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
AIEEE 2011, Diffcult
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Given wave equation is $y(x, t)=e^{\left(-a x^{2}+b t^{2}+2 \sqrt{a b} x t\right)}$
$=e^{-\left[(\sqrt{a x})^{2}+(\sqrt{b} t)^{2}+2 \sqrt{a} x \cdot (\sqrt{b} t)\right]}=e^{-(\sqrt{a x}+(\sqrt{b} t)^{2}}$
$=e^{-(x+\sqrt{\frac{b}{a}} t)^{2}}$
It is a function of type $y=f(x+v t)$
$\Rightarrow$ Speed of wave $=\sqrt{\frac{b}{a}}$
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