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Two-waves-are-represented-by-y-1-Asin-t-and-y-2-Asin-t-What-is-the-resultant-of-the-two-waves-ii-determine-the-amplitude-of-the-resultant-wave-and-under-which-condition-will-it-be-constructive




Question Number 38816 by NECx last updated on 30/Jun/18
Two waves are represented by  y_1 =Asinωt and y_2 =Asin(ωt−δ).  What is the resultant of the two  waves?  ii)determine the amplitude of the  resultant wave and under which  condition will it be constructive or  destructive.
$${Two}\:{waves}\:{are}\:{represented}\:{by} \\ $$$${y}_{\mathrm{1}} ={Asin}\omega{t}\:{and}\:{y}_{\mathrm{2}} ={Asin}\left(\omega{t}−\delta\right). \\ $$$${What}\:{is}\:{the}\:{resultant}\:{of}\:{the}\:{two} \\ $$$${waves}? \\ $$$$\left.{ii}\right){determine}\:{the}\:{amplitude}\:{of}\:{the} \\ $$$${resultant}\:{wave}\:{and}\:{under}\:{which} \\ $$$${condition}\:{will}\:{it}\:{be}\:{constructive}\:{or} \\ $$$${destructive}. \\ $$
Answered by tanmay.chaudhury50@gmail.com last updated on 30/Jun/18
y_1 +y_2 =2Asin(wt−(δ/2))cos((δ/2))  amplitude=2Acos((δ/2))  constructive when cos((δ/2))=±1  cos((δ/2))=cos(nΠ)  δ=2nΠ  destructive when cos((δ/2))=0=cos(n(Π/2))  δ=nΠ
$${y}_{\mathrm{1}} +{y}_{\mathrm{2}} =\mathrm{2}{Asin}\left({wt}−\frac{\delta}{\mathrm{2}}\right){cos}\left(\frac{\delta}{\mathrm{2}}\right) \\ $$$${amplitude}=\mathrm{2}{Acos}\left(\frac{\delta}{\mathrm{2}}\right) \\ $$$${constructive}\:{when}\:{cos}\left(\frac{\delta}{\mathrm{2}}\right)=\pm\mathrm{1} \\ $$$${cos}\left(\frac{\delta}{\mathrm{2}}\right)={cos}\left({n}\Pi\right) \\ $$$$\delta=\mathrm{2}{n}\Pi \\ $$$${destructive}\:{when}\:{cos}\left(\frac{\delta}{\mathrm{2}}\right)=\mathrm{0}={cos}\left({n}\frac{\Pi}{\mathrm{2}}\right) \\ $$$$\delta={n}\Pi \\ $$$$ \\ $$
Commented by NECx last updated on 30/Jun/18
wow..... Thank you so much.
$${wow}…..\:{Thank}\:{you}\:{so}\:{much}. \\ $$

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