Question

The shape of a wave propagating in the positive x or negative x direction is given $y=\dfrac{1}{\sqrt{1+{{x}^{2}}}}$at t = 0 and $y=\dfrac{1}{\sqrt{2-2x+{{x}^{2}}}}$at t = 1 s where x and y are in meters. The shape of the wave disturbance does not change during propagation, then the velocity of the wave is

A. $1{m}/{s}\;$in positive x direction
B. $1{m}/{s}\;$in negative x direction
C. $\dfrac{1}{2}{m}/{s}\;$in positive x direction
D. $\dfrac{1}{2}{m}/{s}\;$in negative x direction

Answer 1

Using the general wave form equation this problem can be solved. Substitute the given values of the time in two separate equations for further calculation and finally compare the same equations to obtain the value of the velocity.

Complete step by step answer:
From given, we have the shape of a wave propagating in the positive x or negative x direction.
$y=\dfrac{1}{\sqrt{1+{{x}^{2}}}}$at t = 0
$y=\dfrac{1}{\sqrt{2-2x+{{x}^{2}}}}$at t = 1 s

The general form of the wave equation is given as follows.
$y=f(x\pm vt)$
Where v is the velocity and t is the time.
We are given the two different equations of the wave propagation at two different values of the time.

When the time becomes zero, the equation of the waveform is given to be as follows.
$y=\dfrac{1}{\sqrt{1+{{x}^{2}}}}$
Firstly, consider the general form of the wave equation and substitute the value of time.

& t=0 \\\ & \Rightarrow y=f(x\pm v(0)) \\\ & \Rightarrow y=f(x) \\\ \end{aligned}$$ Therefore, the equation of the waveform reduces to, $$y=\dfrac{1}{\sqrt{1+{{x}^{2}}}}$$ …… (1) When the time becomes 1 s, the equation of the waveform is given to be as follows. $$y=\dfrac{1}{\sqrt{2-2x+{{x}^{2}}}}$$ Firstly, consider the general form of the wave equation and substitute the value of time. $$\begin{aligned} & t=1 \\\ & \Rightarrow y=f(x\pm v(1)) \\\ & \Rightarrow y=f(x\pm v) \\\ \end{aligned}$$ Therefore, the equation of the waveform reduces to, $$y=\dfrac{1}{\sqrt{1+{{(x-1)}^{2}}}}$$ …… (2) Compare the equations (1) and (2), so we have, $$x\pm v=x-1$$ Therefore, the velocity becomes $$1{m}/{s}\;$$. The direction of the velocity will be opposite to that of the x direction. The shape of the wave disturbance does not change during propagation, then the velocity of the wave is $$1{m}/{s}\;$$in positive x direction, thus, the option (A) is correct. **Note:** The things to be on your finger-tips for further information on solving these types of problems are: The units of the parameters should be taken care of. Even after obtaining the magnitude of the velocity, its direction should be taken into consideration.