The equations of motion for two waves traveli | vedclass

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(A) The resultant wave is given by the superposition principle: $y = y_1 + y_2 = A\sin(\omega t - kx) + A\sin(\omega t - kx - 	heta )$.Using the trig

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$2A\cos \frac{	heta }{2}$

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(A) The resultant wave is given by the superposition principle: $y = y_1 + y_2 = A\sin(\omega t - kx) + A\sin(\omega t - kx - 	heta )$.Using the trigonometric identity $\sin C + \sin D = 2\sin(\frac{C+D}{2})\cos(\frac{C-D}{2})$,we get:$y = 2A \sin(\omega t - kx - \frac{	heta}{2}) \cos(\frac{	heta}{2})$.The resultant amplitude $A_R$ is the coefficient of the sine term:$A_R = 2A \cos(\frac{	heta}{2})$.Alternatively,using the vector addition formula for amplitudes:$A_R = \sqrt{A_1^2 + A_2^2 + 2A_1A_2\cos	heta} = \sqrt{A^2 + A^2 + 2A^2\cos	heta} = \sqrt{2A^2(1 + \cos	heta)}$.Since $1 + \cos	heta = 2\cos^2(\frac{	heta}{2})$,we have $A_R = \sqrt{2A^2 \cdot 2\cos^2(\frac{	heta}{2})} = 2A\cos(\frac{	heta}{2})$.

The equations of motion for two waves traveling in the same direction are given by ${y_1} = A\sin (\omega t - kx)$ and ${y_2} = A\sin (\omega t - kx - \theta )$. The resultant amplitude of the medium particle will be

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