Beyond water
The word wave appears everywhere in physics, but it is also one of those words that people think they understand until they try to explain it. A wave is not just water doing something pretty. It is a pattern of disturbance moving through a medium or, in some cases, through fields.
What really propagates
What matters is that something is propagating without the material itself necessarily traveling along with the same path. This distinction is one of the reasons waves are such a useful concept across different areas of science.
How teaching can unfold
If we slow down and allow intuition to build, the topic becomes much less frightening. We can begin from sound or water, then gradually move toward light, quantum ideas, and more abstract cases.
From familiar to strange
That is how I like to teach: from the familiar toward the stranger, without pretending the stranger is not strange.
Beauty through recurrence
Physics becomes beautiful when people realize that one idea can reappear in many different costumes.