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Question Number 38822 by NECx last updated on 30/Jun/18

The incident wave set up on a string  of length fixed at each end is given  by:   y_1 =Asin(kx−wt)  i)what is the equation of motion  of the reflected wave,y_2 .  ii)obtain the resultant,y=y_1 +y_2   of the two waves.  iii)what type of resultant wave is  this?  iv)for what values of x will the  amplitud of the resultant wave   become zero?  v)for what values of x will y be  maximum?

$${The}\:{incident}\:{wave}\:{set}\:{up}\:{on}\:{a}\:{string} \\ $$$${of}\:{length}\:{fixed}\:{at}\:{each}\:{end}\:{is}\:{given} \\ $$$${by}:\:\:\:{y}_{\mathrm{1}} ={Asin}\left({kx}−{wt}\right) \\ $$$$\left.{i}\right){what}\:{is}\:{the}\:{equation}\:{of}\:{motion} \\ $$$${of}\:{the}\:{reflected}\:{wave},{y}_{\mathrm{2}} . \\ $$$$\left.{ii}\right){obtain}\:{the}\:{resultant},{y}={y}_{\mathrm{1}} +{y}_{\mathrm{2}} \\ $$$${of}\:{the}\:{two}\:{waves}. \\ $$$$\left.{iii}\right){what}\:{type}\:{of}\:{resultant}\:{wave}\:{is} \\ $$$${this}? \\ $$$$\left.{iv}\right){for}\:{what}\:{values}\:{of}\:{x}\:{will}\:{the} \\ $$$${amplitud}\:{of}\:{the}\:{resultant}\:{wave}\: \\ $$$${become}\:{zero}? \\ $$$$\left.{v}\right){for}\:{what}\:{values}\:{of}\:{x}\:{will}\:{y}\:{be} \\ $$$${maximum}? \\ $$

Commented by tanmay.chaudhury50@gmail.com last updated on 30/Jun/18

Answered by tanmay.chaudhury50@gmail.com last updated on 30/Jun/18

i)y_2 =Asin(kx+wt)  ii)y_1 +y_2 =Asin(kx−wt)+Asin(kx+wt)  =2Asin(kx)coswt  iii)standing wave  iv)amplitude=2Asinkx  max amplitude=2A   min amplitude=0  when sinkx=0 =sinnΠ  kx=nΠ    x=((nΠ)/k)=((nΠ)/(2Π))×λ=n(λ/2)  v)max amplitude=2A   when sinkx=±1    sinkx=sin(n+(1/2))Π  kx=(n+(1/2))Π  κ=((2Π)/λ)  2(Π/λ)x=(n+(1/2))Π  x=(n+(1/2))(λ/2)

$$\left.{i}\right){y}_{\mathrm{2}} ={Asin}\left({kx}+{wt}\right) \\ $$$$\left.{ii}\right){y}_{\mathrm{1}} +{y}_{\mathrm{2}} ={Asin}\left({kx}−{wt}\right)+{Asin}\left({kx}+{wt}\right) \\ $$$$=\mathrm{2}{Asin}\left({kx}\right){coswt} \\ $$$$\left.{iii}\right){standing}\:{wave} \\ $$$$\left.{iv}\right){amplitude}=\mathrm{2}{Asinkx} \\ $$$${max}\:{amplitude}=\mathrm{2}{A}\:\:\:{min}\:{amplitude}=\mathrm{0} \\ $$$${when}\:{sinkx}=\mathrm{0}\:={sinn}\Pi \\ $$$${kx}={n}\Pi\:\:\:\:{x}=\frac{{n}\Pi}{{k}}=\frac{{n}\Pi}{\mathrm{2}\Pi}×\lambda={n}\frac{\lambda}{\mathrm{2}} \\ $$$$\left.{v}\right){max}\:{amplitude}=\mathrm{2}{A}\:\:\:{when}\:{sinkx}=\pm\mathrm{1} \\ $$$$ \\ $$$${sinkx}={sin}\left({n}+\frac{\mathrm{1}}{\mathrm{2}}\right)\Pi \\ $$$${kx}=\left({n}+\frac{\mathrm{1}}{\mathrm{2}}\right)\Pi \\ $$$$\kappa=\frac{\mathrm{2}\Pi}{\lambda} \\ $$$$\mathrm{2}\frac{\Pi}{\lambda}{x}=\left({n}+\frac{\mathrm{1}}{\mathrm{2}}\right)\Pi \\ $$$${x}=\left({n}+\frac{\mathrm{1}}{\mathrm{2}}\right)\frac{\lambda}{\mathrm{2}} \\ $$$$ \\ $$$$ \\ $$

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