Question

Two waves $y_1=A\cos (0.5\pi x-100\pi t)$ and $y_2=A\cos (0.46\pi x-92\pi t)$ are travelling in a pipe placed along $x$-axis.
Find the number of times $y_1+y_2=0$ at $x=0$ in $1s$.

• A. 100
• B. 46
• C. 192
• D. 96

Answer 1

Answer: (D)

We have
$y_1+y_2=A\cos 100\pi t+A\cos 92\pi t=0$
$\cos 100\pi t=-\cos 92\pi t$
$\cos (\pi -\theta )=-\cos \theta$
Therefore,
$100\pi t=(2n+1)\pi -92\pi t$
$\Rightarrow t=\frac{2n+1}{192}$
Therefore,
$\Delta t=t_{n+1}-t_n=\frac{2}{192}$
Hence, in $1s,y_1+y_2=0$ for $192/2=96$ times