Think back to the lab we did where you have a string fixed at both ends being ca

Question

Think back to the lab we did where you have a string fixed at both ends being caused to vibrate in its resonant modes (harmonics) by a mechanical wave driver. You have a string of length L=2 meters vibrating in its third harmonic. You notice that this string vibrates at middle C, which you know to be 261.6 Hz. Trying to connect this physics to the music you have learned, you wonder what harmonic the string will vibrate at when its frequency is one octave above middle C. (An octave is defined when the resulting frequency is twice the original.) Changing nothing about the string but its frequency, what harmonic would you need to find to be one octave above middle C? A good explanation here will include relevant concepts, equations, and drawings.

Explanation / Answer

As frequency is doubled, wavelength is halved because speed is kept constant.

wavelength in a string wave is given by

lambda = 2L/n   where n is the harmonic number.

for halving the lambda, n has to be doubled as length is kept same.

n = 3 * 2 = 6

Hence it needs to be at 6th harmonic

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