Question

A wave travelling along the x-axis is described by the equation \[y\left( {x,t} \right) = 0.005cos\left( {\alpha x - \beta t} \right)\] . If the wavelength and time period of the wave are \[0.08m{\text{ }}and{\text{ }}2.0s\] respectively, then α and β in appropriate units are:
(A) $\alpha = 25.00\pi ,\beta = \pi $
(B) $\alpha = \dfrac{{0.08}}{\pi },\beta = \dfrac{{2.0}}{\pi }$
(C) $\alpha = \dfrac{{0.04}}{\pi },\beta = \dfrac{{1.0}}{\pi }$
(D) $\alpha = 12.50\pi ,\beta = \dfrac{\pi }{{2.0}}$

Answer 1

Hint: In order to answer this question, we will write the general equation to mention the variables of the given constant or coefficients. And then in terms of wavelength and time period we will write the formula of $\alpha \,and\,\beta $ ,a s we already have the value of wavelength and time period.

Complete answer:
The given equation along the x-axis:
\[y = 0.005cos\left( {\alpha x - \beta t} \right)\]
Now, as we know the general equation of the wave travelling along the x-axis:
$y = A\cos (kx - \omega t)$
So, as comparing the given equation and the general equation, we have:
$A = 0.005m$ ,
$k = \alpha \,and\,\omega = \beta $
And as we know, $\alpha \,and\,\beta $ in terms of the wavelength and time period:-
$ \Rightarrow \alpha = \dfrac{{2\pi }}{\lambda }$ …….(i)
and, $ \Rightarrow \beta = \dfrac{{2\lambda }}{T}$ …..(ii)
where, $\lambda $ is the wavelength and
$T$ is the time period of the travelling of waves.
Now, we have already the value of wavelength and time period, i.e.:
$\lambda = 0.08m$ and,
$T = 2.0s$
So, we can find the value of $\alpha \,and\,\beta $ easily:
We will substitute the value of $\lambda $ in equation(i)-
$\therefore \alpha = \dfrac{{2\pi }}{\lambda } = \dfrac{{2\pi }}{{0.08}} = 25\pi {m^{ - 1}}$ and,
$\therefore \beta = \dfrac{{2\pi }}{T} = \dfrac{{2\pi }}{2} = \pi rad.{s^{ - 1}}$
Therefore, the value of $\alpha \,and\,\beta $ are $25\pi \,and\,\pi $ respectively.

Hence, the correct option is (A) $\alpha = 25.00\pi ,\beta = \pi $ .

Note:
Progressive wave is a wave that travels continually in the same direction in a medium without changing its amplitude. We shall construct a function that describes the propagation of a wave in a medium and gives the shape of the progressing wave at any point in time throughout its propagation in this section.