Question

Two identical transverse sinusoidal waves travel in opposite directions along a string. The speed of transverse waves in the string is $0.5\text{cm/s}$. Each has an amplitude of $3.0\text{cm}$ and wavelength of $6.0\text{cm}$. The equation for the resultant wave is

• A. $y=6\sin \left(\frac{\pi t}{6}\right)\cos \left(\frac{\pi x}{3}\right)$
• B. $y=6\sin \left(\frac{\pi x}{3}\right)\cos \left(\frac{\pi t}{6}\right)$
• C. Both $\left(a\right)$ and $\left(b\right)$ may be correct
• D. Both $\left(a\right)$ and $\left(b\right)$ are wrong

Answer 1

The correct option is C Both $\left(a\right)$ and $\left(b\right)$ may be correct

Standing wave will be formed by their superposition.
So,
Wave number, $k=\frac{2\pi }{\lambda }=\frac{2\pi }{6}=\frac{\pi }{3}{\text{cm}}^{-1}$
Angular frequency,
$\omega =v\times k=0.5\times \frac{\pi }{3}=\frac{\pi }{6}{\text{s}}^{-1}$
Maximum amplitude of standing wave,
$A_{max}=2A=6\text{cm}$
On putting above data in standard equations of standing wave,
$y=2A\sin \left(\omega t\right)\cos \left(kx\right)$
$y=2A\cos \left(\omega t\right)\sin \left(kx\right)$, we get,
$y=6\sin \left(\frac{\pi t}{6}\right)\cos \left(\frac{\pi x}{3}\right)$
$y=6\sin \left(\frac{\pi x}{3}\right)\cos \left(\frac{\pi t}{6}\right)$
Both forms may be possible.
Exact form will depends on initial condition of string.