A wave travelling along the $x$-axis is described by the equation $y(x, t)=0.005 \cos (\alpha x-\beta t)$. If the wavelength and the time period of the wave are 0.08 m and 2.0 s , respectively, then $\alpha$ and $\beta$ in appropriate units are

(A) $\alpha=25.00 \pi, \beta=\pi$

(B) $\alpha=\frac{0.08}{\pi}, \frac{2.0}{\pi}$

(C) $\alpha=\frac{0.04}{\pi}, \beta=\frac{1.0}{\pi}$

(D) $\alpha=12.50 \pi, \beta=\frac{\pi}{2.0}$

JEE Main · 2008 · 27 Apr 2008 · Offline · Waves & Sound

A wave travelling along the

x

-axis is described by the equation

y (x, t) = 0.005 cos (α x - β t)

. If the wavelength and the time period of the wave are 0.08 m and 2.0 s , respectively, then

α

and

β

in appropriate units are

Answer: (A)

Comparing with the standard travelling wave

y = A cos (k x - ω t)

, the coefficient

α

is the wave number

k = \frac{2 π}{λ}

and

β

is the angular frequency

ω = \frac{2 π}{T}

. With

λ = 0.08 m

k = \frac{2 π}{0.08} = 25 π rad m^{- 1}

, and with

T = 2.0 s

ω = \frac{2 π}{2.0} = π rad s^{- 1}

. So

α = 25 π

and

β = π

in SI units.

JEE Main · 2008 · 27 Apr 2008 · Offline · Single correct option