# The top string of a guitar has a fundamental frequency of 33O Hz when it is allowed to vibrate as a whole, along all its 64.0-cm length from the neck

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The top string of a guitar has a fundamental frequency of 33O Hz when it is allowed to vibrate as a whole, along all its 64.0-cm length from the neck to the bridge. A fret is provided for limiting vibration to just the lower two thirds of the string, If the string is pressed down at this fret and plucked, what is the new fundamental frequency? The guitarist can play a "natural harmonic" by gently touching the string at the location of this fret and plucking the string at about one sixth of the way along its length from the bridge. What frequency will be heard then?

2021-02-28
As the string is vibrating in its fundamental frequency
$$\displaystyle{L}=\frac{\lambda}{{2}}$$
$$\displaystyle\lambda={2}{L}={2}\cdot{0.64}{m}={1.28}{m}$$
The speed of the wave in the string is
$$\displaystyle{V}={f}\lambda={330}\cdot{1.28}={422}{H}{z}$$
a) If the string is stopped by a fret then
L = (2/3)0.64
so $$\displaystyle{\left(\frac{{2}}{{3}}\right)}{0.64}=\frac{\lambda}{{2}}$$
$$\displaystyle\lambda={0.853}{m}$$
frequency $$\displaystyle{f}_{{{1}}}=\frac{{V}}{\lambda}$$
= 422Hz / 0.853 = 495 Hz
b) When plucking the string at one sixth the way of the lengthof the string
$$\displaystyle{2}{L}={3}\lambda$$ ( the string vibrates with third resonance Possibility)
$$\displaystyle\lambda={0.4266}{m}$$
frequency $$\displaystyle{f}_{{{2}}}=\frac{{422}}{{0.27}}{m}={990}{H}{z}$$.