The thing is, a wave is basically some water particles and energy. The water particles don't go away, and as we know from physics, neither does the energy. So how can a wave die when all of its components don't die?
The truth is, a wave doesn't really exist. It's a concept in our head. "Here is some part of water that is higher than the rest, let's call it a wave". And now the wave can be "born" in the sea and "die" at the beach. But in fact nothing was created nor removed. It's just a concept. Everything is in fact interconnected. The disconnected parts that we see are concepts in our head. A tree, the sun, the rain, grass, ... . Nothing stands on its own.
You and me and everyone here, we are just concepts like the wave. What are you composed of? Some DNA from your ancestors, some cultural influence, the plants and animals you eat, the water you drink. After the concept of "you" dies, everything is still here. Everything that you were composed of is still here.
You're not an entity on your own. You are interconnected with everything around you. You are the water you drink and the water is you. Your thoughts are the thoughts of your ancestors, of your fellow humans, etc, and your thoughts are theirs.
So even if the waves die on the beach, the sea is still there. An therefore, the waves are also still there.