In Quincke's experiment, the sound intensity exhibits a minimum...

The minimum intensity is given as I_\min = I and the maximum intensity is I_\max = 9I.

The intensity of sound is proportional to the square of its amplitude. For two waves of amplitudes $A_1$ and $A_2$, the maximum and minimum intensities are given by:

I_\max = k(A_1 + A_2)^2

I_\min = k(A_1 - A_2)^2

Taking the ratio of the maximum to minimum intensity:

\dfrac{I_\max}{I_\min} = \dfrac{(A_1 + A_2)^2}{(A_1 - A_2)^2} = \dfrac{9I}{I} = 9

Taking the square root of both sides:

$\dfrac{A_1 + A_2}{A_1 - A_2} = 3$

Cross-multiplying yields:

$A_1 + A_2 = 3A_1 - 3A_2$

$4A_2 = 2A_1$

$\dfrac{A_1}{A_2} = 2$

The ratio of the amplitudes of the two waves is $2$. The data regarding the displacement of the tube and the speed of sound is extraneous for finding this ratio.