A half-open pipe is constructed to produce a fundamental frequency of 230 Hz when the air temperature is 17 °C. It is used in an overheated building when the temperature is 39 °C. Neglecting thermal expansion in the pipe, what frequency will be heard?
The first temperature is T1 =273 +17 = 290K
The speed of sound at T1 is v1=332•sqrt(T1/273) =
=332•sqrt(290/273) =342.18 m/s.
v= L•f.
The length of pipe is L=v1/f1=342.18 /230=1.49 m
The second temperature is T2 =273 +39 = 312K
The speed of sound at T2 is v2=332•sqrt(T2/273) =
=332•sqrt(312/273) =354.9 m/s.
f2=v2/L=354.9/1.49=238.20 Hz
To calculate the frequency that will be heard in an overheated building, we need to consider the change in temperature and its effect on the speed of sound. The speed of sound in air is given by the formula:
v = √(γ * R * T)
Where:
v = speed of sound
γ = ratio of specific heat capacities for air (approximately 1.4)
R = gas constant for air (approximately 287 J/kgK)
T = temperature in Kelvin
First, let's convert the initial temperature of 17 °C to Kelvin:
T1 = 17 + 273 = 290 K
Next, let's convert the overheated temperature of 39 °C to Kelvin:
T2 = 39 + 273 = 312 K
By plugging these values into the formula, we can find the initial speed of sound, v1, and the overheated speed of sound, v2.
v1 = √(γ * R * T1)
v1 = √(1.4 * 287 * 290)
v2 = √(γ * R * T2)
v2 = √(1.4 * 287 * 312)
Now, we need to consider the half-open pipe. The fundamental frequency of a half-open pipe can be calculated using the formula:
f = v / (2L)
Where:
f = frequency
v = speed of sound
L = length of the pipe
Since we have only been given the frequency for the initial temperature, we can rearrange the formula to solve for the length of the pipe based on the initial frequency:
L = v1 / (2f)
Substituting the values of v1 and the initial frequency of 230 Hz, we can calculate the initial length of the pipe, L1:
L1 = v1 / (2 * 230)
To find the new frequency, we will use the same formula, but this time using the overheated speed of sound, v2, and the initial length of the pipe, L1:
f2 = v2 / (2L1)
Now, let's substitute the values and calculate the new frequency:
f2 = v2 / (2 * L1)
Finally, we can substitute the values of v2 and L1 and calculate the new frequency:
f2 = √(1.4 * 287 * 312) / (2 * L1)
Simplifying this expression will give you the frequency (f2) that will be heard in the overheated building.