Question $1(13$ marks$)$

Two sinusoidal transverse waves with the same amplitude, wavelength and period are traveling along $+$x direction. They interfere with each other. Assume the wave displacements are given as:

$y_1=(0\mathrm{.}5,)sin(8x-16t)m,$ and $,y_2=(0\mathrm{.}5,)sin(8x-16t+\frac{}{4})m$

such that $x$ and $t$ are given in meter and second respectively. Find the followings:

$($a$)$ wavelength $,$

$(2$ marks$)$

$($b$)$ period $T,$

$(2$ marks$)$

$($c$)$ wave speed $v,$

$(2$ marks$)$

$($d$)$ Express the resultant wave displacement function $y^{\text{'}}$ in terms of x and t $.$

$(7$ marks$)$