Question

The equation of a progressive wave for a wire is: \[Y=4\sin \left[ \dfrac{\pi }{2}\left( 8t-\dfrac{x}{8} \right) \right]\] .
If x and y are measured in cm then velocity of wave is:
A. 64cm/s along -x direction
B. 32 cm/s along -x direction
C. 32 cm/s along +x direction
D. 64 cm/s along +x direction

Answer 1

Hint: The given equation is similar with the standard equation of progressive wave with velocity v in +x direction. So, the velocity also will be in the same direction of propagation of the wave. For that students must know the standard equation of progressive wave.

Complete answer:
We know the standard equation of the wave with a velocity v in +x direction is,
\[Y=a\sin 2\pi \left( \dfrac{t}{T}-\dfrac{x}{\lambda } \right)............(1)\]
Where, a is amplitude, t is required time, x is displacement, $\lambda $ is wavelength of that wave and T is the period of the wave.
Given equation of progressive wave for a wire is,
\[Y=4\sin \left[ \dfrac{\pi }{2}\left( 8t-\dfrac{x}{8} \right) \right]\]
We write given equation in form of equation (1) as,
\[\begin{align}
  & Y=4\sin 2\pi \left( \dfrac{8t}{4}-\dfrac{x}{32} \right) \\
 & \Rightarrow Y=4\sin 2\pi \left( \dfrac{t}{0.5}-\dfrac{x}{32} \right)...........(2) \\
\end{align}\]
On comparing equation (1) and (2) we get,
a = 4cm, T = 0.5s and $\lambda =32cm$
Now we find the frequency of the wave which is,
\[\begin{align}
  & n=\dfrac{1}{T}=\dfrac{1}{0.5} \\
 & \Rightarrow n=2Hz \\
\end{align}\]
Using this value of frequency, we calculate the velocity of given wave as,
\[v=n\lambda \]
Substituting the value of, we get
\[v=2\times 32=64cm\]
That is, \[v=64cm/s\] along +x direction

So, the correct answer is “Option D”.

Note:
The questions may ask for wave propagating in –x direction and the standard equation is \[Y=a\sin 2\pi \left( \dfrac{t}{T}+\dfrac{x}{\lambda } \right)\] .
There is only a difference of sign between waves travelling along +x and –x direction. So first of all, try to find the standard equation of the wave which is matching with the given equation. Then write it in that form and compare it with standard one. This method gives you the correct required solution.