Question

The transverse displacement $y(x,t)$ of a wave on a string is given by $y\left(x,t\right)=e^{-\left(a{x}^2+b{t}^2+2\sqrt{ab}xt\right)}$ This represents a

• A. wave moving in - x direction, speed $\sqrt{\frac{b}{a}}$
• B. standing wave of frequency $\sqrt{b}$
• C. standing wave of frequency $\frac{1}{\sqrt{b}}$
• D. wave moving in + x direction, speed $\sqrt{\frac{a}{b}}$

Answer 1

Answer: (A)

$y\left(x,t\right)=e^{-\left(a{x}^2+b{t}^2+2\sqrt{abxt}\right)}=e^{-{\left(\sqrt{ax}+\sqrt{bt}\right)}^2}$
It is a function of type y = $f\left(\omega t+kx\right)$
$\therefore y\left(x,t\right)$ represents wave travelling along -x direction.
Speed of wave $=\frac{\omega }{k}=\frac{\sqrt{b}}{\sqrt{a}}=\sqrt{\frac{b}{a}}.$