Question

Two waves travelling in opposite directions produce a standing wave. The individual wave functions are given by $y_1=4\sin (3x-2t)cm$ and $y_2=4\sin (3x+2t)cm$, 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 $\frac{\pi }{6},\frac{\pi }{2},\frac{5\pi }{6}$, $\frac{7\pi }{6},\dots$

Answer 1

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

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