TOPICS
Search

Helmholtz Differential Equation--Cartesian Coordinates

In two-dimensional Cartesian coordinates, attempt separation of variables by writing

F(x,y)=X(x)Y(y),
(1)

then the Helmholtz differential equation becomes

(d^2X)/(dx^2)Y+(d^2Y)/(dy^2)X+k^2XY=0.
(2)

Dividing both sides by XY gives

1/X(d^2X)/(dx^2)+1/Y(d^2Y)/(dy^2)+k^2=0.
(3)

This leads to the two coupled ordinary differential equations with a separation constant m^2,

1/X(d^2X)/(dx^2)=m^2
(4)
1/Y(d^2Y)/(dy^2)=-(m^2+k^2),
(5)

where X and Y could be interchanged depending on the boundary conditions. These have solutions

X=A_me^(mx)+B_me^(-mx)
(6)
Y=C_me^(isqrt(m^2+k^2)y)+D_me^(-isqrt(m^2+k^2)y)
(7)
=E_msin(sqrt(m^2+k^2)y)+F_mcos(sqrt(m^2+k^2)y).
(8)

The general solution is then

F(x,y)=sum_(m=1)^infty(A_me^(mx)+B_me^(-mx)) ×[E_msin(sqrt(m^2+k^2)y)+F_mcos(sqrt(m^2+k^2)y)].
(9)

In three-dimensional Cartesian coordinates, attempt separation of variables by writing

F(x,y,z)=X(x)Y(y)Z(z),
(10)

then the Helmholtz differential equation becomes

(d^2X)/(dx^2)YZ+(d^2Y)/(dy^2)XZ+(d^2Z)/(dz^2)XY+k^2XYZ=0.
(11)

Dividing both sides by XYZ gives

1/X(d^2X)/(dx^2)+1/Y(d^2Y)/(dy^2)+1/Z(d^2Z)/(dz^2)+k^2=0.
(12)

This leads to the three coupled differential equations

1/X(d^2X)/(dx^2)=l^2
(13)
1/Y(d^2Y)/(dy^2)=m^2
(14)
1/Z(d^2Z)/(dz^2)=-(k^2+l^2+m^2),
(15)

where X, Y, and Z could be permuted depending on boundary conditions. The general solution is therefore

F(x,y,z)=sum_(l=1)^inftysum_(m=1)^infty(A_le^(lx)+B_le^(-lx))(C_me^(my)+D_me^(-my)) ×(E_(lm)e^(-isqrt(k^2+l^2+m^2)z)+F_(lm)e^(isqrt(k^2+l^2+m^2)z)).
(16)

See also

Cartesian Coordinates, Helmholtz Differential Equation

Explore with Wolfram|Alpha

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.

Referenced on Wolfram|Alpha

Helmholtz Differential Equation--Cartesian Coordinates

Cite this as:

Weisstein, Eric W. "Helmholtz Differential Equation--Cartesian Coordinates." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/HelmholtzDifferentialEquationCartesianCoordinates.html

Subject classifications