Question 464: Waves

Question

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

Accepted Answer

The given equation for the stationary wave is:
$  y = 0.04 \cos (\pi x) \sin (50 \pi t)  $
This can be rewritten as:
$ y = 0.02 \sin (50 \pi t + \pi x) + 0.02 \sin (50 \pi t - \pi x) $
This equation represents the combination of two waves:
$ \begin{align*} y_1 &= 0.02 \sin (50 \pi t + \pi x) \quad 	ext{(moving in -ve x-direction)} \ y_2 &= 0.02 \sin (50 \pi t - \pi x) \quad 	ext{(moving in +ve x-direction)} \end{align*} $
Comparing these with the general wave equation $a \sin (\omega t + k x)$, we identify:
Amplitude
: $a = 0.02 \, 	ext{m}$
Time Period $T$
: Calculated as
$   T = \frac{2\pi}{50\pi} = \frac{1}{25} = 0.04 \, 	ext{s}   $
Wavelength $\lambda$
: Given by
$   \lambda = \frac{2\pi}{\pi} = 2 \, 	ext{m}   $
Velocity $v$
: Calculated using
$   v = \frac{50\pi 	imes \lambda}{2\pi} = \frac{100}{2} = 50 \, 	ext{m/s}   $
Thus, based on these calculations, the time period should be $0.04 \, 	ext{s}$ instead of $0.02 \, 	ext{s}$. Hence, the statement in option A is incorrect.

Answer

Time period $=0.02 \mathrm{~s}$

Answer

Wavelength $=2 \mathrm{~m}$

Answer

Velocity $=50 \mathrm{~m} / \mathrm{s}$

Answer

Amplitude $=0.02 \mathrm{~m}$

Waves

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

Choose the correct answer: