For the resonance frequencies: fn = nv/2L If the speed (v) doubles, will my fn also double? Why?
The Correct Answer and Explanation is:
Yes, if the speed vvv doubles, the resonance frequency fnf_nfn will also double. Here’s why:
The formula for the resonance frequencies of a string or air column is given by:fn=nv2Lf_n = \frac{nv}{2L}fn=2Lnv
Where:
- fnf_nfn is the nnn-th resonance frequency,
- nnn is the harmonic number (1 for the fundamental frequency, 2 for the first overtone, and so on),
- vvv is the speed of the wave (speed of sound in the air or wave propagation speed in the string),
- LLL is the length of the string or air column.
Now, if the speed vvv doubles, you are effectively changing the wave’s speed, which directly influences the resonance frequency. The relationship between frequency and speed is linear. That means if the speed of the wave increases, the frequency also increases proportionally.
If vvv becomes 2v2v2v, then the new resonance frequency fn′f’_nfn′ becomes:fn′=n(2v)2L=2×nv2L=2fnf’_n = \frac{n(2v)}{2L} = 2 \times \frac{nv}{2L} = 2f_nfn′=2Ln(2v)=2×2Lnv=2fn
Thus, the resonance frequency will also double. The reason for this is that, with a higher speed, the waves can complete more oscillations per unit of time, resulting in a higher frequency. This happens because the wave is moving faster, allowing it to pass through the medium (the string or air column) in a shorter period of time, which increases the number of complete cycles (or oscillations) that occur in a given time frame.
This principle is fundamental in wave mechanics, where changes in the speed of wave propagation have a direct, proportional effect on frequency.
