Question

Two travelling waves, \[y=0.10\sin (3t+4\pi x)\] metres and \[y=0.2\sin (3t-5\pi x)\]metres meet at t = 0. What is the displacement at x = 4.5 m at this moment?
A. 0.14 m
B. -0.14 m
C. 0.20 m
D. -0.20 m

Answer 1

Hint: While calculating the displacement, both the wave equations should be added first, as the question is to find the displacement value when both the waves meet. After adding the wave equations, substitute the given values of “x” and “t” the required result can be obtained.

Formula used:
\[y=A\sin (wt+kx+\phi )\]

Complete answer:
From given, we have the data,
The wave equation of first wave, \[y=0.10\sin (3t+4\pi x)\]metres
The wave equation of second wave, \[y=0.2\sin (3t-5\pi x)\]metres
x = 4.5 m
t = 0 s
A waveform equation is given by,
\[y=A\sin (wt+kx+\phi )\]
Where A is the amplitude, w is the angular frequency, k is the wavenumber and\[\phi \] is the phase angle.
Firstly consider the equation of the first wave and substitute the given values in it.
\[\begin{align}
  & y=0.10\sin (3t+4\pi x) \\
 & \Rightarrow y=0.10\sin (3\times 0+4\pi \times 4.5) \\
 & \Rightarrow y=0.10\sin \left( 0+4\pi \times \dfrac{9}{2} \right) \\
 & \Rightarrow y=0.10\sin \left( 18\pi \right) \\
\end{align}\]
Here we will make use of the trigonometric angle properties.
\[\sin (n\pi )=0\]
Therefore, the equation of the above waveform becomes,
\[\begin{align}
  & y=0.10\sin (0) \\
 & \Rightarrow y=0 \\
\end{align}\] …… (1)
Now consider the equation of the second wave and substitute the given values in it
\[\begin{align}
  & y=0.2\sin (3t-5\pi x) \\
 & \Rightarrow y=0.2\sin (3\times 0-5\pi \times 4.5) \\
 & \Rightarrow y=0.2\sin \left( 0-5\pi \times \dfrac{9}{2} \right) \\
 & \Rightarrow y=-0.2\sin \left( 22\pi +\dfrac{1}{2}\pi \right) \\
\end{align}\]
Here we will make use of the trigonometric angle properties.
\[\sin (n\pi )=0\]
Therefore, the equation of the above waveform becomes,
\[\begin{align}
  & y=-0.2\sin \left( 0+\dfrac{\pi }{2} \right) \\
 & \Rightarrow y=-0.2\sin \left( \dfrac{\pi }{2} \right) \\
 & \Rightarrow y=-0.2 \\
\end{align}\]…… (2)
Now, compute the sum of the waveforms.
Add the equations (1) and (2) to obtain the net value of the waveforms.
\[\begin{align}
  & y=0+(-0.2) \\
 & \Rightarrow y=-0.2 \\
\end{align}\]
Thus, the value of the displacement is – 0.2 m and the negative sign indicates the direction.

As the value of the displacement at x = 4.5 m and t = 0 is – 0.2 m, thus, the option (D) is correct.

Note:
The things to be on your finger-tips for further information on solving these types of problems are: All the parameters such as the wavenumber, angular frequency and the phase angle must be taken into consideration. In case, if the waveforms are given in the form of sin function and the cos function, then both of the waveform equations should be converted to either of one function.