A variable-length air column is placed just below a vibrating wire that is fixed at the both ends. The length of air column open at one end is gradually increased from zero until the first position of resonance is observed at 20 cm. The wire is 115.4 cm long and is vibrating in its third harmonic.
Find the speed of transverse waves in the wire if the speed of sound in air is 340 m/s. Answer in units of m/s
To find the speed of transverse waves in the wire, we can use the equation:
v = f * λ
Where:
- v is the speed of waves
- f is the frequency of the waves
- λ is the wavelength of the waves
In this case, we know the following information:
- The vibrating wire is fixed at both ends, so it is generating standing waves.
- The length of the wire is 115.4 cm.
- The wire is vibrating in its third harmonic, which means there are three nodes (two antinodes) in the wire.
- The length of the air column open at one end is 20 cm, which is equal to half of the wavelength of the third harmonic.
To find the frequency of the third harmonic, we need to know the fundamental frequency (first harmonic). However, since we are not provided with that information, we can use the relationship between harmonics and the fundamental frequency to estimate the frequency of the third harmonic:
f₃ = 3 * f₁
Now, let's calculate the frequency of the third harmonic:
f₃ = 3f₁
To find the wavelength, we can use the formula for calculating the wavelength in a vibrating string for the third harmonic:
λ = 2L / 3
where L is the length of the wire in meters. Let's convert L to meters first:
L = 115.4 cm = 115.4 / 100 m
Now we can calculate the wavelength:
λ = 2 * (115.4 / 100) / 3
Once we have the frequency and wavelength, we can find the speed of transverse waves in the wire:
v = f₃ * λ
Now we can substitute the values into the equation and calculate the speed of transverse waves in the wire.