Question

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

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

Answer 1

The correct option is C $y=\log (x^2-t^2)-\log (x-t)$

$y=\log \frac{x^2-t^2}{(x-t)}\Rightarrow (\log a-\log b=\log (a/b\left)\right)$
$y=\log (x+t)$
$\frac{\partial y}{\partial x}=\frac{1}{x+t}$
$\frac{{\partial }^{2}y}{{\partial x}^2}=\frac{1}{(x+t{)}^2}$
and
$\frac{\partial y}{\partial t}=\frac{\partial y}{\partial x}\frac{\partial x}{\partial t}$
$\frac{{\delta }^{}x/\delta t}{x+t}=\frac{v}{(x+t)}\frac{{\delta }^{2}y}{\delta t^2}=\frac{v^2}{(x+t{)}^2}\frac{{\delta }^{2}y}{{\delta x}^2}=\frac{1}{v^2}\frac{{\delta }^{2}y}{\delta t^2}Whichisthegeneralformofwaveequation.$