Answer
$T_{app} = t-\frac{vt~cos~\theta}{c}$
Work Step by Step
In time $t$, the knot travels a distance of $vt$
Let $\Delta D$ be the difference in path length of the light from the first burst and the light from the second burst.
We can use trigonometry to find an expression for $\Delta D$:
$\frac{\Delta D}{vt} = cos~\theta$
$\Delta D = vt~cos~\theta$
We can find the time difference of arrival times $T_{app}$:
$T_{app} = t-\frac{\Delta D}{c}$
$T_{app} = t-\frac{vt~cos~\theta}{c}$
