Two transverse waves travel along the same taut string and interfere. Waves $1$ and $2$ are described, respectively, by the function y$1($x$,$ t$)=$ A sin$($kx $-$ t$)$ and y$2($x$,$ t$)=$ Acos$(3$kx $+3$t$).$ The phases $($arguments of the sine and cosine functions$)$ are in radians, as usual, and A $=7\mathrm{.}6$ cm$,$ k $=3\mathrm{.}5$ rad$/$m$,$ $=7\mathrm{.}9$ rad$/$s$.$

$-$ Part $($a$)$

In what direction does each wave travel?

Wave $1$ travels in the positive x$-$direction, while wave $2$ travels in the negative x$-$direction.$$Correct$!$

$-$ Part $($b$)$

Which option correctly describes wave $2$ as a sine wave?

y$2($x$,$t$)=$ A sin$(3$kx$+3$t$+$$/2)$Correct$!$

$-$ Part $($c$)$

Use the trigonometric identity to find the correct total wave function for the interfering waves on the string.

y$($x$,$t$)=2$A cos$($kx $+2$t $+$ $/4)$ sin$(2$kx $+$ t $+$ $/4)$Correct$!$

$$ Part $($d$)$

For A $=7\mathrm{.}6$ cm$,$ k $=3\mathrm{.}5$ rad$/$m$,$ and $=7\mathrm{.}9$ rad$/$s$,$ calculate the total displacement of the string, in centimeters, at the position x $=2\mathrm{.}8$ m at time t $=4\mathrm{.}4$ s$.$