If the particles themselves do not move, then you have no wave, and nothing is transferred.
If the particles are free to move about their rest positions, then you can have a wave, and
it can transfer energy from place to place.
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A wave transfers energy through a medium by causing the particles of the medium to vibrate back and forth in place. While the particles themselves may not travel with the wave, the energy created by the wave is passed along by the movement of the particles.
Water waves transfer energy by displacing water particles as the wave travels through the water, causing them to move in a circular motion. However, the water particles themselves do not travel with the wave - they only oscillate in place. This transfer of energy without the transfer of matter is known as wave propagation.
Energy in a wave moves by causing particles in the medium to vibrate and transfer energy from one particle to another. As the wave propagates, the energy is carried forward through the medium without the particles themselves actually moving with the wave. This transfer of energy allows the wave to continue propagating through the medium.
As the energy wave moves through the water, the still water particles start to oscillate or move in a circular motion. The particles transfer energy to neighboring particles causing them to also move in a wave-like pattern. This movement is what we perceive as the wave passing through the water.
Mechanical waves do not transfer matter; they transfer energy and momentum. This means that while the wave causes particles in the medium to oscillate, the particles themselves do not travel with the wave as it propagates.
When a wave passes by, particles in the medium oscillate back and forth in the direction perpendicular to the wave's propagation. This movement of particles is what allows the wave to transfer energy through the medium.