Step 1: Understanding the Question:
This problem explores how changing the physical dimensions and surface properties of a room affects its reverberation time.
Reverberation time is the duration required for the sound pressure level to decay by 60 dB.
Step 2: Key Formula or Approach:
The governing principle for reverberation in standard enclosures is Sabine’s Formula:
\[ T = \frac{0.161 \cdot V}{A} \]
where \(T\) is the reverberation time, \(V\) is the volume of the room, and \(A\) is the total absorption in metric sabins.
Step 3: Detailed Explanation:
Let the initial volume be \(V_1\) and the initial total absorption be \(A_1\).
The initial reverberation time is given by \(T_1 = \frac{0.161 \cdot V_1}{A_1}\).
According to the conditions specified in the question:
1. The new volume \(V_2 = 2V_1\) (Volume is doubled).
2. The new total absorption \(A_2 = \frac{A_1}{2}\) (Absorption is halved).
Now, let's calculate the new reverberation time \(T_2\) by substituting these values into the formula:
\[ T_2 = \frac{0.161 \cdot (2V_1)}{(A_1 / 2)} \]
Mathematically, dividing by a fraction is the same as multiplying by its reciprocal:
\[ T_2 = \frac{0.161 \cdot 2V_1 \cdot 2}{A_1} \]
\[ T_2 = 4 \cdot \left( \frac{0.161 \cdot V_1}{A_1} \right) \]
By comparing this to the initial equation, we can see that \(T_2 = 4 \cdot T_1\).
Physically, this makes sense because a larger volume means sound waves have to travel further before hitting a wall, and reduced absorption means they lose less energy per collision.
Both factors independently contribute to making the sound "linger" longer in the room.
Step 4: Final Answer:
By doubling the volume and halving the absorption, the reverberation time increases by a total factor of four.