As a crest moves through a point, in what ways is the motion of that point like the motion of a ball thrown straight upward?

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A crest in a transverse wave causes displacement perpendicular from the direction of energy transfer. The particles in a transverse wave (and here, I assume a matter wave like water or earth because light is a different subject altogether) move up and down as the energy is transferred to the right or left.

Since a particle in the medium is oscillating perpendicular to the energy transfer, it moves up, away from the equilibrium line and down to the equilibrium line and down below the equilibrium line and back up again.

This is exactly like throwing a ball upward. It moves upward with some velocity and some kinetic energy and as it rises, it loses kinetic energy and velocity until the velocity is at zero, and then it heads downward again. For material waves, gravity is the factor that causes this.

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