A half-open pipe is constructed to produce a fundamental frequency of 224 Hz when the air temperature is 15 °C. It is used in an overheated building when the temperature is 38 °C. Neglecting thermal expansion in the pipe, what frequency will be heard?
The first temperature is T1 =273 +15 = 288K
The speed of sound at T1 is v1=332•sqrt(T1/273) =
=332•sqrt(288/273) =341 m/s.
v= L•f.
The length of pipe is L=v1/f1=341/224=1.52 m
The second temperature is T2 =273 +38 = 311K
The speed of sound at T2 is v2=332•sqrt(T2/273) =
=332•sqrt(311/273) =354.4 m/s.
f2=v2/L=354.4/1.52=233.13 Hz
To determine the frequency that will be heard in the overheated building, we need to consider the effect of temperature on the speed of sound in air.
The speed of sound in air can be calculated using the formula:
V = sqrt(γ * R * T)
where:
- V is the speed of sound in air
- γ is the adiabatic index (for air, it is approximately 1.4)
- R is the gas constant for air (approximately 287 J/(kg·K))
- T is the temperature in Kelvin
First, let's convert the temperatures to Kelvin:
Temperature in Kelvin (T1) = 15 °C + 273.15 = 288.15 K
Temperature in Kelvin (T2) = 38 °C + 273.15 = 311.15 K
Next, we calculate the ratio of the speeds of sound (V1 and V2) using the formula:
V2 / V1 = sqrt(T2 / T1)
Now we can find the new fundamental frequency (F2) using the formula:
F2 = F1 * (V2 / V1)
where:
- F1 is the initial fundamental frequency (224 Hz)
- F2 is the new fundamental frequency
Let's plug in the values and calculate:
F2 = 224 Hz * (sqrt(311.15 K / 288.15 K))
Calculating the equation, we find:
F2 ≈ 224 Hz * 1.046
Therefore, the frequency that will be heard in the overheated building will be approximately:
F2 ≈ 234.3 Hz