How do waves transfer energy without transferring matter?

Waves transfer energy by causing particles to oscillate, but the particles themselves do not move along with the wave.

Waves, whether they are sound waves, light waves, or any other type of wave, are essentially disturbances that travel through a medium. This medium can be a solid, liquid, or gas, or even empty space in the case of light waves. The key point to understand is that while the wave itself moves through the medium, the particles of the medium do not travel with the wave.

Let's take a sound wave as an example. When a sound wave travels through the air, it causes the air particles to vibrate back and forth. These vibrations are passed on to neighbouring particles, which in turn pass them on to their neighbours, and so on. This creates a chain reaction of vibrations that travels through the air, carrying the energy of the sound wave. However, each individual air particle only moves a tiny amount back and forth - it does not travel along with the wave.

Similarly, in a water wave, the water particles move in a circular motion, going up and down as the wave passes. But again, they do not move along with the wave - they stay in roughly the same place, while the wave itself moves forward.

In both these examples, the energy of the wave is transferred through the medium, but the particles of the medium stay where they are. This is how waves can transfer energy without transferring matter. It's a bit like a Mexican wave at a football match - the wave moves around the stadium, but the people creating the wave stay in their seats.