Two pipes of equal length are each open at one end. Each has a fundamental frequency of 715 Hz at 300 K. In one pipe the air temperature is increased to 313 K. If the two pipes are sounded together, what beat frequency results?
To find the beat frequency resulting from two pipes sounded together, we need to first calculate the fundamental frequency of each pipe at the new temperature, and then find the difference between these frequencies.
The fundamental frequency of a pipe is given by the formula:
f = v / (2L)
Where f is the frequency, v is the speed of sound in air, and L is the length of the pipe.
Given that the two pipes are equal in length and open at one end, their fundamental frequencies at 300 K would be the same. Let's calculate this frequency:
f1 = v / (2L)
Next, we need to calculate the fundamental frequency of the first pipe at the new temperature of 313 K:
f1' = v' / (2L)
To find v', the new speed of sound at 313 K, we can use the formula:
v' = v * sqrt(T' / T)
Where v is the speed of sound at 300 K, T' is the new temperature in Kelvin (313 K), and T is the initial temperature in Kelvin (300 K).
Now, let's calculate v':
v' = v * sqrt(T' / T)
v' = v * sqrt(313 / 300)
Next, we can use f1' = v' / (2L) to find the new fundamental frequency of the first pipe at 313 K.
Now, let's calculate the beat frequency:
Beat frequency = |f1 - f1'|
Finally, we have the beat frequency resulting from two pipes sounded together at different temperatures.