1 \( - ve\,x - \) direction with frequency \(1\,Hz\)
2 \( + ve\,x - \) direction with frequency \(\pi \) \(Hz\) and the wavelength \(\lambda = 0.2\,m\)
3 \( + ve\,x - \) direction with frequency \(1 \) \(Hz\) and the wavelength \(\lambda = 0.2\,m\)
4 \( + ve\,x - \) direction with frequency \(0.25 \) \(m\) and the wavelength \(\lambda = 0.2\,m\)
Explanation:
The given wave equation
\(y = 0.25\sin (10\pi x - 2\pi t)\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\left( 1 \right)\)
The minus (-) between \((10 \pi x)\) and \((2 \pi t)\) implies that the wave is travelling along positive \(x\) direction.
Now comparing Eq. (1) with standard wave equation
\(y = a\sin (kx - \omega t)\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\left( 2 \right)\)
We have
\(a = 0.25m,\omega = 2\pi ,k = 10\pi m\)
\(\therefore \;\;\;{\mkern 1mu} {\kern 1pt} \frac{{2\pi }}{T} = 2\pi \Rightarrow f = 1Hz\)
\({\text{ Also, }}\lambda = \frac{{2\pi }}{k} = \frac{{2\pi }}{{10\pi }} = 0.2\;m\)