Question

A travelling wave on a string is given by y $ = $A sin$\left[ {\alpha x + \beta t + \dfrac{\pi }{6}} \right]$. The displacement and velocity of oscillation of a point $\alpha = 0.56/cm,\,\,\beta = 12/\sec \,A = 7.5cm,x = 1cm$ and $t = 1s$ is
A. $4.6cm,\,\,46.5cm{s^{ - 1}}$
B. $3.75cm,\,\,77.94cm{s^{ - 1}}$
C. $1.76cm,\,\,7.5cm{s^{ - 1}}$
D. $7.5cm,\,\,75cm{s^{ - 1}}$

Answer 1

Hint: The travelling waves are the wave in which the position of maximum and minimum amplitude travel through the medium. Mathematically, A travelling wave is a periodic function of one dimensional space that moves with constant speed. The standard equation of a travelling wave is given as, $y\left( {x,t} \right) = A\,\sin \,\left( {wt \pm kx + \phi } \right)$.

Complete step by step solution:
Given that $\alpha = 0.56/cm,\,\,\beta = 12/\sec \,A = 7.5cm,x = 1cm$and $t = 1s$
The displacement of travelling wave is,
$y = A\,\,\sin \,\,\,\left( {\alpha x + \beta t + \dfrac{\pi }{6}} \right)......\left( i \right)$
Putting the above values in equation (i)
$y = 7.5\,\,\sin \left[ {\left( {0.56} \right)\left( 1 \right) + \left( {12} \right) \times \left( 1 \right) + \dfrac{\pi }{6}} \right]$
$y \simeq 7.5 \times 0.5$
$y \simeq 3.75cm$

The velocity of the travelling wave $v = \dfrac{{dy}}{{dt}}$
Putting the value of y from equation (i) into $v = \dfrac{{dy}}{{dt}}$
So, $v = \dfrac{{dy}}{{dt}}\left[ {A\,\sin \left( {\alpha x + \beta t + \dfrac{\pi }{6}} \right)} \right]$
$\Rightarrow$ \[v = A\dfrac{d}{{dt}}\left[ {\sin \left( {\alpha x + \beta t + \dfrac{\pi }{6}} \right)} \right]\]
$\Rightarrow$ \[v = A\beta \,\cos \left( {\alpha x + \beta t + \dfrac{\pi }{6}} \right)\]
$\Rightarrow$ $v = 7.5 \times 12 \times \cos \left[ {\left( {0.56} \right)\left( 1 \right) + 12 \times 1 + \dfrac{\pi }{6}} \right]$
$\Rightarrow$ $v \simeq 7.5 \times 12 \times 0.8$
$\Rightarrow$ $v \simeq 77.94cm/s$

Hence, option (B) is correct.

Additional Information: If displacement equation of travelling wave is given say y then the velocity will be $v = \dfrac{{dy}}{{dt}}.$

Note: The travelling waves transport energy from one area of space to another, whereas standing waves do not transport energy.