2 conflicting Standing wave equations

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marinacalder
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Hi!

I thought that the standing wave trig equation could be derived by adding 2 waves ...

y1=asin(wt-kx) y2=(-)sin(wt+kx)

-ve because of the phase change at the boundary.

but in many places I have seen

y1=asin(wt-kx) y2=asin(wt+kx)

i.e. positive.

could someone please explain how?
Many thanks
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Eimmanuel
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(Original post by marinacalder)
Hi!

I thought that the standing wave trig equation could be derived by adding 2 waves ...

y1=asin(wt-kx) y2=(-)sin(wt+kx)

-ve because of the phase change at the boundary.

but in many places I have seen

y1=asin(wt-kx) y2=asin(wt+kx)

i.e. positive.

could someone please explain how?
Many thanks

What you are describing is “two ways” of forming a standing wave.

In the many places that you have seen where they describe the forming of standing wave by superposition of two progressive waves travelling in opposite directions.

This means that there is a wave that travels to the right describes by y1=Asin(ωtkx) (this wave can be originated from a source on the left) and the other wave that travels to the left is described by y2=Asin(ωt + kx) (this wave can be originated from a source on the right). The standing wave is described mathematically by 2Asin(ωt)cos(kx).

What you have described is another way of forming standing wave. Say that you send a transverse wave to the right described by y1=Asin(ωtkx) and the wave undergo a 180° phase change at the boundary. The reflected wave is described by y1r= −Asin(ωt + kx). The superposition of the two waves formed the standing wave. The standing wave is described mathematically by 2Acos(ωt)sin(−kx) = −2Acos(ωt)sin(kx).

They are “equivalent” even though the final form of the mathematical description may be different. The evolution of the waveform may be different. The physics for the two standing waves (2Asin(ωt)cos(kx) and −2Acos(ωt)sin(kx)) is the same.

The situation is similar to whether you want to describe the oscillation of a mass that is attached to spring by sin or cos function.
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marinacalder
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(Original post by Eimmanuel)
What you are describing is “two ways” of forming a standing wave.

In the many places that you have seen where they describe the forming of standing wave by superposition of two progressive waves travelling in opposite directions.

This means that there is a wave that travels to the right describes by y1=Asin(ωtkx) (this wave can be originated from a source on the left) and the other wave that travels to the left is described by y2=Asin(ωt + kx) (this wave can be originated from a source on the right). The standing wave is described mathematically by 2Asin(ωt)cos(kx).

What you have described is another way of forming standing wave. Say that you send a transverse wave to the right described by y1=Asin(ωtkx) and the wave undergo a 180° phase change at the boundary. The reflected wave is described by y1r= −Asin(ωt + kx). The superposition of the two waves formed the standing wave. The standing wave is described mathematically by 2Acos(ωt)sin(−kx) = −2Acos(ωt)sin(kx).

They are “equivalent” even though the final form of the mathematical description may be different. The evolution of the waveform may be different. The physics for the two standing waves (2Asin(ωt)cos(kx) and −2Acos(ωt)sin(kx)) is the same.

The situation is similar to whether you want to describe the oscillation of a mass that is attached to spring by sin or cos function.
Wow! That explaination was fantastic , I honestly can’t thank you enough!! Repping as much as poss
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