Question

The reverberation times in a cinema theatre are $3s$, $2s$ when it is empty, filled with the audience respectively. The reverberation time when the theatre is half filled with audience is:
A. $2.3 s$
B. $2.4 s$
C. $2.5 s$
D. $2.6 s$

Answer 1

Hint:When sound produced in a closed space, it will be bouncing with the walls, flooring and surface of the objects present in that room and will eventually fade away. The person in that room will continue to receive continuous reflections of diminishing sound. This continuation of sound before it fades away to a negligible extent is called reverberation. Reverberation time is considered as the measure of time required to fade away sound in the room completely. The reverberation time depends upon the size of the room and material used to construct the room.

Formula Used:
The formula for calculating reverberation time is:
$\dfrac{{R{T_1}}}{{R{T_2}}} = \dfrac{{{A_1} + {A_2}}}{{{A_1}}}$

Complete step by step answer:
The reverberation time is the time required to decrease the intensity of sound in the cinema hall to one millionth of its initial intensity. To calculate the reverberation time for the cinema hall, we need to keep in mind a few things.

Step 1: Let us first assume a few things for simplifying our calculation.
The area of the cinema hall is not given, so let us assume the area of an empty cinema hall as ${A_1}$. Similarly, we will take ${A_2}$ as an area of cinema hall filled with audience.We can say, the area of the cinema hall when completely filled with the audience is ${A_1} + {A_2}$.
Now, we need to calculate the reverberation time for a half-filled theatre, for which we first need to calculate the area of theatre half filled with the audience.Hence, the area of the theatre when it is half filled will be taken as ${A_1} + \frac{{{A_2}}}{2}$. Now, as given the reverberation time of an empty theatre is, $R{T_1} = 3$sec.Similarly, as provided the reverberation time when theatre is filled with the audience, \[R{T_2} = 2\] sec.

Let us assume the reverberation time when the theatre is half filled with audience be $R{T_3}$.
As we know, the reverberation time is inversely proportional to area.
$RT \propto \dfrac{1}{A}$
Let us substitute the given values and we get,
$ \dfrac{{R{T_1}}}{{R{T_2}}} = \dfrac{{{A_1} + {A_2}}}{{{A_1}}}$……………….(1)

Step 2: Let us substitute the given values in the equation (1) to obtain the reverberation time for a theatre half filled with audience.
As we know, $R{T_1} = 3\sec $ and $R{T_2} = 2\sec $
The equation (1) will be written as,
$\dfrac{3}{2} = \dfrac{{{A_1} + {A_2}}}{{{A_1}}}$
\[
\Rightarrow 3{A_1} = 2{A_1} + 2{A_2}\]
$\Rightarrow 3{A_1} - 2{A_1} = 2{A_2}$
$\Rightarrow {A_1} = 2{A_2}$………………….(2)

Step 3: Now, reverberation time for theatre half filled with audience is,
$\dfrac{{R{T_1}}}{{R{T_3}}} = \dfrac{{{A_1} + \dfrac{{{A_2}}}{2}}}{{{A_1}}}$ …….(3) (Because, the reverberation time is inversely proportional to area of the theatre.)
As we know, $R{T_1} = 3\sec $and ${A_1} = 2{A_2}$
Substitute these values in equation (3), we get,
$\dfrac{3}{{R{T_3}}} = \dfrac{{2{A_2} + \dfrac{{{A_2}}}{2}}}{{2{A_2}}}$
$\because {A_1} = 2{A_2}$
\[
\Rightarrow \dfrac{3}{{R{T_3}}} = \dfrac{{\dfrac{{5{A_2}}}{2}}}{{2{A_2}}}\]
$\Rightarrow \dfrac{3}{{R{T_3}}} = \dfrac{5}{4}$
$\Rightarrow R{T_3} = \dfrac{{3 \times 4}}{5}$
$\Rightarrow R{T_3} = \dfrac{{12}}{5}$
$\therefore R{T_3} = 2.4\sec $

The reverberation time when theatre is half filled with audience be $R{T_3}$= 2.4 sec. So, the correct answer is option (B).

Note:The right amount of reverberation is good for cinema halls and orchestra hall, as it enhances the sound quality. Excess of reverberation may cause difficulties for a listener in that room. As excess of reverberation leads to hearing of both direct and reflected sounds by a listener which will make it difficult for a listener to understand what the speaker is speaking.