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Q36P

Expert-verifiedFound in: Page 474

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:**

** **

**${{\mathbf{y}}}_{{\mathbf{1}}}{\left(x,t\right)}{\mathbf{=}}{\left(4.00\mathrm{mm}\right)}{\mathbf{sin}}{\left(2\mathrm{\pi x}-400\mathrm{\pi t}\right)}\phantom{\rule{0ex}{0ex}}\phantom{\rule{0ex}{0ex}}{{\mathbf{y}}}_{{\mathbf{2}}}{\left(x,t\right)}{\mathbf{=}}{\left(4.00mm\right)}{\mathit{s}}{\mathit{i}}{\mathit{n}}{\left(2\pi x-400\pi t+0.7\pi \right)}\phantom{\rule{0ex}{0ex}}\phantom{\rule{0ex}{0ex}}{{\mathbf{y}}}_{{\mathbf{3}}}{\left(x,t\right)}{\mathbf{=}}{\left(4.00mm\right)}{\mathit{s}}{\mathit{i}}{\mathit{n}}{\left(2\pi x-400\pi t+\pi \right)}\phantom{\rule{0ex}{0ex}}\phantom{\rule{0ex}{0ex}}{{\mathbf{y}}}_{{\mathbf{4}}}{\left(x,t\right)}{\mathbf{=}}{\left(4.00mm\right)}{\mathit{s}}{\mathit{i}}{\mathit{n}}{\left(2\pi x-400\pi t+1.7\pi \right)}$**

**What is the amplitude of the resultant wave?**

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

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

**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.**

From the given wave equations, we can observe that all the waves are traveling along the same direction. However, the phase difference between the 1^{st} and 3^{rd}wave 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.

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