The length of a closed pipe is 160 mm calculate the wavelength and the frequency of first overtone and third harmonic
To calculate the wavelength and frequency of the first overtone and third harmonic in a closed pipe, we can use the formula:
c = 2Lf
Where:
c = speed of sound (approximately 343 m/s)
L = length of the pipe (converted to meters)
f = frequency
First, let's convert the length of the pipe from mm to meters:
L = 160 mm = 160/1000 m = 0.16 m
1) First overtone (second harmonic):
In the first overtone, we have one additional antinode compared to the fundamental mode. This means that the length of the pipe is half the wavelength.
Wavelength (λ) = 2L = 2 * 0.16 m = 0.32 m
To calculate the frequency, we rearrange the formula:
f = c / 2L = 343 m/s / (2 * 0.16 m) ≈ 1067.19 Hz
Therefore, the wavelength of the first overtone is 0.32 m and the frequency is approximately 1067.19 Hz.
2) Third harmonic:
In the third harmonic, there are two additional antinodes compared to the fundamental mode. This means that the length of the pipe is one-third of the wavelength.
Wavelength (λ) = 3L = 3 * 0.16 m = 0.48 m
To calculate the frequency:
f = c / λ = 343 m/s / 0.48 m ≈ 714.58 Hz
Therefore, the wavelength of the third harmonic is 0.48 m and the frequency is approximately 714.58 Hz.
To calculate the wavelength and frequency of the first overtone and third harmonic of a closed pipe, we need to know the fundamental frequency. The fundamental frequency (f₁) of a closed pipe is determined by the length of the pipe and the speed of sound in the medium it is in.
Given:
Length of the closed pipe (L) = 160 mm = 0.16 meters
The formula to calculate the fundamental frequency of a closed pipe is:
f₁ = (v / 4L)
Where:
f₁ = fundamental frequency
v = speed of sound
L = length of the closed pipe
The speed of sound in air at room temperature is approximately 343 meters per second.
Using the formula, we can calculate the fundamental frequency (f₁):
f₁ = (343 / 4 * 0.16)
f₁ = 2151.875 Hz (rounded to 3 decimal places)
Now, let's calculate the wavelength and frequency of the first overtone (2nd harmonic) and the third harmonic.
Wavelength (λ₁) of the first harmonic (fundamental frequency) is given by:
λ₁ = v / f₁
λ₁ = 343 / 2151.875
λ₁ ≈ 0.159 meters (rounded to 3 decimal places)
To find the wavelength of the first overtone (2nd harmonic), we need to consider that the first overtone is twice the frequency of the fundamental frequency. Therefore, the frequency of the first overtone (f₂) is 2 * f₁.
Wavelength (λ₂) of the first overtone (2nd harmonic) is given by:
λ₂ = v / f₂
λ₂ = 343 / (2 * 2151.875)
λ₂ ≈ 0.080 meters (rounded to 3 decimal places)
To find the frequency of the first overtone (2nd harmonic):
f₂ = 2 * f₁
f₂ = 2 * 2151.875
f₂ ≈ 4303.75 Hz (rounded to 3 decimal places)
Since the third harmonic is thrice the frequency of the fundamental frequency, the frequency of the third harmonic (f₃) is 3 * f₁.
Wavelength (λ₃) of the third harmonic is given by:
λ₃ = v / f₃
λ₃ = 343 / (3 * 2151.875)
λ₃ ≈ 0.053 meters (rounded to 3 decimal places)
To find the frequency of the third harmonic:
f₃ = 3 * f₁
f₃ = 3 * 2151.875
f₃ ≈ 6455.625 Hz (rounded to 3 decimal places)
In summary:
- Wavelength and frequency of the first overtone (2nd harmonic):
Wavelength (λ₂) ≈ 0.080 meters
Frequency (f₂) ≈ 4303.75 Hz
- Wavelength and frequency of the third harmonic:
Wavelength (λ₃) ≈ 0.053 meters
Frequency (f₃) ≈ 6455.625 Hz