Answer
$\Delta f=\frac{1}{t}$
(see step by step solution)
Work Step by Step
The active medium in a particular laser that generates laser light at a wavelength of $\lambda=694\;nm$ is $L=6.00\;cm$ long and $d=1.00\;cm$ in diameter.
We treat the medium as an optical resonance cavity analogous to a closed organ pipe.
For closed organ pipe, the general formula wavelengths is
$\lambda=\frac{4L\mu}{2n-1}$
where, $n=1,2,3,4................$. The value of $n$ also corresponds to the number of nodes. $\mu$ is the refractive index of the material.
Then the general formula for the frequency is given by
$f=\frac{c}{\lambda}$
or, $f=\frac{c(2n-1)}{4L\mu}$
Partially differentiating both sides keeping $L$, $\mu$ constant, we obtain
$\Delta f=\frac{2c}{4L\mu}\Delta n$
or, $\Delta f=\frac{\Delta n}{t}$
where, $t=\frac{2\mu L}{c}$ is the travel time of laser light for one round trip back and forth along the laser axis.
Here, $\Delta n=1$. Therefore we obtain
$\Delta f=\frac{1}{t}$
Thus the $\Delta f$ is just the inverse of the travel time of laser light for one round trip back and forth along the laser axis.
