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Two waves \mathrm{y}_{1}=\mathrm{A}_{1} \sin \left(\omega \mathrm{t}-\beta_{1}\right) and \mathrm{y}_{2}=\mathrm{A}_{2} \sin \left(\omega \mathrm{t}-\beta_{2}\right) superimpose to form a resultant wave. The amplitude of this resultant wave is given by

Option: 1

\mathrm{A_{1}+A_{2}}


Option: 2

\mathrm{\left|A_{1}-A_{2}\right|}


Option: 3

\mathrm{\sqrt{A_{1}^{2}+A_{2}^{2}+2 A_{1} A_{2} \cos \left(\beta_{1}-\beta_{2}\right)}}


Option: 4

\mathrm{\sqrt{A_{1}^{2}+A_{2}^{2}-2 A_{1} A_{2} \cos \left(\beta_{1}-\beta_{2}\right)}}


Answers (1)

best_answer

\mathrm{Phase\: diff =\omega \mathrm{t}-\beta_{1}-\omega \mathrm{t}+\beta_{2}=\beta_{2}-\beta_{1}}

\mathrm{\therefore }  Amplitude of resultant wave

\mathrm{=\sqrt{A_{1}^{2}+A_{2}^{2}+2 A_{1} A_{2} \cos \left(\beta_{2}-\beta_{1}\right)} }
\mathrm{=\sqrt{A_{1}^{2}+A_{2}^{2}+2 A_{1} A_{2} \cos \left(\beta_{1}-\beta_{2}\right)} \quad[\text { As } \cos (-\theta)=+\cos \theta] }

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