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?
For a tube open at one end, as in (b), there must be an odd number of quarter-wavelengths, if standing waves are to occur. See
http://www.tutorvista.com/content/physics/physics-iii/waves/vibrations-in-pipes.php
For a tube open at both ends, as in (a), there must be an even number of quarter-wavelengths.
To find the three longest wavelengths for standing sound waves in a tube, we can use the formula:
λ = 2L/n
where:
λ = wavelength
L = length of the tube
n = mode number (1, 2, 3, ...)
First, let's calculate the wavelengths for a tube that is open at both ends:
(a) Open at both ends:
Given:
L = 124 cm
We need to find the three longest wavelengths, so let's calculate for mode numbers 1, 2, and 3:
For n = 1:
λ₁ = 2L/1
λ₁ = 2(124 cm)/1
λ₁ = 248 cm
For n = 2:
λ₂ = 2L/2
λ₂ = 2(124 cm)/2
λ₂ = 124 cm
For n = 3:
λ₃ = 2L/3
λ₃ = 2(124 cm)/3
λ₃ ≈ 82.7 cm
Therefore, for a tube that is open at both ends, the three longest wavelengths are:
1. λ₁ = 248 cm
2. λ₂ = 124 cm
3. λ₃ ≈ 82.7 cm
Next, let's calculate the wavelengths for a tube that is open at one end and closed at the other:
(b) Open at one end, closed at the other:
Given:
L = 124 cm
For n = 1:
λ₁ = 2L/1
λ₁ = 2(124 cm)/1
λ₁ = 248 cm
For n = 3:
λ₃ = 2L/3
λ₃ = 2(124 cm)/3
λ₃ ≈ 82.7 cm
For n = 5:
λ₅ = 2L/5
λ₅ = 2(124 cm)/5
λ₅ ≈ 49.6 cm
Therefore, for a tube that is open at one end and closed at the other, the three longest wavelengths are:
1. λ₁ = 248 cm
2. λ₃ ≈ 82.7 cm
3. λ₅ ≈ 49.6 cm
These 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 tube that is open at both ends, we can use the formula:
λ = 2L/n
Where:
λ is the wavelength
L is the length of the tube
n is the harmonic number, which represents the number of half-wavelengths that fit within the tube length.
For a 124-cm-long tube that is open at both ends, the possible harmonic numbers are odd integers (n = 1, 3, 5, ...).
(a) Open at both ends:
Using the formula above, we can calculate the three longest wavelengths for this case:
1. For n = 1: λ = 2(124 cm)/1 = 248 cm
2. For n = 3: λ = 2(124 cm)/3 ≈ 82.67 cm
3. For n = 5: λ = 2(124 cm)/5 ≈ 49.6 cm
Therefore, the three longest wavelengths for standing sound waves in a 124-cm-long tube that is open at both ends are 248 cm, 82.67 cm, and 49.6 cm.
(b) Open at one end, closed at the other:
For a tube that is open at one end and closed at the other, the possible harmonic numbers are odd integers minus one (n = 2, 4, 6, ...).
Using the same formula as before, we can calculate the three longest wavelengths for this case:
1. For n = 2: λ = 2(124 cm)/2 = 124 cm
2. For n = 4: λ = 2(124 cm)/4 = 62 cm
3. For n = 6: λ = 2(124 cm)/6 ≈ 41.33 cm
Therefore, the three longest wavelengths for standing sound waves in a 124-cm-long tube that is open at one end and closed at the other are 124 cm, 62 cm, and 41.33 cm.