San José State University |
---|
applet-magic.com Thayer Watkins Silicon Valley & Tornado Alley USA |
---|
Generalized Helmholtz Equation |
---|
The equation under consideration is
If k is constant then the solution is φ(x)=A·sin(kx)+B·cos(kx). This suggests that the solution to the general equation may be a function of ∫_{0}^{x}k(ζ)dζ.
Consider a change of the independent variable such that
The first derivative of φ with respect to x is given by
And the second derivative is
When this expression is substituted into the Helmholtz equation under consideration the result is
Note that
Let d(1/k)/dx be denoted as g(z). The transformed equation is then
HOME PAGE OF Thayer Watkins |