Question

Two waves passing through a region are represented by $y _{1}=(2.0\, cm ) \sin \left[\left(3.14\, cm ^{-1}\right) x -\left(157\, s ^{-1}\right) t \right]$ and $y_{2}=(2.5\, cm ) \sin \left[\left(1.57\, cm ^{-1}\right) x-\left(314 s ^{-1}\right) t\right]$ The magnitude displacement of the particle at $x=4.5\, cm$ at time $t=5.0\, ms$ is__________ $cm$

Answer 1

$y_{1}=(2.0\, cm ) \sin \left[\left(3.14 cm ^{-1}\right)(4.5 cm )\right.$
$\left.-\left(157 s ^{-1}\right)\left(5.0 \times 10^{-3} s \right)\right]$
$=(2.0\, cm ) \sin \left[4.5 \pi-\frac{\pi}{2} \times 100 \times 5.0 \times 10^{-3}\right] $
$=(2.0\, cm ) \sin \left[4.5 \pi-\frac{\pi}{4}\right]$
$=(2.0\, cm ) \sin \left(4 \pi+\frac{\pi}{4}\right)=\frac{2.0\, cm }{\sqrt{2}}$
$y_{2}=(2.5\, cm ) \sin \left[\left(1.57\, cm ^{-1}\right)(4.5\, cm )\right.$
$\left.-\left(314\, s ^{-1}\right)\left(5.0 \times 10^{-3}\right)\right]$
$=(2.5\, cm ) \sin \left[2.25 \pi-\frac{\pi}{2}\right]$
$=(2.5\, cm ) \sin \left[2 \pi-\frac{\pi}{4}\right]$
$=-\frac{2.5\, cm }{\sqrt{2}}$
Net displacement:
$y=y_{1}+y_{2}=-\frac{0.5}{\sqrt{2}}=-(0.5) \times 0.70=-0.35\, cm$