trapskrumcu

2022-09-25

Four sound waves are to be sent through the same tube of air, in the same direction:

${s}_{1}(x,t)=(9.00nm)\mathrm{cos}(2\pi x-700\pi t)$

${s}_{2}(x,t)=(9.00nm)\mathrm{cos}(2\pi x-700\pi t+0.7\pi )$

${s}_{3}(x,t)=(9.00nm)\mathrm{cos}(2\pi x700\pi t+\pi )$

${s}_{4}(x,t)=(9.00nm)\mathrm{cos}(2\pi x-700\pi t+1.7\pi )$

What is the amplitude of the resultant wave?

${s}_{1}(x,t)=(9.00nm)\mathrm{cos}(2\pi x-700\pi t)$

${s}_{2}(x,t)=(9.00nm)\mathrm{cos}(2\pi x-700\pi t+0.7\pi )$

${s}_{3}(x,t)=(9.00nm)\mathrm{cos}(2\pi x700\pi t+\pi )$

${s}_{4}(x,t)=(9.00nm)\mathrm{cos}(2\pi x-700\pi t+1.7\pi )$

What is the amplitude of the resultant wave?

cercimw

Beginner2022-09-26Added 8 answers

The sound waves are

${s}_{1}(x,t)=(9.00nm)\mathrm{cos}(2\pi x-700\pi t)$

${s}_{2}(x,t)=(9.00nm)\mathrm{cos}(2\pi x-700\pi t+0.7\pi )$

${s}_{3}(x,t)=(9.00nm)\mathrm{cos}(2\pi x700\pi t+\pi )$

${s}_{4}(x,t)=(9.00nm)\mathrm{cos}(2\pi x-700\pi t+1.7\pi )$

From super position The net wave is

$S(x,t)={s}_{1}(x,t)+{s}_{2}(x,t)+{s}_{3}(x,t)+{s}_{4}(x,t)$

By using the sum identities

$\mathrm{cos}x+\mathrm{cos}y=2\mathrm{cos}(\frac{x+y}{2})\mathrm{cos}(\frac{x-y}{2})$

$S(x,t)=({s}_{1}(x,t)+{s}_{3}(x,t))+({s}_{2}(x,t)+{s}_{4}(x,t))$

$=(18nm)\mathrm{cos}(2\pi x-700\pi t+\pi /2)\mathrm{cos}(\pi /2)+(18nm)\mathrm{cos}(2\pi x-700\pi t+1.2\pi )\mathrm{cos}(\pi /2)=0$

Result:

S=0

${s}_{1}(x,t)=(9.00nm)\mathrm{cos}(2\pi x-700\pi t)$

${s}_{2}(x,t)=(9.00nm)\mathrm{cos}(2\pi x-700\pi t+0.7\pi )$

${s}_{3}(x,t)=(9.00nm)\mathrm{cos}(2\pi x700\pi t+\pi )$

${s}_{4}(x,t)=(9.00nm)\mathrm{cos}(2\pi x-700\pi t+1.7\pi )$

From super position The net wave is

$S(x,t)={s}_{1}(x,t)+{s}_{2}(x,t)+{s}_{3}(x,t)+{s}_{4}(x,t)$

By using the sum identities

$\mathrm{cos}x+\mathrm{cos}y=2\mathrm{cos}(\frac{x+y}{2})\mathrm{cos}(\frac{x-y}{2})$

$S(x,t)=({s}_{1}(x,t)+{s}_{3}(x,t))+({s}_{2}(x,t)+{s}_{4}(x,t))$

$=(18nm)\mathrm{cos}(2\pi x-700\pi t+\pi /2)\mathrm{cos}(\pi /2)+(18nm)\mathrm{cos}(2\pi x-700\pi t+1.2\pi )\mathrm{cos}(\pi /2)=0$

Result:

S=0

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