Question

Two wave functions in a medium along x-direction are given by $y_1=\frac{1}{2+{\left(2x-3t\right)}^2}m$ and $y_2=-\frac{1}{2+{\left(2x+3t-6\right)}^2}m$ , where $x$ is in metres and $t$ is in seconds. Then

• A. at $x=\frac{3}{2}$ $m$ ,the resultant displacement will be zero at all time.
• B. at $t=1s$ , the resultant displacement will be zero everywhere.
• C. Both waves travel along the same direction.
• D. Both waves travel in the opposite directions.

Answer 1

Answer: (A), (B), (D)

The correct options are
A at $x=\frac{3}{2}$ $m$ ,the resultant displacement will be zero at all time.
B at $t=1s$ , the resultant displacement will be zero everywhere.
D Both waves travel in the opposite directions.

Resultant displacement, $y=y_1+y_2$
For resultant displacement to be zero, i.e.,
$y=0$
$y=y_1+y_2$
$y_1=-y_2$
${\left(2x-3t\right)}^2={\left(2x+3t-6\right)}^{2}Onsolving(x-\frac{3}{2})(t-1)=0$
Therefore at $x=\frac{3}{2},$ resultant displacement is zero for all values of t.

And from equations it is clear that they are travelling in opposite direction.