The trouble is if you want a rigorous and definitive answer to the question, you need to include all facets. You can not avoid Maths if you want that.I just made the relationship between a wave and the medium (or field ?) clear to understand without any mathematics. The OP suggested that they are one and the same thing, I just gave an example to show that they are not.
In QM a particle is just an "excitation" in a field right ? Is that analogous to "disturbances" in my above example ?
What you have written can be enough for a ‘chat’ but how far is this thread supposed to take us?
You pays yer money and you takes yer pick in a question like this.
There is no all embracing answer, I think.