Question

Five waveforms moving with equal speeds on the x-axis
$y_1=8\sin (\omega t+kx);y_2=6\sin (\omega t+\frac{\pi }{2}+kx);y_3=4\sin (\omega t+\pi +kx);$

$y_4=2\sin (\omega t+\frac{3\pi }{2}+kx);$

$y_5=4\sqrt{2}\sin (\omega t-kx+\frac{\pi }{4})$ are superimposed on each other. The resulting wave is:

• A. $8\sqrt{2}\cos kx\sin (\omega t+\frac{\pi }{4})$
• B. $8\sqrt{2}\sin (\omega t-kx+\frac{\pi }{4})$
• C. $8\sqrt{2}\sin kx\cos (\omega t+\frac{\pi }{4})$
• D. $8\sin (\omega t+kx)$

Answer 1

The correct option is A $8\sqrt{2}\cos kx\sin (\omega t+\frac{\pi }{4})$

Lets first find resultant of four waves using phase diagram

first wave is representated along$+ve$ X-axis with vector of length equal to amplitude of the wave.

resultant of those four can be written as (by adjusting X and Y component)

$R=\sqrt{4^2+4^2}=4\sqrt{2}$

and $\sin \theta =\frac{4}{4\sqrt{2}}=\frac{1}{\sqrt{2}}\therefore \theta =\frac{\pi }{4}\therefore y_1+y_2+y_3+y_4=4\sqrt{2}\sin (wt+kx+\frac{\pi }{4})$

now add fifth wave

$\therefore y=(y_1+y_2+y_3+y_4)+y_5=4\sqrt{2}\sin (wt+kx+\frac{\pi }{4})+4\sqrt{2}\sin (wt-kx+\frac{\pi }{4})=4\sqrt{2}[2\times \sin (wt+\frac{\pi }{4})cos\left(kx\right)]$

By using $\sin C+\sin D=2\sin \frac{C+D}{2}\cos \frac{C+D}{2}$

$\therefore y=8\sqrt{2}\sin (wt+\frac{\pi }{4})\cos \left(kx\right)$