Vibrations from a $622-$Hz tuning fork set up standing waves in a string clamped at both ends. The wave speed for the string is $388$ m$/$s$.$ The standing wave has four loops and an amplitude of $1\mathrm{.}90$ mm$.($a$)$ What is the length of the string? $($b$)$ Write an equation for the displacement of the string as a function of position and time.