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Expert-verified Found in: Page 474 ### Fundamentals Of Physics

Book edition 10th Edition
Author(s) David Halliday
Pages 1328 pages
ISBN 9781118230718 # Four waves are to be sent along the same string, in the same direction:${{\mathbf{y}}}_{{\mathbf{1}}}\left(x,t\right){\mathbf{=}}\left(4.00\mathrm{mm}\right){\mathbf{sin}}\left(2\mathrm{\pi x}-400\mathrm{\pi t}\right)\phantom{\rule{0ex}{0ex}}\phantom{\rule{0ex}{0ex}}{{\mathbf{y}}}_{{\mathbf{2}}}\left(x,t\right){\mathbf{=}}\left(4.00mm\right){\mathbit{s}}{\mathbit{i}}{\mathbit{n}}\left(2\pi x-400\pi t+0.7\pi \right)\phantom{\rule{0ex}{0ex}}\phantom{\rule{0ex}{0ex}}{{\mathbf{y}}}_{{\mathbf{3}}}\left(x,t\right){\mathbf{=}}\left(4.00mm\right){\mathbit{s}}{\mathbit{i}}{\mathbit{n}}\left(2\pi x-400\pi t+\pi \right)\phantom{\rule{0ex}{0ex}}\phantom{\rule{0ex}{0ex}}{{\mathbf{y}}}_{{\mathbf{4}}}\left(x,t\right){\mathbf{=}}\left(4.00mm\right){\mathbit{s}}{\mathbit{i}}{\mathbit{n}}\left(2\pi x-400\pi t+1.7\pi \right)$What is the amplitude of the resultant wave?

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

See the step by step solution

## Step 1: The given data

$\mathrm{i}.{\mathrm{y}}_{1}\left(\mathrm{x},\mathrm{t}\right)=\left(4.00\mathrm{mm}\right)\mathrm{sin}\left(2\mathrm{\pi x}-400\mathrm{\pi t}\right)\phantom{\rule{0ex}{0ex}}\mathrm{ii}.{\mathrm{y}}_{1}\left(\mathrm{x},\mathrm{t}\right)=\left(4.00\mathrm{mm}\right)\mathrm{sin}\left(2\mathrm{\pi x}-400\mathrm{\pi t}+0.7\mathrm{\pi }\right)\phantom{\rule{0ex}{0ex}}\mathrm{iii}.{\mathrm{y}}_{1}\left(\mathrm{x},\mathrm{t}\right)=\left(4.00\mathrm{mm}\right)\mathrm{sin}\left(2\mathrm{\pi x}-400\mathrm{\pi t}+\mathrm{\pi }\right)\phantom{\rule{0ex}{0ex}}\mathrm{iv}.{\mathrm{y}}_{1}\left(\mathrm{x},\mathrm{t}\right)=\left(4.00\mathrm{mm}\right)\mathrm{sin}\left(2\mathrm{\pi x}-400\mathrm{\pi t}+1.7\mathrm{\pi }\right)$

## Step 2: 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.

## Step 3: 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 between the 1st and 3rdwave is $\pi$, therefore, there would be no resulting wave from the superposition of these two waves. Similarly, it can be observed that the 2nd and 4th 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. ### Want to see more solutions like these? 