Two pipes, equal in length, are each open at one end. Each has a fundamental frequency of 475 Hz at 297 K. In one pipe the air temperature is increased to 304 K. If the two pipes are sounded together, what beat frequency results?
The speed of sound is higher by a factor sqrt(475/297) = 1.265 at the higher temperature.
Frequency = (wave speed)/(wavelength)
Fundamental wavelength will not change with T because it depends upon pipe length.
If the cooler-air pipe has a fundamental frequency of 475 Hz, it will be 475*1.265 = 600.1 Hz at the higher temperature.
The beat frequency will be 600 - 475 = 125 Hz.
To find the beat frequency resulting from sounding two pipes together, we need to calculate the frequency difference between the two pipes when they are at different temperatures.
The fundamental frequency of a closed or open pipe can be calculated by using the formula:
f = (n * v) / (2L)
Where:
f is the frequency,
n is the harmonic number (1 for the fundamental frequency),
v is the speed of sound in air (approximately 343 m/s at room temperature),
and L is the length of the pipe.
In this case, we have two pipes that are equal in length, so the fundamental frequencies of the pipes will be the same at 297 K. Let's calculate the fundamental frequency of each pipe at this temperature.
For the first pipe, we have:
f₁ = (1 * v) / (2L)₁
For the second pipe, we have:
f₂ = (1 * v) / (2L)₂
Since the lengths of the pipes are the same, we can simplify this to:
f₁ = f₂ = f₀ (say)
Now, the beat frequency (f_beat) is the difference between the frequencies of the two pipes when they are at different temperatures.
We can find the frequency of the second pipe at 304 K using the formula:
f₂' = (1 * v') / (2L)
Where:
f₂' is the frequency of the second pipe at 304 K,
v' is the speed of sound in air at 304 K,
and L is the length of the pipe.
The change in temperature does not affect the length of the pipe, so both pipes are still equal in length.
Since we know that f₁ = f₂ = f₀, the beat frequency can be calculated as:
f_beat = |f₁ - f₂'|
Now, let's substitute the given values into the equations and calculate the result: