Question

The volume of a cinema hall is given to be \[100\;{\text{m}} \times 60\;{\text{m}} \times 20\;{\text{m}}\] and the total acoustic absorption in it is\[6800{\text{ metric sabine}}\]. Determine the reverberation time in the cinema hall.
A) \[1.5\;{\text{s}}\]
B) \[2\;{\text{s}}\]
C) \[3\;{\text{s}}\]
D) \[4\;{\text{s}}\]

Answer 1

Hint: First find out the volume of the entire hall and then use the Sabine formula to get the required reverberation time. Convert the given data into proper units and then apply the formula.

Complete step by step answer:
As we know the definition of reverberation time in simple words depicts the time taken for a sound wave to completely fade out after multiple reflections within the boundaries of a space. This depreciation in the sound wave quality occurs because of the absorption of sound inside the space because of large obstacles inside, for example walls in a cinema hall or inside an auditorium. In an auditorium or big hall, excessive reverberation is not desirable. To reduce the reverberation, the roof and walls of the auditorium are generally covered with sound-absorbent materials like compressed fibre board, rough plaster or draperies. These materials have the absorption property, which is denoted in the metric Sabine unit.
The value of one metric Sabin means in one square meter of the absorbing material, the acoustic absorption is $100\% $.
According to the Sabine’s Equation,
The reverberation time is mathematically written as-
\[T = 0.161 \times \dfrac{V}{{{A_t}}}.....\left( 1 \right)\]
Where, \[T\]is the total reverberation time taken for the sound wave.
\[V\] is the volume of the space
\[{A_t}\] is the total acoustic absorption of the material used to reduce the reverberation effect,
Now here we have,
Volume of the cinema hall, \[V = \;100 \times 60 \times 20 = 120000{\text{ }}{{\text{m}}^{\text{3}}}\]
And given value of \[{A_t} = 6800\;{\text{metric sabine}}\]
So by putting the values given above in equation \[\left( 1 \right)\] we get the value of \[T\]as-
\[T = 0.161 \times \dfrac{{120000}}{{6800}}\]
Solving the above equation we get \[T = {\text{2}}{\text{.84}}\;{\text{seconds}}\], which is approximately equal to \[3\;{\text{seconds}}\].
Hence, option (C) is correct.

Note: As we know that the Sabine formula changes if the measurement is given in feet. The Sabine formula was derived by Prof. W.C Sabine where he conducted the experiments in rooms of variable sizes.