The transverse displacement y (x, t) of a wave on a string is giv… - Vidyadip

MCQ

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{a}{b}} $
✓
wave moving in $-x$ direction with speed $\sqrt {\frac{b}{a}} $
C
standing wave of frequency $\sqrt b $
D
standing wave of frequency $\frac{1}{{\sqrt b }}$

✓

Answer

Correct option: B.

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

b
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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