The wave function (in SI units) of a wave is \( \psi(x,t)=10^{3}\sin\pi\left(3\times10^{6}x-9\times10^{14}t\right) \). The speed of the wave is
Answer & explanation
Correct answer: option 2
The correct answer is Option (2) → \( 3\times10^{8}\,\mathrm{m\,s^{-1}} \)
A wave of the form \( \sin(kx-\omega t) \) travels with speed \( v=\omega/k \), where \( k \) is the wave number and \( \omega \) the angular frequency. Comparing with the given wave function:
\( k=3\pi\times10^{6}\,\mathrm{m^{-1}} \)
\( \omega=9\pi\times10^{14}\,\mathrm{rad\,s^{-1}} \)
\( \begin{aligned} v &= \frac{\omega}{k} \\ &= \frac{9\pi\times10^{14}\,\mathrm{rad\,s^{-1}}}{3\pi\times10^{6}\,\mathrm{m^{-1}}} \\ &= 3\times10^{8}\,\mathrm{m\,s^{-1}} \end{aligned} \)
This is the speed of light in vacuum.