In general a wave is a moving self-sustained disturbance of a medium, and that medium can be either a field (e.g. the gravitational field) or a substance (a solid or fluid). Here, the focus is on waves in a material media, and these are known as mechanical waves .
First we need to consider
Consider an object - a bell, a rope, or even the earth. Each is a vast collection of atoms forming an essentially continous elastic medium. Within limits, if the atoms in a material are pushed together, they repel, and, if separated, they attract. Electrically interacting, the atoms behave as if they are connected to one another via springs. As a result of this, once displaced, the medium tries to return to its equilibrium state. Only the state of being displaced moves through the medium as wave. The disturbance of a medium under the influence of a restoring force (like the springs) is common to all mechanical waves. Thus we need to consider
For a summary on waves, watch Waves: Light, Sound, and the nature of Reality
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