$15\mathrm{.}68$ A string with both ends held fixed is vibrating in its third harmonic. The waves have a speed of

$192$

$$m$/$s

and a frequency of $240$ Hz$.$ The amplitude of the standing wave at an antinode is $0\mathrm{.}400$ cm$.($a$)$ Calculate the amplitude at points on the string a distance of $($i$)40\mathrm{.}0$ cm; $($ii$)20\mathrm{.}0$ cm; and $($iii$)10\mathrm{.}0$ cm from the left end of the string. $($b$)$ At each point in part $($a$),$ how much time does it take the string to go from its largest upward displacement to its largest downward displacement? $($c$)$ Calculate the maximum transverse velocity and the maximum transverse acceleration of the string at each of the points in part $($a$).$