A closed organ pipe has a fundamental frequency of 380 Hz at 30C. What would be its next higher resonant frequency at 18C?
To determine the next higher resonant frequency of a closed organ pipe at a different temperature, we can use the formula:
f2 = ((v2 / v1) * f1) + f1
where:
f2 = the next higher resonant frequency at the new temperature
f1 = the fundamental frequency at the original temperature
v1 = velocity of sound at the original temperature
v2 = velocity of sound at the new temperature
First, let's calculate the velocity of sound at both temperatures using the formula:
v = 331.5 + (0.6 * T)
where:
v = velocity of sound in m/s
T = temperature in degrees Celsius
For the original temperature of 30°C:
v1 = 331.5 + (0.6 * 30)
v1 = 331.5 + 18
v1 = 349.5 m/s
For the new temperature of 18°C:
v2 = 331.5 + (0.6 * 18)
v2 = 331.5 + 10.8
v2 = 342.3 m/s
Now, substitute these values into the formula for f2:
f2 = ((v2 / v1) * f1) + f1
f2 = ((342.3 / 349.5) * 380) + 380
f2 ≈ 371.16 Hz
Therefore, the next higher resonant frequency at 18°C would be approximately 371.16 Hz.