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 $1 \,s$.

• 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=(2 n+1) \pi-92 \pi t $
$\Rightarrow t=\frac{2 n+1}{192}$
Therefore,
$\Delta t =t_{n+1}-t_{n}=\frac{2}{192}$
Hence, in $1 s , y_{1}+y_{2}=0$ for $192 / 2=96$ times