“Helmholtz Differential Equation--Cartesian Coordinates”的概念、定义、习题解题方法及翻译-数学辞典

单词

Helmholtz Differential Equation--Cartesian Coordinates

释义

Helmholtz Differential Equation--Cartesian Coordinates

In 2-D Cartesian Coordinates, attempt Separation of Variables by writing

(1)

then the Helmholtz Differential Equation becomes

(2)

Dividing both sides by

gives

(3)

This leads to the two coupled ordinary differential equations with a separation constant

			(4)
			(5)

where

and

could be interchanged depending on the boundary conditions. These have solutions

The general solution is then

(8)

In 3-D Cartesian Coordinates, attempt Separation of Variables by writing

(9)

then the Helmholtz Differential Equation becomes

(10)

Dividing both sides by

gives

(11)

This leads to the three coupled differential equations

where

, and

could be permuted depending on boundary conditions. The general solution is therefore


	(15)

References

Morse, P. M. and Feshbach, H. Methods of Theoretical Physics, Part I. New York: McGraw-Hill, pp. 501-502, 513-514 and 656, 1953.

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