The transverse displacement y\left ( x,t \right ) of a wave on a string is given by

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}}

 

Answers (1)

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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