Question

The vibrations of a string of length $60cm$ fixed at both ends are represented by the equation $y=4\sin \left[\frac{\pi x}{15}\right]\cos \left(96\pi t\right)$ where $x$ and $y$ are in cm and $t$ in sec. The equation of component waves whose superposition gives the above wave are

• A. $y_1=2\sin (96\pi t+\frac{\pi x}{5})$ , $y_2=-2\sin (96\pi t-\frac{\pi x}{5})$
• B. $y_1=2\sin (\frac{\pi x}{5}+96\pi t)$ , $y_2=-2\sin (\frac{\pi x}{5}-96\pi t)$
• C. $y_1=2sm(\frac{\pi x}{15}+96\pi t)$ , $y_2=-2\sin (96\pi t-\frac{\pi x}{15})$
• D. $y_1=2\sin (\frac{\pi \kappa }{15}+96\pi t)$ $y_2=-2\sin (\frac{\pi x}{15}-96\pi t)$

Answer 1

The correct option is C $y_1=2sm(\frac{\pi x}{15}+96\pi t)$ , $y_2=-2\sin (96\pi t-\frac{\pi x}{15})$

$y=4sin\left(\frac{\pi x}{5}\right)cos\left(96\pi t\right)$

$=2\left[sin\right(\frac{\pi x}{15}+96\pi t)+sin(\frac{\pi x}{15}-96\pi t\left)\right]=y_1+y_2$