Solution

Given,

y₁ = a₁ sin (ωt),
and, y₂ = a₂ sin(ωt)

Let I₁ and I₂ be their intensities, then

\(I_1\propto a_1^2\),
\(I_2\propto a_2^2\)

Or, \(\frac{I_1}{I_2}=\left(\frac{a_1}{a_2}\right)^2\)

When, I₁ = 2I₂ we get,

\(\frac{2I_2}{I_2}=\left(\frac{a_1}{a_2}\right)^2\)
Or, \(2=\left(\frac{a_1}{a_2}\right)^2\)
Or, \(\frac{a_1}{a_2}=\sqrt2\)
i.e., \(\frac{a_1}{a_2}=\sqrt2:1\)

Thus, the ratio of these amplitudes would be √2 : 1