A sound source is fixed at the origin and continuously emits sound...

The sound source is stationary and fixed at the origin $(0,0)$.

The listener travels along the line $x = 336 \text{ m}$, which means the listener's velocity is directed along the y-axis (i.e., $\vec{v}_L = \pm 26 \hat{j} \text{ m s}^{-1}$).

When the listener is positioned at $y = 0$, their coordinates are $(336, 0)$. The line of sight joining the source to the listener is along the x-axis.

The angle $\theta$ between the listener's velocity vector (along the y-axis) and the line of sight (along the x-axis) is $90^{\circ}$.

The component of the listener's velocity along the direction of sound propagation is:
$v_{L,\parallel} = v_L \cos(90^{\circ}) = 0$

According to the Doppler effect for a stationary source and a moving observer, the observed frequency $f'$ is given by:
f' = f_0 \left(\dfrac{v \pm v_{L,\parallel}}{v} \right)

Substituting the given values:
f' = 660 \left(\dfrac{330 \pm 0}{330} \right) =$660 \text{ Hz}$

Since the listener's velocity has no component along the line joining the source and the listener at that instant, there is no Doppler shift. The observed frequency is equal to the emitted frequency.

Answer: $660 \text{ Hz}$