Answer
(a) $\lambda_1 = 3.00~m$
$f_1 = 20.7~Hz$
(b) $\lambda_3 = 1.00~m$
$f_3 = 62.0~Hz$
(c) $\lambda_4 = 0.750~m$
$f_4 = 82.7~Hz$
Work Step by Step
(a) We can find the wavelength of the fundamental, which occurs when $n = 1$.
$\lambda_n = \frac{2L}{n}$
$\lambda_1 = \frac{2L}{1}$
$\lambda_1 = (2)(1.50~m)$
$\lambda_1 = 3.00~m$
We can find the frequency of the fundamental.
$f_1 = \frac{v}{\lambda_1}$
$f_1 = \frac{62.0~m/s}{3.00~m}$
$f_1 = 20.7~Hz$
(b) We can find the wavelength of the second overtone, which is the third harmonic. This occurs when $n = 3$.
$\lambda_n = \frac{2L}{n}$
$\lambda_3 = \frac{2L}{3}$
$\lambda_3 = \frac{(2)(1.50~m)}{3}$
$\lambda_3 = 1.00~m$
We can find the frequency of the second overtone.
$f_3 = \frac{v}{\lambda_3}$
$f_3 = \frac{62.0~m/s}{1.00~m}$
$f_3 = 62.0~Hz$
(c) We can find the wavelength of the fourth harmonic, which occurs when $n = 4$.
$\lambda_n = \frac{2L}{n}$
$\lambda_4 = \frac{2L}{4}$
$\lambda_4 = \frac{(2)(1.50~m)}{4}$
$\lambda_4 = 0.750~m$
We can find the frequency of the fourth harmonic.
$f_4 = \frac{v}{\lambda_4}$
$f_4 = \frac{62.0~m/s}{0.750~m}$
$f_4 = 82.7~Hz$
