Answer to Question #89509 in Physics for Bianca

Question #89509
Does y = sin (kx − ωt) satisfies the wave equation? Justify your answer
1
Expert's answer
2019-05-13T10:29:33-0400

The wave equation:


v22yx2=2yt2v^2 \frac{\partial^2 y}{\partial x^2}=\frac{\partial^2 y}{\partial t^2}

We have:


yx=kcos(kxωt)\frac{\partial y}{\partial x}=k \cos{(kx − ωt)}

2yx2=k2sin(kxωt)\frac{\partial^2 y}{\partial x^2}=-k^2 \sin (kx − ωt)

yt=ωcos(kxωt)\frac{\partial y}{\partial t}=-\omega \cos{(kx − ωt)}

2yt2=ω2sin(kxωt)\frac{\partial^2 y}{\partial t^2}=-\omega^2 \sin (kx − ωt)

Thus,


v2(k2sin(kxωt))=ω2sin(kxωt)v^2 (-k^2 \sin (kx − ωt) )=-\omega^2 \sin (kx − ωt)

We have the wave equation with


v=ωkv=\frac{\omega}{k}


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