Question
The wave described by
$$y = 0.25\sin \left( {10\,\pi x - 2\,\pi t} \right),$$
where, $$x$$ and $$y$$ are in metre and $$t$$ in second, is a wave travelling along the
A.
negative $$x$$-direction with frequency $$1\,Hz$$
B.
positive $$x$$-direction with frequency $$\pi \,Hz$$ and wavelength $$\lambda = 0.2\,m$$
C.
positive $$x$$-direction with frequency $$1\,Hz$$ and wavelength $$\lambda = 0.2\,m$$
D.
negative $$x$$-direction with amplitude $$0.25\,m$$ and wavelength $$\lambda = 0.2\,m$$
Answer :
positive $$x$$-direction with frequency $$1\,Hz$$ and wavelength $$\lambda = 0.2\,m$$
Solution :
$$y = 0.25\sin \left( {10\,\pi x - 2\,\pi t} \right)$$
Compare the above equation with
$$y = A\sin \left( {kx - \omega t} \right)$$
As $${\omega t}$$ and $$kx$$ have opposite sign, wave travels along positive $$x.$$
$$\eqalign{
& {\text{As,}}\,\,2\,\pi t = \omega t \cr
& \Rightarrow \omega = 2\pi = 2\pi \nu \cr
& \Rightarrow \nu = 1\,Hz \cr
& {\text{Also,}}\,\,kx = 10\,\pi x \cr
& k = 10\,\pi = \frac{{2\,\pi }}{\lambda } \cr
& \lambda = \frac{{2\,\pi }}{{10\,\pi }} = 0.2 \cr} $$
Hence, option (C) is correct.