The formula that relates the fringe width with the wavelength of the source is given as,

⇒β=λD/d

So, we can notice that keeping the distance between the screen and the slits, that is, D and the distance between the slits, that is, d constant, the fringe width becomes directly proportional to the wavelength of the light. Therefore, we get,

⇒ β ∝ λ…… (1)

The formula that relates the frequency with the wavelength of the source is given by,

⇒ f = c/λ Where λ is the wavelength of the light, c is the speed of the light in air and f is the frequency of the light/source.

So, here, the speed of the light in air is constant, so, the frequency becomes inversely proportional to the wavelength of the light. Hence, we get,

⇒f ∝ 1/λ Hence we can write,

⇒ λ ∝ 1/f…… (2) Let us compare the equations (1) and (2), thus,

we get, ⇒ β ∝ 1/f

Finally, the relation between the fringe width and the frequency of the source is obtained. So, we can say that the fringe width and the frequency of the source are inversely proportional to each other. Now, we need to find the change in the fringe width when the frequency doubles.

⇒f′=3f

So in place of f′ we can write 1/β′

Hence we get, ⇒1/β′=3 x 1/β

On taking inverse, we get ⇒ β′ = β/3