Question

A wave generator at one end of a very long string creates a wave given by $y_1=\left(6.0cm\right)\cos \frac{\pi }{2}[\left(2.00m^{-1}\right)x+\left(8.00s^{-1}\right)t]$
and a wave generator at the other end creates the wave
$y_2=\left(6.0cm\right)\cos \frac{\pi }{2}\left[\right(2.00m^{-1})x-(8.00s^{-1}\left)t\right]$
for $x>0$, what is the location of the node (in $cm$) having the smallest value of $x$.

Answer 1

From given equation
$k=\pi \text{and}\omega =4\pi$
$f=\frac{\omega }{2\pi }=\frac{4\pi }{2\pi }=2Hz$
$\lambda =\frac{2\pi }{k}=\frac{2\pi }{\pi }=2m$
$v=\lambda .f=2\times 2=4m/\sec$.
$y_1+y_2=6\cos \frac{\pi }{2}[2x+8t]+6\cos \frac{\pi }{2}[2x-8]$
$=6\cos \{(\pi x+4\pi t)+\cos (\pi x-4\pi t\left)\right\}$
As we know,
$\cos \alpha +\cos \beta =2\cos \left(\frac{\alpha +\beta }{2}\right)\cos \left(\frac{\alpha +\beta }{2}\right)$
$y=6\left[2\cos \right(\pi x).\cos (4\pi t\left)\right]$
$=12\cos \left(\pi x\right)\cos \left(4\pi t\right)$


Hence,
$\cos \left(\pi x\right)=0$

$\pi x=n\pi +\frac{\pi }{2}$

$\therefore x=(n+\frac{1}{2})\text{m}$

For $n=0,x=0.5\text{m}\text{or}50\text{cm}$