Question
The wave described by $$y = 0.25\sin \left( {10\pi x - 2\pi t} \right),$$ where $$x$$ and $$y$$ are in meters and $$t$$ in seconds, is a wave travelling along the :
A.
$$- ve$$ $$x$$ direction with frequency $$1\,Hz.$$
B.
$$+ve$$ $$x$$ direction with frequency $$\pi \,Hz$$ and wavelength $$\lambda = 0.2\,m.$$
C.
$$+ve$$ $$x$$ direction with frequency $$1\,Hz$$ and wavelength $$\lambda = 0.2\,m$$
D.
$$-ve$$ $$x$$ direction with amplitude $$0.25\,m$$ and wavelength $$\lambda = 0.2\,m$$
Answer :
$$+ve$$ $$x$$ direction with frequency $$1\,Hz$$ and wavelength $$\lambda = 0.2\,m$$
Solution :
$$y = 0.25\sin \left( {10\pi x - 2\pi t} \right)$$
Comparing this equation with the standard wave equation
$$\eqalign{
& y = a\sin \left( {kx - \omega t} \right) \cr
& {\text{We}}\,{\text{get,}}\,\,k = 10\pi \Rightarrow \frac{{2\pi }}{\lambda } = 10\pi \Rightarrow \lambda = 0.2\,m \cr
& {\text{And}}\,\omega = 2\pi \,\,{\text{or,}}\,\,2\pi v = 2\pi \Rightarrow v = 1\,Hz. \cr} $$
The sign inside the bracket is negative, hence the wave travels in $$+ ve$$ $$x$$-direction.