How does a wave pass thru a medium?

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A wave passing through a medium, such as a rope, causes particles to oscillate up and down rather than moving along with the wave. Each particle reaches a crest or trough at specific intervals, corresponding to the wave's frequency. Not all particles become crests and troughs simultaneously; rather, specific particles that reach these positions will do so repeatedly with each wavelength. This oscillation continues as the wave propagates through the medium. Understanding this behavior is crucial for grasping wave dynamics in various contexts.
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i mean i know particles move up and down but i want to go into a bit more detail.consider a constant fequency wave passing thru a rope.now what i want to know is that when the wave passes do all the particles bcome crest and trough turn by turn or do some specific particles (those which bcame crest and trough on the first wavelength) bcome crests and trough again and again after each wavelength?
kindly xplain
 
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Each particle reached by the wave oscillates up and down (for a transverse wave on a rope), between two extreme positions.
 
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