Question

Two waves travelling in opposite directions produce a standing wave. The individual wave functions are given by $y_{1}=4 \sin (3 x-2 t) cm$ and $y_{2}=4 \sin (3 x+2 t) cm$, where $x$ and $y$ are in $cm$ :

• A. The maximum displacement of the motion at x=2.3 cm is 4.63 cm
• B. The maximum displacement of the motion at x=2.3 cm is 5.32 cm
• C. Nodes are formed at $x$ values given by $0, \frac{\pi}{3}, \frac{2 \pi}{3}, \pi, \frac{4 \pi}{3}$,
• D. Antinodes are formed at $x$ values given by $\frac{\pi}{6}, \frac{\pi}{2}, \frac{5 \pi}{6}$, $\frac{7 \pi}{6}, \ldots$

Answer 1

Answer: (A), (C), (D)

Resulting wave equation will be
$y= 8 \sin 3 x \cos 2 t-R \cos 2 t$
at $x=2.3$ we use $R=8 \sin (6.9)=4.63\, cm$
Nodes $(R=0)$ are obtained where $\sin (3 x)=0$
$\Rightarrow x=0, \frac{\pi}{3}, \frac{2 \pi}{3}, \ldots$
Antinodes $(R=8\, cm )$ are obtained where $\sin (3 x)=\pm 1$
$\Rightarrow x=\frac{\pi}{6}, \frac{\pi}{2}, \frac{5 \pi}{6},...$