To find the frequency of the wave, we need to start by analyzing the given displacement equation of the wave:

$$y = C \sin \left( \frac{2 \pi}{\lambda} (at - x) \right)$$

Where:

• $y$ is the displacement of the wave
• $C$ is the amplitude
• $$\frac{2 \pi}{\lambda}$$ is the wave number (denoting how many wavelengths fit into a unit length)
• $\lambda$ is the wavelength
• $a$ is some constant (likely representing the speed of the wave)
• $t$ is time
• $x$ is the distance

By comparing with the standard form of a travelling wave, we have:

$$y = C \sin \left( k (at - x) \right)$$

Where $k$ is the wave number:

$$k = \frac{2 \pi}{\lambda}$$

From the standard wave equation, the argument of the sine function is usually written as:

$k (at - x)$

This implies that $ka$ in the wave equation represents the angular frequency $\omega$ of the wave:

$\omega = k a$

Substituting $$k = \frac{2 \pi}{\lambda}$$ into $\omega$:

$$\omega = \left( \frac{2 \pi}{\lambda} \right) a = \frac{2 \pi a}{\lambda}$$

Angular frequency $\omega$ is related to the frequency $f$ by:

$\omega = 2 \pi f$

So:

$$2 \pi f = \frac{2 \pi a}{\lambda}$$

Solving for the frequency $f$:

$$f = \frac{a}{\lambda}$$

Therefore, the correct answer is:

Option D: $\frac{a}{\lambda}$