In an organ pipe with one end closed, a standing wave is formed. If the wave speed of longitudinal wave is v in the air contained by organ pipe and length of pipe is ℓ , then equation of wave can be represented by
Question
In an organ pipe with one end closed, a standing wave is formed. If the wave speed of longitudinal wave is v in the air contained by organ pipe and length of pipe is ℓ , then equation of wave can be represented by
Solution 1
In an organ pipe with one end closed, the standing wave formed is a result of the superposition of the incident wave and the reflected wave. The condition for the formation of the standing wave is that the length of the pipe should be an odd multiple of the quarter wavelength.
The wavelength (λ) of the wave can be represented as:
λ = 4ℓ/n, where n is an odd integer (1, 3, 5, ...)
The frequency (f) of the wave can be calculated using the wave speed (v) and the wavelength (λ):
f = v/λ
Substituting the value of λ from the first equation into the second equation, we get:
f = nv/4ℓ
The equation of the wave can be represented as:
y(x, t) = A sin(kx - ωt + φ)
where:
- y(x, t) is the displacement of the wave at position x and time t
- A is the amplitude of the wave
- k is the wave number, which is equal to 2π/λ
- ω is the angular frequency, which is equal to 2πf
- φ is the phase constant
Substituting the values of k and ω from the above equations, we get:
y(x, t) = A sin((2πn/4ℓ)x - (2πnv/4ℓ)t + φ)
This is the equation of the wave in an organ pipe with one end closed.
Solution 2
In an organ pipe with one end closed, the standing wave formed is a result of the superposition of the incident wave and the reflected wave. The condition for the formation of the standing wave is that the length of the pipe should be an odd multiple of the quarter wavelength.
The wavelength (λ) of the wave can be represented as:
λ = 4ℓ/n, where n is an odd integer (1, 3, 5, ...)
The frequency (f) of the wave can be calculated using the wave speed (v) and the wavelength (λ):
f = v/λ
Substituting the expression for λ from above, we get:
f = nv/4ℓ
The equation of the wave can be represented as:
y(x, t) = A sin(kx - ωt + φ)
where:
- y(x, t) is the displacement of the wave at position x and time t
- A is the amplitude of the wave
- k is the wave number, which is equal to 2π/λ
- ω is the angular frequency, which is equal to 2πf
- φ is the phase constant
Substituting the expressions for λ and f from above, we get:
y(x, t) = A sin((2πn/4ℓ)x - (2πnv/4ℓ)t + φ)
This is the equation of the wave in an organ pipe with one end closed.
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