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 \(\displaystyle{f}_{{{2}}}=\frac{{422}}{{0.27}}{m}={990}{H}{z}\).

\(\displaystyle{L}=\frac{\lambda}{{2}}\)

\(\displaystyle\lambda={2}{L}={2}\cdot{0.64}{m}={1.28}{m}\)

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 \(\displaystyle{f}_{{{2}}}=\frac{{422}}{{0.27}}{m}={990}{H}{z}\).