Question

A wave travelling along the $x-$axis is described by the equation $y(x,t)=0.008\cos (\alpha x-\beta t)$. If the wavelength and time period of the wave are $0.16\text{m}$ and $6.0$ respectively, then $\alpha$ and $\beta$ in appropriate units are

• A. $\alpha =25.00\pi ,\beta =\pi$
• B. $\alpha =12.50\pi ,\beta =\frac{\pi }{2}$
• C. $\alpha =12.50\pi ,\beta =\frac{\pi }{3}$
• D. $\alpha =\frac{0.08}{\pi },\beta =\frac{2}{\pi }$

Answer 1

The correct option is C $\alpha =12.50\pi ,\beta =\frac{\pi }{3}$

Given,
$y(x,t)=0.008\cos (\alpha x-\beta t)....\left(1\right)$
Wavelength $\left(\lambda \right)=0.16\text{m}$
Time period $\left(T\right)=6.0\text{s}$
Comparing Eq.$\left(1\right)$ with standard equation of a travelling wave,
$y=A\sin (kx-\omega t+\varphi )$
We get,
$A=0.008\text{m},k=\alpha {\text{m}}^{-1},\omega =\beta {\text{s}}^{-1}$
we know that wave number is, $k=\frac{2\pi }{\lambda }$ and angular frequency, $\omega =\frac{2\pi }{T}$
$\Rightarrow \alpha =\frac{2\pi }{\lambda }$
$\Rightarrow \alpha =\frac{2\pi }{0.16}=12.5\pi \text{units}$
Similarly
$\beta =\frac{2\pi }{T}$
$\Rightarrow \beta =\frac{2\pi }{6}=\frac{\pi }{3}\text{units}$