# The transverse displacement $\dpi{100} y\left ( x,t \right )$ of a wave on a string is given by$\dpi{100} y\left ( x,t \right )= e^{-\left ( ax^{2}+bt^{2} +2\sqrt{abxt}\right )}$ This represents a Option 1) $wave\: moving\: in\: +x-direction\: with \: speed \sqrt{\frac{a}{b}}$ Option 2) $wave\: moving\: in\: -x\: -direction\: with \: speed \sqrt{\frac{b}{a}}$ Option 3) $standing\: wave\: of\: f\! requency\:\sqrt{b}$ Option 4) $standing\: wave\: of\: f\! requency\:\frac{1}{\sqrt{b}}$

As we learnt in

Relation between phase difference and path difference -

Phase difference $\left ( \Delta \phi \right )$

$= \frac{2\pi }{\lambda }\times path \: dif\! \! ference\left ( \Delta x \right )\\\lambda =wave\; length$

-

$y(x,t)=e^{-(ax^{2}+bt^{2}+2\sqrt{abxt})}=e^\sqrt{ax}+\sqrt{bt})^{2}$

It is function of type $y=f(\omega t+kx)=f\left(t-\frac{x}{v} \right )$

$v\rightarrow$ Wave velcoity

$\therefore\ \; V=\frac{\omega}{K}=\frac{\sqrt{b}}{\sqrt{a}}=\sqrt{\frac{b}{a}}$

Correct option is 2.

Option 1)

$wave\: moving\: in\: +x-direction\: with \: speed \sqrt{\frac{a}{b}}$

This is an incorrect option.

Option 2)

$wave\: moving\: in\: -x\: -direction\: with \: speed \sqrt{\frac{b}{a}}$

This is the correct option.

Option 3)

$standing\: wave\: of\: f\! requency\:\sqrt{b}$

This is an incorrect option.

Option 4)

$standing\: wave\: of\: f\! requency\:\frac{1}{\sqrt{b}}$

This is an incorrect option.

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