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. $y=e^{2 x} \sin t$

Answer 1

Answer: (C)

we have
$y=\log \frac{x^{2}-t^{2}}{x-t}=\log (x +t)$
$\left[\right.$ As $\left.\log a-\log b=\log \frac{a}{b}\right]$
$\Rightarrow \frac{\partial y}{\partial x}=\frac{1}{(x +t)}$
$\Rightarrow \frac{\partial^{2} y}{\partial x^{2}}=-\frac{1}{(x +t)^{2}}$
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.