How does a wavelength of the fourth harmonic on a rope with both ends fixed compare with the lenght of the rope?
To find the wavelength of the fourth harmonic on a rope with both ends fixed, we need to understand the concept of harmonics. Harmonics are the different modes of vibration that a rope can undergo when it is plucked or struck.
For a rope with both ends fixed, the fundamental frequency (first harmonic) has the longest wavelength. The wavelength of the fundamental frequency is twice the length of the rope. Therefore, the formula for the wavelength of the nth harmonic on a rope with both ends fixed is:
λ = (2L) / n,
where λ is the wavelength, L is the length of the rope, and n is the harmonic number.
In this case, we want to find the wavelength of the fourth harmonic. So we plug in n = 4 into the formula:
λ = (2L) / 4.
Simplifying the equation, we have:
λ = L / 2.
This tells us that the wavelength of the fourth harmonic is equal to half the length of the rope.
Therefore, the wavelength of the fourth harmonic on a rope with both ends fixed is half the length of the rope.