Consider a room that is 6m long, 5m wide and 3m high and whose walls, floor and ceiling have the same sound absorption coefficient (α=0.01). When all machines are in use, the sound pressure level (SPL) within the room amounts to 90 dB. A carpet is to be placed on the floor to improve the working environment. If the noise level within the room is to be lowered to 80 dB, what should the carpet’s sound absorption coefficient (αcarpet) be?
0.019
0.194
0.388
0.038
more than 1 (cannot be reached)
0.777
To solve this problem, we need to understand the concept of sound absorption coefficient and how it affects noise levels in a room.
The sound absorption coefficient, represented by the symbol α, measures the amount of sound energy absorbed by a material. It ranges from 0 (perfectly reflective) to 1 (perfectly absorbent).
In this case, the room is 6m long, 5m wide, and 3m high. The walls, floor, and ceiling all have the same sound absorption coefficient α = 0.01. The initial sound pressure level (SPL) within the room when all machines are in use is 90 dB.
Now, we want to lower the noise level within the room to 80 dB by placing a carpet on the floor. We need to determine the sound absorption coefficient (αcarpet) required for the carpet to achieve this reduction in noise level.
To solve this problem, we can use the Sabine equation, which relates the sound absorption coefficient, room dimensions, and initial and final sound pressure levels. The equation is as follows:
SPLfinal = SPLinitial + 10 × log10(1/αtotal)
Where:
SPLfinal is the desired final sound pressure level (80 dB)
SPLinitial is the initial sound pressure level (90 dB)
αtotal is the total sound absorption coefficient of the room and carpet.
We can rearrange the equation to solve for αtotal:
1/αtotal = 10 ^ ((SPLfinal - SPLinitial) / 10)
Let's calculate the value:
1/αtotal = 10 ^ ((80 - 90) / 10) = 10 ^ (-1) = 0.1
Now, since the walls, floor, and ceiling all have the same sound absorption coefficient α = 0.01, we can express the total sound absorption coefficient of the room and carpet as:
αtotal = αexisting walls + αcarpet
0.1 = 0.01 + αcarpet
Simplifying the equation:
αcarpet = 0.1 - 0.01 = 0.09
Therefore, the carpet's sound absorption coefficient (αcarpet) should be 0.09.
None of the given options match exactly with the calculated value of 0.09. However, assuming that the provided options are rounded, the closest match would be 0.019.
To determine the sound absorption coefficient (αcarpet) needed to lower the noise level within the room to 80 dB, we can use the Sabine's equation:
L1 = L2 + 10 * log10(A2/A1)
Where:
L1 is the initial sound pressure level (SPL) within the room (90 dB),
L2 is the desired SPL within the room (80 dB),
A1 is the initial total absorption in the room (including walls, floor, and ceiling),
A2 is the total absorption in the room after the carpet is placed.
Let's calculate A1:
A1 = α * surface area
Where α is the sound absorption coefficient (α = 0.01) and the surface area is the sum of the areas of the walls, floor, and ceiling:
surface area = 2 * (length * height + width * height) + length * width
= 2 * (6m * 3m + 5m * 3m) + 6m * 5m
= 2 * (18m^2 + 15m^2) + 30m^2
= 2 * (33m^2) + 30m^2
= 66m^2 + 30m^2
= 96m^2
So, A1 = 0.01 * 96m^2 = 0.96m^2.
Now, let's calculate A2 using the same formula:
A2 = αcarpet * (surface area + carpet area)
The carpet area is length * width = 6m * 5m = 30m^2.
So, A2 = αcarpet * (96m^2 + 30m^2) = αcarpet * 126m^2.
Using the Sabine's equation, we can rearrange it to find αcarpet:
L1 - L2 = 10 * log10(A2/A1)
10 = 10 * log10(αcarpet * 126m^2 / 0.96m^2)
1 = log10(αcarpet * 131.25)
10^1 = αcarpet * 131.25
10 = αcarpet * 131.25
αcarpet = 10 / 131.25
αcarpet ≈ 0.0763
Therefore, the carpet's sound absorption coefficient (αcarpet) should be approximately 0.0763, which is not provided in the given options.