What are the three longest wavelengths for standing sound waves in a 126-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 tube, we need to consider the modes of vibration that can exist in the respective conditions. Let's proceed with (a) open at both ends and (b) open at one end, closed at the other.
(a) Open at Both Ends:
For a tube that is open at both ends, the wavelength of the standing waves can be determined by using the formula:
λ = 2L/n
Where:
λ = Wavelength
L = Length of the tube
n = Mode of vibration (1, 2, 3, ...)
We need to find the three longest wavelengths by considering the first three modes of vibration (n = 1, 2, 3).
1. For the first mode of vibration (n = 1):
λ1 = 2L/1 = 2L
2. For the second mode of vibration (n = 2):
λ2 = 2L/2 = L
3. For the third mode of vibration (n = 3):
λ3 = 2L/3
We have found the three longest wavelengths for a tube that is open at both ends.
(b) Open at One End, Closed at the Other:
For a tube that is open at one end and closed at the other, the formula to determine the wavelength of the standing waves is slightly different:
λ = 4L/n
Using the same approach as in the previous case, we can find the three longest wavelengths for the first three modes of vibration (n = 1, 2, 3).
1. For the first mode of vibration (n = 1):
λ1 = 4L/1 = 4L
2. For the second mode of vibration (n = 2):
λ2 = 4L/2 = 2L
3. For the third mode of vibration (n = 3):
λ3 = 4L/3
Now we have determined the three longest wavelengths for a tube that is open at one end and closed at the other.
In summary:
(a) Open at both ends: The three longest wavelengths are 2L, L, and 2L/3.
(b) Open at one end, closed at the other: The three longest wavelengths are 4L, 2L, and 4L/3.
Remember to substitute the given length of the tube (126 cm) into the formulas to obtain the actual values of the wavelengths.
To find the three longest wavelengths for standing sound waves in a 126-cm-long tube, we can use the formula:
λ = 2L/n
Where:
λ is the wavelength,
L is the length of the tube, and
n is the harmonic number.
(a) Open at both ends:
For a tube that is open at both ends, the harmonic numbers (n) can be any positive integer (n = 1, 2, 3, ...).
The three longest wavelengths for this case can be found by determining the first three harmonics:
First harmonic (n = 1): λ₁ = 2L/1 = 2 * 126 cm = 252 cm
Second harmonic (n = 2): λ₂ = 2L/2 = 2 * 126 cm = 252 cm
Third harmonic (n = 3): λ₃ = 2L/3 = 2 * 126 cm / 3 = 84 cm
So, the three longest wavelengths for a tube open at both ends are 252 cm, 252 cm, and 84 cm.
(b) Open at one end, closed at the other:
For a tube that is open at one end and closed at the other, only the odd harmonic numbers (n = 1, 3, 5, ...) are allowed.
Similarly, we calculate the first three harmonics:
First harmonic (n = 1): λ₁ = 2L/1 = 2 * 126 cm = 252 cm
Second harmonic (n = 3): λ₂ = 2L/3 = 2 * 126 cm / 3 = 84 cm
Third harmonic (n = 5): λ₃ = 2L/5 = 2 * 126 cm / 5 = 50.4 cm
So, the three longest wavelengths for a tube open at one end and closed at the other are 252 cm, 84 cm, and 50.4 cm.