A vertical tube with a tap at the base is filled with water, and a tuning fork vibrates over its mouth. As the water level is lowered in the tube, resonance is heard when the water level has dropped 18 cm, and again after 54 cm of distance exists from the water to the top of the tube. What is the frequency of the tuning fork?

To find the frequency of the tuning fork, we need to understand the concept of resonance in the context of this vertical tube experiment.

In this experiment, resonance occurs when the frequency of the tuning fork matches the natural frequency of the air column in the tube. When this happens, a loud sound is heard. Resonance can only occur when the column of air between the water surface and the top of the tube has a length that is an integer multiple of half the wavelength of the sound wave produced by the tuning fork.

Given that resonance is heard at two different water levels: one when the water level has dropped 18 cm, and again after 54 cm of distance exists from the water to the top of the tube, we can calculate the wavelength of the sound wave produced by the tuning fork at these two water levels.

Let's assume the speed of sound in air is approximately 343 m/s.

At the first resonance, when the water level has dropped 18 cm, the height of the air column is equal to the wavelength of the sound wave. Therefore, the wavelength can be calculated as follows:

Wavelength = 2 * height of the air column
= 2 * 0.18 m
= 0.36 m

At the second resonance, when the water level has dropped 54 cm, the height of the air column is equal to three times the wavelength of the sound wave. Therefore, the wavelength can be calculated as follows:

Wavelength = 2 * height of the air column
= 2 * 0.54 m
= 1.08 m

Now, we can use the formula for the speed of a wave (v) = frequency (f) × wavelength (λ) to calculate the frequency (f) of the tuning fork at the two different water levels:

For the first resonance:
frequency (f1) = v / λ1
= 343 m/s / 0.36 m
= 952.78 Hz

For the second resonance:
frequency (f2) = v / λ2
= 343 m/s / 1.08 m
= 317.59 Hz

Therefore, the frequency of the tuning fork is approximately 952.78 Hz at the first resonance and approximately 317.59 Hz at the second resonance.