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Since I feel this question was not clearly answered, I am rewording it here.

How does energy reaches from left to right in a transverse wave when the vibration of particles is only in "up and down" direction?

I'm visualising it like this: the first particle gets the energy, it gets excited and jumps up. Then it transfers the energy to the second particle and settles back. The second particle now gets excited and jumps up and transfers the energy to the third one and so on. Am I thinking in the right way?

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Yes, I think your description is pretty reasonable. Suppose you represent your medium as made up of lots of thin strips:

enter image description here

Energy is transfered along the medium as one strip rubs against another:

enter image description here

This diagram is supposed to show friction (the red line) pulling the right strip upwards as the left strip moves up - possibly not one of my more effective diagrams.

Be a little cautious about taking this too literally as what actually happens will depend on what is doing the waving. Waves in a skipping rope are obviously different from waves in the sea. Nevertheless the basic principle applies, that if you consider some infinitesimally thin part of the wave it exerts a force on its neigbouring infinitesimally thin bit.

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The vibration of the particles isn't only up and down. In the other question Wendy Krieger said ocean waves are transverse waves. Take a look at the Wikipedia Wind wave article where you can see this gif:

enter image description here GNUFDL image by Kraaiennest, see Wikipedia

See the red test particles? They don't just go up and down, they go round and round. They have an angular momentum. Also note that the wave isn't localized in that this motion still occurs deep under the water. It diminishes with depth, but nevertheless the wave takes "many paths".

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