Mineral wool is a material used in construction for the purposes of insulation and soundproofing. A sheet of 11 cm in thickness causes the original intensity to decrease to 63% of its original value.
You are trying to build a soundproof room that reduces sound coming in and out by 95%. That is, the intensity after crossing the walls is only 5% of the initial value.
What should be the thickness of the mineral wool used in the walls of your soundproof room?
im assuming I need to find the absorption constant from the 63% decrease and 11 cm thickness, and then use that value to find the thickness necessary for the soundproof room
I found a solution to this question in a previous post.
To find the necessary thickness of the mineral wool for your soundproof room, you need to determine the absorption constant based on the given information.
First, let's calculate the absorption constant for the sheet of mineral wool with a thickness of 11 cm and causing a decrease in intensity to 63% of its original value.
The decrease in intensity can be expressed as a fraction or a decimal. In this case, it is 63%, which can be written as 0.63.
The absorption coefficient (denoted by α) is defined as the fraction of sound absorbed by the material. It can be represented as:
α = 1 - fraction of sound transmitted
Since the question mentions a decrease in intensity, we can infer that the transmitted fraction represents the remaining intensity after going through the material. Therefore, the fraction of sound transmitted can be calculated as:
Transmitted fraction = 1 - 0.63
Now, we can substitute the transmitted fraction in the expression for α:
α = 1 - (1 - 0.63)
Simplifying, we get:
α = 0.63
So, the absorption coefficient for the mineral wool with a thickness of 11 cm is 0.63.
Now, to find the necessary thickness for your soundproof room, where the sound intensity after crossing the walls is only 5% of the initial value, we can use the following formula:
Fraction of sound transmitted = (1 - α)^number of layers
In this case, the fraction of sound transmitted is 5%, which can be written as 0.05.
Now, let's rearrange the formula to solve for the number of layers:
number of layers = log(base 1-α) (fraction of sound transmitted)
Using logarithms with base (1 - α), we can solve for the number of layers.
number of layers = log(base 0.37) (0.05) (Approximately, log base 0.37 of 0.05)
Once you have the number of layers, you can multiply it by the thickness of one layer (11 cm in this case) to determine the total thickness required for the mineral wool in your soundproof room.
Please note that the approximation (log base 0.37 of 0.05) assumes that the absorption constant remains the same for each layer.
Keep in mind that other factors like the construction of the walls, additional materials, etc., can influence the overall soundproofing effectiveness. Consulting with a professional or acoustic engineer is advisable for precise calculations in real-world scenarios.