a homogeneous cylinder at an arbitrary point has not been seen from others. In order to solve the problem, the author has taken the following steps: (1) expanding Green's function $ {e^{ik\left\vert {r' - r} \right\vert}}/\left\vert {r' - r} \right\vert$ in the cylindrical coordinate system; (2) transforming Green's function $ {e^{ik\left\vert {r' - r} \right\vert}}/\left\vert {r' - r} \right\vert$ into Green's function $ 1/\left\vert {r' - r} \right\vert$ by setting the wave number $ k$ to be zero and integrating the separated azimuthal function $ {\cos ^n}\left( {\phi ' - \phi } \right)$; (3) using the integral recursion relation for the function $ {r'^{2m + 1}}/{\left[ {{{\left( {z' - z} \right)}^2} + {{r'}^2} + {r^2}} \right]^{\left( {4m + 1} \right)/2}}$ with respect to $ r'$ and those for the functions $ 1/{\left[ {{{\left( {z' - z} \right)}^2} + {r^2}} \right]^{\left( {2m - 1} \right)/2}}$ and $ 1/{\left[ {{{\left( {z' - z} \right)}^2} + {r^2} + {a^2}} \right]^{\left( {4m - 2l - 1} \right)/2}}$ with respect to $ z'$ , then we can complete the integrals for the function $ 1/\left\vert {r' - r} \right\vert$ and obtain the analytical expression of the potential for the cylinder in the cylindrical coordinate system. For numerical comparison, we have calculated the potentials for the cylinder and the prolate or oblate spheroid with equivalent volume and same high aspect ratio at far field point. The results are satisfactory.


References [Enhancements On Off] (What's this?)

  • [1] O. D. Kellogg, Foundations of Potential Theory, Dover, New York, 1929
  • [2] William Duncan MacMillan, The theory of the potential, MacMillan’s Theoretical Mechanics, Dover Publications, Inc., New York, 1958. MR 0100172
  • [3] Wan-xian Wang, Expansion of the Green’s function in a cylindrical coordinate system, Quart. Appl. Math. 48 (1990), no. 3, 499–501. MR 1074964, https://doi.org/10.1090/qam/1074964
  • [4] G. N. Watson, A treatise on the theory of Bessel functions, Cambridge Mathematical Library, Cambridge University Press, Cambridge, 1995. Reprint of the second (1944) edition. MR 1349110
  • [5] T. M. MacRobert, Spherical Harmonics, Pergamon Press, 1927

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DOI: https://doi.org/10.1090/qam/1096238
Article copyright: © Copyright 1991 American Mathematical Society


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