Big misconceptions with the fundamentals of “ waves”

Question

I don’t really know whether my question/request suits this website as it is not a precise question which targets a precise idea. I still hope to find help, however. My main problem is with ”waves”, as I have never succeeded to comprehend any part of this branch of physics. I first wish a rigid definition of the term wave with examples that clarify this definition.

Secondly, I have a few questions such as

1) If a particle of the medium of propagation oscillates due to the arrival of this unknown agent “ wave”, does it oscillate only once or it enters in a periodic motion? (Wave propagating through a rope for instance, waves in water, wave train.) If it is enough for the particle to propagate once, then what is the period of the waves?

2) do waves really exist or we represent the changes in the medium with a wave.

3) on standing transverse waves , are they two traveling waves interfering ? I get that the difference in phase is \pi radians, then how come anti nodes are the interference of two waves in the same phase.

4) How do other oscillations (pendulums , alternating current) relate to waves. 5)How standing waves happen in a trumpet and in a flute

Answer 1

When used in a broad sense, term wave encompasses many phenomena, some of which involve directed displacement of a matter, others periodic oscillations of the matter, and yet others could be said not to involve any matter at all.

Some examples of the waves involving directed displacement of matter are shock waves and solitons. The sea waves close to the shore are another obvious example of such waves.

In a more narrow sense waves mean periodic movement existing in the media, where at every point we have periodic oscillations in time, and the phase of these oscillations is periodic in space. Waves in water far from the shore in calm weather are a good example. If one places a floating object on such a wave, the object will be moving up and down, but not really moving in respect to the shore. Obviously, such a movement involves more than one oscillation.

Finally, when we talk about electromagnetic waves, there is no matter involved in the conventional sense of the word - the oscillations are those of a massless electromagnetic field, but they share the characteristics of the waves described in the previous paragraph: periodic oscillations in every point and periodicity in space.

Not sure, if I have answered all the questions - please indicate in the comments, and I will try to expand on it.

Remark
The reason for calling shock waves and solitons waves is that the general solutions of wave equation $$ \frac{\partial^2 u(x,t)}{\partial t^2} = v^2 \frac{\partial^2 u(x,t)}{\partial t^2} $$ in free space have the form of a running wave $$f(x-vt).$$ Even if $f(.)$ is not a periodic function, it still makes sense to call such a solution a wave, and extend the term to similar running wave solutions of other equations (such as Korteveg-de Vries equation.)