Question

Find the wrong statement from the following about the equation of stationary wave given by $Y=0.04cos(\pi x)sin(50\pi t)m$ where $t$ is in second. Then for the stationary wave.

• A. Time period $=0.02s$
• B. Wavelength $=2$
• C. Velocity $=50m/s$
• D. Amplitude $=0.02m$

Answer 1

Answer: (A)

Key Idea The displacement of a wave in term of time period is given by
$y=A\sin \left[2\pi \left(\frac{t}{T}-\frac{x}{\lambda }\right)\right]$
This equation in terms of speed of wave (v),
becomes $y=A\sin \left[\frac{2\pi }{\lambda }(vt-x)\right]$
The given equation of wave is
$y=0.04\cos (\pi x)\sin (50\pi t)$
$=0.02\sin (50\pi t+\pi x)+0.02\sin (50\pi t-\pi x)$
$[\because 2\sin A\cdot \cos B=\sin (A+B)\cdot \cos (A-B)]$
Thus, the given wave is the combination of two waves,
$y_1=0.02\sin (50\pi t+\pi x)$ (in -ve $x$ -direction)
and $y_2=0.02\sin (50\pi t-\pi x)$ (in + ve $x$ -direction)
Comparing them with the general equation of wave $a\sin (\omega t+kx)$, we get
Amplitude, $a=0.02m$
Time period, $T=\frac{2\pi }{50\pi }=\frac{1}{25}=0.04s$
Wavelength, $\lambda =\frac{2\pi }{\pi }=2m$
Velocity, $v=\frac{50\pi \times \lambda }{2\pi }$
$=\frac{100}{2}=50m{s}^{-1}$
So, option (a) is wrong.