Question

The transverse displacement y(x, t) of a wave on a string is given by \[y(x,t)={{e}^{-(a{{x}^{2}}+b{{t}^{2}}+2\sqrt{ab}xt)}}\]. This represents a:
A. wave moving in +x direction with speed \[\sqrt{\dfrac{a}{b}}\]
B. wave moving in –x direction with speed \[\sqrt{\dfrac{b}{a}}\]
C. standing wave of frequency \[\sqrt{b}\]
D. standing wave of frequency \[\dfrac{1}{\sqrt{b}}\]

Answer 1

Hint: The given equation of the transverse displacement of a wave on a string should be represented in terms of the standard equation of the transverse displacement equation by rearranging the terms. The coefficient of ‘t’, that is, the wave speed will be the required speed value.
Formula used:
\[y(x,t)=f(x+vt)\]

Complete step by step answer:
From the given information, we have the data as follows.
The transverse displacement y(x, t) of a wave on a string is given by,
\[y(x,t)={{e}^{-({{(\sqrt{a}x)}^{2}}+{{(\sqrt{b}t)}^{2}}+2\sqrt{ab}xt)}}\]
The standard equation of the transverse displacement y(x, t) of a wave is given as follows.
\[y(x,t)=f(x+\upsilon t)\]
Rewrite the given equation of the transverse displacement y(x, t) of a wave on a string.
\[\begin{align}
  & y(x,t)={{e}^{-({{(\sqrt{a}x)}^{2}}+{{(\sqrt{b}t)}^{2}}+2\sqrt{ab}xt)}} \\
 & \Rightarrow y(x,t)={{e}^{-{{(\sqrt{a}x+\sqrt{b}t)}^{2}}}} \\
\end{align}\]
Therefore, the expression of the transverse displacement y(x, t) of a wave on a string in terms of the standard equation is given as follows.
\[\therefore y(x,t)={{e}^{-{{\left( x+\sqrt{\dfrac{b}{a}}t \right)}^{2}}}}\]
Upon comparing the obtained equation with the standard equation, the coefficient of ‘t’, that is the wave speed is given as \[\sqrt{\dfrac{b}{a}}\].
The sign between the x and t parameters of the equation is positive, thus, the wave moves along the negative direction of the x-axis.
\[\therefore \]The transverse displacement y(x, t) of a wave on a string is given by \[y(x,t)={{e}^{-(a{{x}^{2}}+b{{t}^{2}}+2\sqrt{ab}xt)}}\], represents a wave moving in –x direction with speed\[\sqrt{\dfrac{b}{a}}\].

So, the correct answer is “Option B”.

Note: The direction of a wave can be computed as, if there is a positive sign between the ‘x’ and ‘t’ parameters of the equation, then, the wave is said to travel along the negative direction. Similarly, if there is a negative sign between the ‘x’ and ‘t’ parameters of the equation, then, the wave is said to travel along the positive direction.