Question

An auditorium of dimensions $\left( {100 \times 40 \times 10} \right){m^3}$ contains $1000{m^2}$ curtains of absorption coefficient $0.2{m^{ - 2}},2000{m^2}$ of carpets of absorption coefficient $0.7{m^{ - 2}}$ .If 1000 men of absorption coefficient .9 per person are sitting in the hall, then reverberation time is
A. 2.7s
B. 7.2s
C. 3.5s
D. 3.7s

Answer 1

Hint:The reverberation time of an auditorium is defined as the time it takes for sound to decay by 60dB. For example, if the sound in an auditorium took 10 seconds to decay from 100dB to 40dB, the reverberation time would be 10 seconds then reverberation time of an auditorium can be written as the T60 time.

Complete Step by Step Answer:
Sabine found that the time of reverberation depends upon the size of the auditorium, volume of the Sound and the kind of music or sound for which auditorium is to be used. Acoustically good auditorium is that where time of reverberation is negligibly small. In Case of a speech, a series of notes are produced in an auditorium each one has its own intensity.

The rate of decreasing intensity of an impulse should be allowing the other without confusion. Hence there should not be any confusion.The reverberation time is directly proportional to the volume of the auditorium and inversely proportional to the averaged absorption coefficient surface area of the walls and the materials inside the auditorium. So,
$T \propto \dfrac{V}{{\sum {as} }}$
Where, V is the volume of the auditorium, ‘a’ is the absorption coefficient of an area ‘s’.
According to the Sabine theory,
\[T = \dfrac{{0.161V}}{{\sum {as} }}\]
Different absorbing surfaces of area represent as ${s_1}$, ${s_2}$ ,${s_3}$ ,${s_4}$ etc., having absorption coefficients ${a_1}$, ${a_2}$, ${a_3}$, ${a_4}$ etc., then,
$T = \dfrac{{0.161V}}{{{a_1}{s_1} + {a_2}{s_2} + ....}}$
As given in the question, Volume of the auditorium is
$
V = (100 \times 40 \times 10){m^3} \\
\Rightarrow V = 4000{m^3} \\
$
Absorption coefficient(a1) of curtains with surface area(s1) as per question is
$
{a_1} = 0.2{m^{ - 2}} \\
\Rightarrow{s_1} = 1000{m^2} \\
$
Absorption coefficient(a2) of carpets with surface area(s2) as per question is
$
{a_2} = 0.7{m^{ - 2}} \\
\Rightarrow{s_2} = 2000{m^2} \\
$
Absorption coefficient(a3) per persons for (s3) persons as per question is
$
{a_3} = 0.9 \\
\Rightarrow{s_3} = 1000 \\
$
Putting these values we get:
$T = \dfrac{{0.161 \times 40000}}{{(0.2 \times 1000) + (0.7 \times 2000) + (0.9 \times 1000)}}$
$
T = \dfrac{{0.161 \times 4 \times {{10}^4}}}{{2500}} \\
\therefore T = 2.57s \\ $
Hence, option A is the correct answer.

Note: It should be noted that if we know sound absorption coefficient of a substance or material and also amount of it present in a room, we can easily estimate the total absorption in a room. The total absorption area is calculated as the sum of all surface areas in the room, each multiplied by its respective absorption coefficient. Moreover if the size of the room as well as total absorption is known, the reverberation time can be calculated (through Sabine’s equation) even before the room is built.