trapskrumcu

2022-09-25

Four sound waves are to be sent through the same tube of air, in the same direction:
${s}_{1}\left(x,t\right)=\left(9.00nm\right)\mathrm{cos}\left(2\pi x-700\pi t\right)$
${s}_{2}\left(x,t\right)=\left(9.00nm\right)\mathrm{cos}\left(2\pi x-700\pi t+0.7\pi \right)$
${s}_{3}\left(x,t\right)=\left(9.00nm\right)\mathrm{cos}\left(2\pi x700\pi t+\pi \right)$
${s}_{4}\left(x,t\right)=\left(9.00nm\right)\mathrm{cos}\left(2\pi x-700\pi t+1.7\pi \right)$
What is the amplitude of the resultant wave?

cercimw

The sound waves are
${s}_{1}\left(x,t\right)=\left(9.00nm\right)\mathrm{cos}\left(2\pi x-700\pi t\right)$
${s}_{2}\left(x,t\right)=\left(9.00nm\right)\mathrm{cos}\left(2\pi x-700\pi t+0.7\pi \right)$
${s}_{3}\left(x,t\right)=\left(9.00nm\right)\mathrm{cos}\left(2\pi x700\pi t+\pi \right)$
${s}_{4}\left(x,t\right)=\left(9.00nm\right)\mathrm{cos}\left(2\pi x-700\pi t+1.7\pi \right)$
From super position The net wave is
$S\left(x,t\right)={s}_{1}\left(x,t\right)+{s}_{2}\left(x,t\right)+{s}_{3}\left(x,t\right)+{s}_{4}\left(x,t\right)$
By using the sum identities
$\mathrm{cos}x+\mathrm{cos}y=2\mathrm{cos}\left(\frac{x+y}{2}\right)\mathrm{cos}\left(\frac{x-y}{2}\right)$
$S\left(x,t\right)=\left({s}_{1}\left(x,t\right)+{s}_{3}\left(x,t\right)\right)+\left({s}_{2}\left(x,t\right)+{s}_{4}\left(x,t\right)\right)$
$=\left(18nm\right)\mathrm{cos}\left(2\pi x-700\pi t+\pi /2\right)\mathrm{cos}\left(\pi /2\right)+\left(18nm\right)\mathrm{cos}\left(2\pi x-700\pi t+1.2\pi \right)\mathrm{cos}\left(\pi /2\right)=0$
Result:
S=0

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