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Step-by-Step Solution
Step 1: Understand the Concept of Reverberation Time
Reverberation time ($T_R$) is defined as the time taken for the sound intensity in a room (or auditorium) to drop to one millionth of its original value. It describes how long sound persists after its source stops emitting.
Step 2: Recall the Formula for Reverberation Time
According to Sabine’s formula, the reverberation time is given by:
$$
T_R = \frac{K \, V}{\alpha \, S},
$$
where:
$V$ is the volume of the room.
$S$ is the total surface area of the room.
$K$ is a constant.
$\alpha$ is the average absorption coefficient of the surfaces.
Step 3: Relate Volume and Surface Area to the Room’s Dimensions
Consider the room to be approximately cubic with side length $l$. Then:
$V = l^3$ (volume of a cube).
$S = 6 l^2$ (surface area of a cube).
Substituting into the reverberation time formula:
$$
T_R = \frac{K \, l^3}{\alpha \, (6\,l^2)} = \frac{K}{6\,\alpha}\, l.
$$
Hence, $T_R$ is directly proportional to $l$.
Step 4: Doubling the Dimensions
If all dimensions of the room are doubled, the new side length becomes $2l$. Therefore,
$$
\text{New } T_R \propto 2l.
$$
Step 5: Calculate the New Reverberation Time
Since $T_R \propto l$, doubling $l$ doubles the reverberation time. The initial reverberation time is given as 1 second, so the new reverberation time becomes:
$$
T_R' = 2 \times 1 \,\text{ second} = 2 \,\text{seconds}.
$$
Answer
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