Question

Two waves having the intensities in the ratio $9:1$ produce interfaces. The ratio of maximum to minimum intensity is equal to:
(A) $4:1$
(B) $9:1$
(C) $2:1$
(D) $10:8$

Answer 1

Hint: In order to answer this question, first we will assume the two intensities and that to find the intensity, we always need to have the amplitude. As we know that the intensity is directly proportional to the square of the amplitude.

Complete answer:
Let the intensities of two waves be ${I_1}\,and\,{I_2}$ .
Now, as per the question-
The ratio of the intensities of two waves is $9:1$ (given).
$\because {I_1}:{I_2} = 9:1$
We can also write the upper ratio of intensities as:
$\dfrac{{{I_1}}}{{{I_2}}} = \dfrac{9}{1}$
So, now as we know that, the intensity is directly proportional to the square of the amplitude, i.e.
$\therefore I\alpha {a^2}$
we can also write the upper proportional equation as:
$ \Rightarrow \dfrac{{{a_1}}}{{{a_2}}} = \sqrt {\dfrac{{{I_1}}}{{{I_2}}}} $
and as we already know, $\dfrac{{{I_1}}}{{{I_2}}} = \dfrac{9}{1}$
$ \Rightarrow \dfrac{{{a_1}}}{{{a_2}}} = \sqrt {\dfrac{9}{1}} = \dfrac{3}{1}$
So, the ratio of amplitude is, ${a_1}:{a_2} = 3:1$ .
Now, the maximum intensity in the terms of amplitudes:-
$
  \therefore {I_{\max }} = {({a_1} + {a_2})^2} \\
  \,\,\,\,\,\,\,\,\,\,\,\,\, = {(3 + 1)^2} = 16 \\
 $
Similarly, the minimum intensity in the terms of amplitudes:-
$
  \therefore {I_{\min }} = {({a_1} - {a_2})^2} \\
  \,\,\,\,\,\,\,\,\,\,\,\,\, = {(3 - 1)^2} = 4 \\
 $
Now, we have both the maximum and minimum intensity, so we can easily find the ratio of them:
$\therefore \dfrac{{{I_{\max }}}}{{{I_{\min }}}} = \dfrac{{16}}{4} = \dfrac{4}{1}$
We can also write as:
${I_{\max }}:{I_{\min }} = 4:1$
Therefore, the ratio of maximum to minimum intensity is equal to $4:1$ .

Hence, the correct option is (A) $4:1$ .

Note:
The perceived response to the physical feature of intensity is called loudness. It's a more complicated subjective quality linked with a wave. The larger the amplitude, the greater the intensity, and the louder the sound in general. The term "loud" refers to sound waves with a large amplitude.