how could you prove experimentally that energy is associated with wave ?

I can suggest a really easy experiment that you can do at home and I am sure that you have done it before.

Do you have a magnifying glass and a piece of paper? If so, hold the magnifying glass in sunlight and focus the light on a point in the piece of paper. After a few seconds, you will see smokes coming out and the paper starts burning.

How does that tell you energy is associated with wave?

Light is an electromagnetic wave. When you focus the light on the paper, the paper heats up. This means that heat energy is provided to the paper. And who is providing the heat? It's the light. The energy associated with the light wave is absorbed by the paper and converted into heat energy. So, from this you can infer that the light wave must have energy associated with it (otherwise conservation of energy is violated. Because from where else the paper gets heat energy?)