Transverse waves on a string have wave speed $,$ amplitude $0\mathrm{.}0700$ m$,$ and wavelength $0\mathrm{.}320$ m$.$ The waves travel in the $,$ and at the end of the string has its maximum upward displacement. $($a$)$ Find the frequency, period, and wave number of these waves. $($b$)$ Write a wave function describing the wave. $($c$)$ Find the transverse displacement of a particle at at time $.($d$)$ How much time must elapse from the instant in part $($c$)$ until the particle at next has maximum upward displacement?