Question 170: Waves

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 $a:$

Accepted Answer

Given wave equation is
$y\left( {x,t} \right){ = _e}\left( { - a{x^2} + b{t^2} + 2\sqrt {ab} \,xt} \right)$
$ = {e^{ - \left[ {{{\left( {\sqrt {ax} } \right)}^2} + {{\left( {\sqrt {bt} } \right)}^2} + 2\sqrt a x.\sqrt b t} \right]}}$
$ = {e^{ - {{\left( {\sqrt a x + \sqrt b t} \right)}^2}}}$
$ = {e^{ - {{\left( {x + \sqrt {{b \over a}} t} \right)}^2}}}$
It is a function of type $y = f\left( {x + vt} \right)$
$ \Rightarrow $ Speed of wave $ = \sqrt {{b \over a}} $

Answer

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

Answer

standing wave of frequency $\sqrt b $

Answer

standing wave of frequency ${1 \over {\sqrt b }}$

Answer

wave moving in $+x$ direction speed $\sqrt {{a \over b}} $

Waves

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:

Choose the correct answer: