What are the three longest wavelengths for standing sound waves in a 124-cm-long tube that is (a) open at both ends and (b) open at one end, closed at the other?

To find the three longest wavelengths for standing sound waves in a 124-cm-long tube, we'll use the formulas for open and closed tubes.

(a) For a tube that is open at both ends:
In this case, the standing waves that can be formed will have nodes at both ends of the tube. The wavelength can be calculated using the formula:
λ = 2L / n
where λ is the wavelength, L is the length of the tube, and n is the harmonic number.

To find the longest wavelengths, we need to consider the fundamental frequency (n = 1), the first overtone (n = 2), and the second overtone (n = 3).

1. Fundamental Frequency (n = 1):
λ = 2L / n = 2 * 124 cm / 1 = 248 cm

2. First Overtone (n = 2):
λ = 2L / n = 2 * 124 cm / 2 = 124 cm

3. Second Overtone (n = 3):
λ = 2L / n = 2 * 124 cm / 3 ≈ 82.7 cm

Therefore, the three longest wavelengths for a tube that is open at both ends are 248 cm, 124 cm, and approximately 82.7 cm.

(b) For a tube that is open at one end and closed at the other:
In this case, the standing waves that can be formed will have a node at the closed end and an antinode at the open end. The wavelength can be calculated using the formula:
λ = 4L / (2n - 1)

1. Fundamental Frequency (n = 1):
λ = 4L / (2n - 1) = 4 * 124 cm / (2 * 1 - 1) = 4 * 124 cm / 1 = 496 cm

2. First Overtone (n = 2):
λ = 4L / (2n - 1) = 4 * 124 cm / (2 * 2 - 1) = 4 * 124 cm / 3 ≈ 165.3 cm

3. Second Overtone (n = 3):
λ = 4L / (2n - 1) = 4 * 124 cm / (2 * 3 - 1) = 4 * 124 cm / 5 ≈ 99.2 cm

Therefore, the three longest wavelengths for a tube that is open at one end and closed at the other are 496 cm, approximately 165.3 cm, and approximately 99.2 cm.

To determine the three longest wavelengths for standing sound waves in a tube, we need to calculate the fundamental frequency and then find the corresponding harmonics.

(a) For a tube open at both ends:
The fundamental frequency (first harmonic) can be calculated using the formula:

f1 = v / (2L)

where v is the speed of sound and L is the length of the tube. In this case, L = 124 cm = 1.24 meters.

Let's assume the speed of sound in air is approximately 343 meters per second.

Substituting these values into the formula, we get:

f1 = 343 / (2 * 1.24)

Now, we can find the wavelength of the fundamental frequency (λ1) using the formula:

λ1 = v / f1

Substituting the values, we find:

λ1 = 343 / (343 / (2 * 1.24)) = 2 * 1.24 meters

To find the second harmonic (first overtone), we multiply the fundamental frequency by 2:

f2 = 2 * f1

λ2 = λ1 / 2

Similarly, for the third harmonic (second overtone):

f3 = 3 * f1

λ3 = λ1 / 3

(b) For a tube open at one end and closed at the other:
The fundamental frequency (first harmonic) can be calculated using the formula:

f1 = v / (4L)

Using the same values as in part (a), we find:

f1 = 343 / (4 * 1.24)

λ1 = v / f1

λ1 = 343 / (343 / (4 * 1.24)) = 4 * 1.24 meters

To find the second harmonic (first overtone) in this case, we multiply the fundamental frequency by 3:

f2 = 3 * f1

λ2 = λ1 / 3

Similarly, for the third harmonic (second overtone):

f3 = 5 * f1

λ3 = λ1 / 5

Therefore, the three longest wavelengths for a tube open at both ends are 2 * 1.24 meters, 1 * 1.24 meters, and (2/3) * 1.24 meters.

For a tube open at one end and closed at the other, the three longest wavelengths are 4 * 1.24 meters, (4/3) * 1.24 meters, and (4/5) * 1.24 meters.