Question

A transverse wave is travelling on a string with velocity $v$. The shape of string at $t=1\, s$ is given by $y=\frac{5}{x^{2}+6 x+9}$ and at $t=2 \,s$, it is given by $y=\frac{5}{x^{2}+12 x+36}$, then fill the value of $(v+c)$, where $c$ denotes the direction of motion of wave ($+1$ for positive $x$-direction $\&-1$ for $-v e\,x$-direction).

Answer 1

$y(\text { at } t=1)=\frac{5}{x^{2}+6 x+9}$
[ by assuming wave is moving in +ive $x$ direction]
$y(x, t)=\frac{5}{\left((x-v(t-1))^{2}+6(x-v(t-1))+9\right)}$
$y($ at $t=2)=\frac{5}{(x-v)^{2}+6(x-v)+9}$
$=\frac{5}{x^{2}+x(6-2 v)+v^{2}-6 v+9}$
by comparing with given equation at $t=2$
$6-2 v=12 $
$\Rightarrow v=-3 \,m / s $
$\Rightarrow$ -ive $\times $ -direction
$\Rightarrow $ speed $=3 m / s $
$\Rightarrow $ Ans. $ V+C=3+(-1)=2$