Originally Posted by

**CaptainBlack** Absorb the constants to give:

$\displaystyle

\tan(\delta .x)=m/x

$,

where $\displaystyle m=h/k$, also put $\displaystyle y=\delta .x$, then we are interested in the solutions of:

$\displaystyle \tan(y)=n/y$,

where now $\displaystyle n=\delta .h/k$.

Now when $\displaystyle n \ne 0$ this obviously (if I have done this right) has a solution

in each of the intervals $\displaystyle (0, \pi/2)$, $\displaystyle (-\pi/2, 0)$, $\displaystyle (((k+1)\pi, (2k+3)\pi /2)$,

$\displaystyle (-(2k+3)\pi/2, -(k+1)\pi)\ k=0,\ 1,\ ..$.

Also for large $\displaystyle k$ the roots approach $\displaystyle (k+1) \pi$

(with the mirror image roots when $\displaystyle k$ is negative).

Now numerical methods can be used to find the root in any particular

interval (since we know the interval that a root lies in I would use binary

search for this, though Newton-Raphson is probably faster)

RonL