Assume the length of an organ pipe is L=24cm. You have excited the wave in the tube, and you observe that it has displacement nodes at 4cm, 12cm, and 20cm measured from one end of the pipe. In which harmonic, n, is the air in the pipe oscillating? What is the frequency of the musical tone produced by the organ pipe in the situation described above.

The air in the pipe is oscillating in the third harmonic, n=3. The frequency of the musical tone produced by the organ pipe is given by the formula f = (n*v)/2L, where v is the speed of sound in air. Therefore, the frequency of the musical tone produced by the organ pipe is (3*v)/48 Hz.

To determine the harmonic, n, in which the air in the pipe is oscillating, we need to find the relationship between the length of the organ pipe and the position of the displacement nodes.

In a closed-end pipe (like this one), the displacement nodes form at intervals that are equal to one-fourth of a wavelength. Therefore, we can identify the length of the pipe that corresponds to one complete wavelength by subtracting the distance between the first two nodes: 12 cm - 4 cm = 8 cm.

Now, we need to figure out the number of half-wavelengths that fit into the overall length of the organ pipe. Dividing the length of the pipe (24 cm) by the length of one half-wavelength (8 cm), we get:

24 cm / 8 cm = 3 half-wavelengths.

Since each half-wavelength corresponds to one complete wave, the air in the pipe is oscillating in the third harmonic.

To determine the frequency of the musical tone produced by the organ pipe, we can use the formula:

frequency (f) = velocity (v) / wavelength (λ)

We know the speed of sound in air is approximately 343 m/s, and we have determined the length of one complete wavelength (2 * 8 cm = 16 cm).

First, let's convert the length of the wavelength to meters:

16 cm = 0.16 m

Now, we can calculate the frequency:

f = 343 m/s / 0.16 m

f ≈ 2143.75 Hz

Therefore, the frequency of the musical tone produced by the organ pipe is approximately 2143.75 Hz.

To determine the harmonic in which the air in the pipe is oscillating, we need to consider the relationship between the length of the pipe and the positions of the displacement nodes. The harmonic number, n, is related to the number of nodes and antinodes in the pipe.

In this case, we have three displacement nodes at 4cm, 12cm, and 20cm from one end of the pipe. Let's calculate the distance between these nodes to identify the pattern:

Distance between first and second node = 12cm - 4cm = 8cm
Distance between second and third node = 20cm - 12cm = 8cm

We can observe that the distance between the nodes is the same, so this indicates a fundamental frequency or the first harmonic. Therefore, the air in the pipe is oscillating in the first harmonic.

To find the frequency of the musical tone produced by the organ pipe, we can use the formula:

f = v / λ

Where:
f is the frequency (what we want to find)
v is the velocity of sound in air
λ is the wavelength

The velocity of sound in air at room temperature is approximately 343 meters per second (m/s). However, we need to convert the length of the pipe to meters since the speed of sound is given in meters per second.

L = 24cm = 24 / 100 = 0.24m

Now, we can calculate the wavelength using the formula:

λ = 2L / n

Where:
L is the length of the pipe
n is the harmonic number

Substituting the values, we get:

λ = 2 * 0.24m / 1 = 0.48m

Finally, we can substitute the calculated values into the formula for frequency:

f = 343 m/s / 0.48m

Calculating this expression, we find that the frequency of the musical tone produced by the organ pipe is approximately 714.58 Hz.