A dry water well is 1.5 m in radius and 5.0 m deep.
In order from lowest to highest, the fifth resonant frequency (harmonic) of this well is closest to ...
Hint: The water well is not just an open-closed tube
To find the resonant frequency (harmonic) of the well, we first need to understand the basic concept of resonance. Resonance occurs when an object is forced to vibrate at its natural frequency, which causes it to vibrate with maximum amplitude. In the case of the water well, we can assume that the resonant frequencies are related to the standing waves that can form within the well.
We can calculate the resonant frequencies using the formula for the frequency of a standing wave in a closed-open tube, which can be approximated for our purposes:
f = (n * vs) / (4L),
where:
f is the frequency,
n is the harmonic number (1 for the fundamental frequency, 2 for the second harmonic, etc.),
vs is the speed of sound,
and L is the length of the tube.
However, the well is not a simple closed-open tube but a cylindrical structure. Nonetheless, we can approximate the resonant frequencies by considering the first few modes of vibration in the well.
For a cylindrical structure with hard walls like the well, the first few resonant frequencies can be approximated as follows:
f1 ≈ (c / 2π) * (1 / h), (1st harmonic)
f2 ≈ (c / 2π) * (1 / (2h)), (2nd harmonic)
f3 ≈ (c / 2π) * (1 / (3h)), (3rd harmonic)
f4 ≈ (c / 2π) * (1 / (4h)), (4th harmonic)
f5 ≈ (c / 2π) * (1 / (5h)), (5th harmonic)
where:
f1, f2, f3, f4, f5 are the respective frequencies of the first, second, third, fourth, and fifth harmonic,
c is the speed of sound, and
h is the depth of the well.
Given that the well has a radius of 1.5 m and a depth of 5.0 m, we can use these values to calculate the resonant frequencies closest to the fifth harmonic:
f5 ≈ (c / 2π) * (1 / (5 * 5.0)).
The speed of sound in air is approximately 343 m/s.
Substituting the values into the formula, we can calculate the fifth resonant frequency:
f5 ≈ (343 / (2π)) * (1 / 25) ≈ 2.18 Hz (rounded to two decimal places).
Therefore, the fifth resonant frequency (harmonic) of the water well is closest to 2.18 Hz.