Question
At $$t = 0,$$ the shape of a travelling pulse is given by
$$y\left( {x,0} \right) = \frac{{4 \times {{10}^{ - 3}}}}{{8 - {{\left( x \right)}^2}}}$$
where $$x$$ and $$y$$ are in metres. The wave function for the travelling pulse if the velocity of propagation is $$5\,m/s$$ in the $$x$$ direction is given by
A.
$$y\left( {x,t} \right) = \frac{{4 \times {{10}^{ - 3}}}}{{8 - \left( {{x^2} - 5t} \right)}}$$
B.
$$y\left( {x,t} \right) = \frac{{4 \times {{10}^{ - 3}}}}{{8 - {{\left( {x - 5t} \right)}^2}}}$$
C.
$$y\left( {x,t} \right) = \frac{{4 \times {{10}^{ - 3}}}}{{8 - {{\left( {x + 5t} \right)}^2}}}$$
D.
$$y\left( {x,t} \right) = \frac{{4 \times {{10}^{ - 3}}}}{{8 - \left( {{x^2} + 5t} \right)}}$$
Answer :
$$y\left( {x,t} \right) = \frac{{4 \times {{10}^{ - 3}}}}{{8 - {{\left( {x - 5t} \right)}^2}}}$$
Solution :
$$\eqalign{
& y\left( {x,t} \right) = f\left( {x - vt} \right) \cr
& y = \left( {x,0} \right) = \frac{{4 \times {{10}^{ - 3}}}}{{8 - {x^2}}} \cr} $$
For a travelling wave in the $$x$$-direction
$$y\left( {x,t} \right) = \frac{{4 \times {{10}^{ - 3}}}}{{8 - {{\left( {x - 5t} \right)}^2}}}$$