An organ pipe that is closed at one end has a fundamental frequency of 122 Hz. There is a leak in the church roof, and some water gets into the bottom of the pipe, as shown in the Figure. The organist then finds that this organ pipe has a frequency of 300 Hz. What is the depth of the water in the pipe?
To find the depth of the water in the pipe, we can use the formula for the frequency of a closed-end organ pipe:
fn = (2n - 1)v / 4L
where fn is the frequency of the nth harmonic, v is the speed of sound, L is the length of the pipe, and n is the harmonic number.
In this case, the fundamental frequency is 122 Hz, so n = 1. We can rearrange the formula to solve for L:
L = (2n - 1)v / 4fn
Substituting the given values, we have:
L = (2 * 1 - 1) * v / (4 * 122)
L = v / 488
Now, let's consider the situation with water in the pipe. The frequency is 300 Hz, so we need to find the new length of the pipe, L'. We can use the same formula as before:
L' = (2 * 1 - 1) * v / (4 * 300)
L' = v / 1200
Since the pipe is closed at one end, the length is given by:
L' = (n + 1/4) * λ/2
where λ is the wavelength of the sound wave.
Let's divide L' by L to simplify the equation:
L' / L = ((n + 1/4) * λ/2) / (v / 488)
Since n = 1, we have:
L' / L = ((1 + 1/4) * λ/2) / (v / 488)
L' / L = (5/4) * λ / (2 * v / 488)
L' / L = (5/2) * λ / (v / 244)
Now, we know that the speed of sound v is given by the equation:
v = √(γ * R * T)
where γ is the adiabatic index for air and is approximately 1.4, R is the gas constant for air, and T is the temperature in Kelvin.
Assuming standard conditions (273 K), we can calculate the speed of sound v.
Finally, we can substitute the values into the equation:
L' / L = (5/2) * λ / (v / 244)
Solving for λ, we get:
λ = (L' / L) * (v / 244) * (2/5)
Since the pipe is closed at one end, the fundamental frequency corresponds to the wavelength of four times the length of the pipe. Therefore:
λ = 4L
Substituting the values, we can solve for the initial length of the pipe, L:
4L = (L' / L) * (v / 244) * (2/5)
L = (L' / 4) * (244 / v) * (5/2)
Now, we have the equation to find the depth of the water in the pipe, denoted as d:
L + d = (L' / 4) * (244 / v) * (5/2)
Simplifying further, we get:
d = (L' / 4) * (244 / v) * (5/2) - L
Substituting the given values, we can calculate the depth of the water in the pipe.