Question

Four waves are to be sent along the same string, in the same direction:

$$\begin{gathered} {\mathbf{y}}_{\mathbf{1}}\left(x,t\right)\mathbf{=}\left(4.00\mathrm{mm}\right)\mathbf{sin}\left(2\mathrm{\pi x}-400\mathrm{\pi t}\right) \\ {\mathbf{y}}_{\mathbf{2}}\left(x,t\right)\mathbf{=}\left(4.00mm\right)\mathit{s}\mathit{i}\mathit{n}\left(2\pi x-400\pi t+0.7\pi \right) \\ {\mathbf{y}}_{\mathbf{3}}\left(x,t\right)\mathbf{=}\left(4.00mm\right)\mathit{s}\mathit{i}\mathit{n}\left(2\pi x-400\pi t+\pi \right) \\ {\mathbf{y}}_{\mathbf{4}}\left(x,t\right)\mathbf{=}\left(4.00mm\right)\mathit{s}\mathit{i}\mathit{n}\left(2\pi x-400\pi t+1.7\pi \right) \end{gathered}$$

What is the amplitude of the resultant wave?

Short answer

There is no resultant wave. Hence the amplitude is zero.

Step-by-step solution

The given data

$$\begin{gathered} \mathrm{i}.{\mathrm{y}}_1\left(\mathrm{x},\mathrm{t}\right)=\left(4.00\mathrm{mm}\right)\sin \left(2\mathrm{\pi x}-400\mathrm{\pi t}\right) \\ \mathrm{ii}.{\mathrm{y}}_1\left(\mathrm{x},\mathrm{t}\right)=\left(4.00\mathrm{mm}\right)\sin \left(2\mathrm{\pi x}-400\mathrm{\pi t}+0.7\mathrm{\pi }\right) \\ \mathrm{iii}.{\mathrm{y}}_1\left(\mathrm{x},\mathrm{t}\right)=\left(4.00\mathrm{mm}\right)\sin \left(2\mathrm{\pi x}-400\mathrm{\pi t}+\mathrm{\pi }\right) \\ \mathrm{iv}.{\mathrm{y}}_1\left(\mathrm{x},\mathrm{t}\right)=\left(4.00\mathrm{mm}\right)\sin \left(2\mathrm{\pi x}-400\mathrm{\pi t}+1.7\mathrm{\pi }\right) \end{gathered}$$

Understanding the concept of superposition principle

Using the principle of superposition of the waves, we can find the amplitude of the resultant wave. We can also observe the phase difference between the waves to determine their resultant wave.

Calculation of the amplitude of the wave

From the given wave equations, we can observe that all the waves are traveling along the same direction. However, the phase difference betweenthe1^st and 3^rdwave is $\pi$, therefore, there would be no resulting wave from the superposition of these two waves. Similarly, it can be observed that the 2^nd and 4^th wave have phase difference equals to $\pi$. Therefore, again, there would be no resulting wave from superposition of these two waves. Hence, we can conclude that, there would be no wave generated as a resultant of the superposition of these four waves.

So the amplitude of the resulting wave would be zero.