A wire having a mass per unit length m and tension T resonates at a frequency of 420 Hz. The next higher frequency at which the wire resonates is 490 Hz. What is the length of the wire?

We know that the resonant frequencies occur at increasing multiples of the fundamental, f0. Since the difference between the first frequency listed (420 Hz) and the next higher one (490 Hz) is 70 Hz, that must be the fundamental frequency. Thus the first frequency represented the sixth harmonic, and the second one was the seventh. The length of the wire is half the wavelength of the fundamental frequency.

L=\frac {\lambda_0} 2 = \frac {\( \frac {v}{f_0}\)} {2} =\frac 1{2f_0} \sqrt{\frac Tm}=\frac 1{140} \sqrt{\frac Tm}