Well, waves are very logical to me. And to many others as well. To make it short, is it because you don't find them logical that they do not exist?
Here, we have well-defined theoretical frameworks (electromagnetism, quantum mechanics, non-linear optics, general relativity, etc.) that have been tested experimentally. Predictions match observations. No matter you like it or not for this or that reason, it works and waves are all over he place.
Just one little thing: matter waves exist. Please check any quantum physics textbook. Wave-particle duality lies at the basic of the quantum world (and all particle and nuclear physics experiments run for more than one century) . Once again, you may not like it, but the theory predicts things that are confirmed experimentally. So, until proved wrong, it is there.
Why do not waves exist? There are, only the word wave is an adjective, which describes the object. Object can be anything you like.
And it is quite true that this is the theory. And the THEORY is proven, but who has proved the existence of waves? The theory and assumption for me are different things.
In conclusion: my goal is not to argue with you and I'm not against theories. But if initially it is not right to perceive reality, then eventually it will accumulate with such "facts" that it will be difficult to correct later. Try to retrain a person who is not properly taught something to his childhood.
And the fact that you call the waves around us is just the reaction of something to the effect. As I said before, threw a stone into the water got fat. The impact of stone. The reaction of the shape and state of the water has changed. But you can not separate a wave from water, or you can ... except theoretically and in quantum physics using the dualism of concepts.
Physics is more than words. Physics aims to describe reality, make predictions about the results of any experiment (in the broad sense) that should be carried out. Whilst waves were originally connected to fluid dynamics, the concept is more general and applies to varied fields.
For instance, one hears sound. Sound is a vibration that propagates and that our brain materalizes somehow. It is thus a wave. In the physics sense of it. I am sure you would not argue that sound does not exist.
PS: I am fine with the arguing/no arguing part. We are jus chatting and there is nothing wrong with that :)