Question

Which of the following functions represent a travelling wave ?

• A. ${\left(x-vt\right)}^2$
• B. Ln $\left(x+vt\right)$
• C. ${e^{-\left(x-\pi \right)}}^2$
• D. $\frac{1}{x+vt}$

Answer 1

The correct option is A ${\left(x-vt\right)}^2$

The condition for the traveling wave is given by

$\frac{d^{2}y}{d{t}^2}=v^2\frac{d^{2}y}{d{x}^2}$. . . . . .$\left(1\right)$

Let consider option $\left(A\right)$,
$y=(x-vt{)}^2$

$\Rightarrow \frac{dy}{dt}=-2(x-vt)$

$\Rightarrow \frac{d^{2}y}{d{t}^2}=2v^2$. . . . .$\left(2\right)$

$\Rightarrow \frac{dy}{dx}=2(x-vt)$

$\Rightarrow \frac{d^{2}y}{d{x}^2}=2$. . . . . . .$\left(3\right)$

From equation $\left(1\right),\left(2\right)$ and $\left(3\right)$:

$2v^2=2v^2$ [Satisfying the condition for travelling wave]

Rest of the options are not satisfying the condition for the travelling wave.

Hence, the correct option is $\left(A\right)$.