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Question

Equations of a stationary wave and a travelling wave are y1=asinkxcosωt and y2=asin(ωtkx). The phase difference between two points x1=π3k and x2=3π2k is ϕ for the first wave and ϕ2 for the second wave. Find the ratio ϕ1ϕ2.

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Solution

Phase difference between two points in a standing wave =nπ

Where n= number of nodes between two points

given points are x1=π3Kx2=3π2K

Equation of standing wave

y1=asinkxcoswt

at node positions kx=nπ

x=nπK

where n=0,1,2,3...

So nodes are at x=πK,2πK,3πK,...

So there is only one node between ${ x }_{ 1 }and{ x }_{ 2 }hencephasedifference{ \phi }_{ 1 }=\pi$

For the travelling wave

ϕ2=2πλΔx

For equation

y2=asinkxcoswt

So K=2πλϕ2=K[x2x1]=K[3π2Kπ3K]=π[3213]=76πϕ1ϕ2=π76π=67


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