Question 208: 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: