Answer
(a) $f_1 = 12~Hz$
$v = 24~m/s$
(b) The image shows the standing wave pattern for the fourth harmonic, which is 48 Hz.
Work Step by Step
(a) Let $f_{n+1} = 48~Hz$ and let $f_n = 36~Hz$. We can find the fundamental frequency:
$f_{n+1} = (n+1)~f_1$ and $f_n = n~f_1$
$f_{n+1}- f_n = (n+1)~f_1 - n~f_1 = f_1$
Therefore, $f_1 = f_{n+1}-f_n$
$f_1 = 48~Hz-36~Hz = 12~Hz$
We can find the wave speed.
$v = f_1~\lambda_1$
$v = (f_1)(2L)$
$v = (12~Hz)(2)(1.0~m)$
$v = 24~m/s$
(b) The image shows the standing wave pattern for the fourth harmonic, which is 48 Hz.
