Question

Which of the following function correctly represent the travelling wave equation for finite values of $x$ and $t$ :

• A. $y=x^2-l^2$
• B. $y=\cos x^2\sin t$
• C. $y=\log \left(x^2-t^2\right)-\log (x-t)$
• D. $y=e^{2x}\sin t$

Answer 1

Answer: (C)

we have
$y=\log \frac{x^2-t^2}{x-t}=\log (x+t)$
$[$ As $\log a-\log b=\log \frac{a}{b}]$
$\Rightarrow \frac{\partial y}{\partial x}=\frac{1}{(x+t)}$
$\Rightarrow \frac{{\partial }^{2}y}{\partial x^2}=-\frac{1}{(x+t{)}^2}$
and $\frac{\partial y}{\partial t}=\frac{(\partial x/\partial t)}{(x+t)}=\frac{v}{(x+t)}$
$\Rightarrow \frac{{\partial }^{2}y}{\partial t^2}=-\frac{v^2}{(x+t{)}^2}$
$\Rightarrow \frac{{\partial }^{2}y}{\partial x^2}=\frac{1}{v^2}\frac{{\partial }^{2}y}{\partial t^2}$
Which is the general form of wave equation.