Helmholtz Differential Equation--Polar Coordinates
المؤلف:
Morse, P. M. and Feshbach, H
المصدر:
Methods of Theoretical Physics, Part I. New York McGraw-Hill
الجزء والصفحة:
...
18-7-2018
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Helmholtz Differential Equation--Polar Coordinates
In two-dimensional polar coordinates, the Helmholtz differential equation is
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(1)
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Attempt separation of variables by writing
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(2)
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then the Helmholtz differential equation becomes
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(3)
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Multiply both sides by 
to obtain
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(4)
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The solution to the second part of (4) must be periodic, so the differential equation is
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(5)
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which has solutions
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(6)
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Plug (5) back into (4)
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(7)
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This has solution
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(8)
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where 
and 
are Bessel functions of the first and second kinds, respectively. Combining the solutions gives the general solution
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![F(r,theta)=sum_(m=0)^infty[A_mcos(mtheta)+B_msin(mtheta)]×[C_mJ_m(kr)+D_mY(kr)].](http://mathworld.wolfram.com/images/equations/HelmholtzDifferentialEquationPolarCoordinates/NumberedEquation9.gif)
REFERENCES:
Morse, P. M. and Feshbach, H. Methods of Theoretical Physics, Part I. New York McGraw-Hill, pp. 502-504, 1953.
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