How can I visualise waves?

Question

A wave is a sustained disturbance produced at a point in a medium(or field) and is transmitted to other parts of the medium(or field) without the actual translatory motion of the particles.

This definition of a wave helps me to understand mechanical waves. I imagine a stream of particles arranged in a line. When one of the particles vibrates about its mean position, then the adjacent particle(s) will also do the same (but with a phase difference) since they are connected by inter particular forces of attraction(bond forces). Thus at a broad level we can see a wave propagating while at the particulate level we can see the Simple Harmonic Motion of individual particles. But I am struggling to understand other waves like matter waves, electron waves and light waves. In high school books they are shown as curved lines as if a particles is always following that path. But I believe that that is not true. Which model should I imagine to understand these waves(especially matter waves)? What exactly is "oscillating" in matter waves?

Addendum

After doing a lot of research, I arrived at a conclusion.

Imagine two devices that can measure any physical quantity. Keep one of the devices at any position in the field region and the other device immediately after it. If for any position, the value of a physical quantity displayed oscillates between minimum and maximum values and the two devices do not simultaneously display the same value then there exists a 'wave' of that physical quantity in that region. Amplitude is given by the difference of maximum value(global) and mean value(global) displayed. Wavelength is given by the minimum distance in which the other device has to be placed so that the two devices simultaneously show the same value.

Are my conclusions correct?

Answer 1

In high school books they are shown as curved lines as if a particles is always following that path. But I believe that that is not true.

It is very tempting to think of a wave (e.g., an electromagnetic wave – a photon) as a particle travelling along a wave-like (sinus) curve. However, this is indeed not true. A wave doesn't mean a particle travelling in a wave-like trajectory – instead, there is no necessity to include particles when talking about waves¹.

When talking about waves (I will focus on EM waves here, but these principles apply to other types of waves too), it is necessary (or at least very helpful, in my opinion) to introduce fields. If you want to go into more detail, I highly recommend reading What is a field, really?. For now, it might suffice to think of a field as something like a sheet or cloth that it everywhere, except it doesn't really exist².

An EM wave then would be an oscillation in this "sheet", the electromagnetic field³. In the scope of our analogy, this oscillation is not so much different from the mechanical waves that you are familiar with. The key difference is that the oscillation doesn't happen in a physical object but instead in what is called a field.

I should also note that my analogy shouldn't be taken too literally, but I believe it is at least a somewhat intuitional introduction to EM waves. Someone else might answer in more technical terms.

If you wish an actual visualization, there exist many such as this:

^(Source)

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¹ You may have heard of wave-particle-duality which simplified means that elementary particles such as photons or electrons can either behave like a wave or a particle. It doesn't make much sense so "mix" both of these descriptions. So one has either a particle-like or a wave-like behavior.

² At least in the sense that it isn't a physical object made out of elementary particles – it is rather an abstract concept, but I consider it a not-so-bad analogy to think of a field as some plane.

³ It really is an oscillation in both the magnetic and electric field. The oscillations in the magnetic field are perpendicular to those in the electric field, as shown in the animation below. You can also read more about this in the linked article.

Answer 2

An electromagnetic wave is a periodic variation in the strength of the electromagnetic field. It is important not to think of the sinusoidal shape of the wave as representing a displacement in space. If you draw the wave on paper with just an x and y axis, where x is the direction in which the wave is propagating, the y direction represents the value of the change to the electric field. To put it crudely to make the point clear, if it were possible to move a field-measuring device along the x-axis, the strength displayed on the dial of the meter would go up and down in line with the peaks and troughs of the wave. To consider a simplified, classical, before and after picture, in the absence of the wave the electromagnetic field has a steady value at every point in space. When the wave is superimposed, those values move up and down with a given frequency.

As for the matter waves of quantum mechanics, they are mathematical functions that allow us to calculate the properties of matter on a small scale with astonishing degrees of agreement with experimental results. Quite what they 'really are' is still an open question. Of course, we might find that they 'really are' nothing other than mathematical functions that allow us to make physically accurate models.

Answer 3

Let us start with the case of mechanical waves. Matter particles remain in place but the wave is propagated by a series of simple harmonic vibrations. This is not the case when we discuss wave particle duality and by that I mean matter waves and electromagnetic waves. So mechanical waves are totally unrelated to wave particle dual matter waves in terms of motion of particles.

You must also recall two important basics.

Matter is anything that has mass and takes up space.

Electromagnetic waves are not matter strictly speaking.

Discussing about a beam of light, you can say two things.

1. light has no mass and has no particulate nature. it is the result of perpendicular oscillations of magnetic and electric fields. OR
2. light is a stream of photons with no wave nature. Hence now light has some properties of matter but is it completely matter? NO. Can you say that in wave nature, photons are vibrating? No. Can you say that in wave theory photons are following a sinusoidal path? No.

(Strictly speaking even the formula of E=hf is also a unification of mass-particle duality because in particulate approach photons do not oscillate. and the frequency in formula is the one taken from wave approach of electromagnetic radiation.)

Now come to case of a beam of electrons. Two observations arise.

1. Electrons have a definite mass and volume and are moving in a straight line with no oscillations. All matter properties hold true.

2. Electrons have a De Broglie Wavelength associated with them. They behave as waves. So can we confidently say that electron beam wave is a stream of electrons following a sinusoidal path? No.

Even De Broglie's formula also has a problem. It leads to the conclusion that photons have mass. And hence photon is matter. But that is(if I am not wrong) false.

To imagine visualizing matter waves is like visualizing photons moving in a wave nature rather than a stream as predicted by particulate nature. In short if we can imagine the motion of photons in a wave only then can we be able to imagine motion of matter particles in a matter wave.

Just as @Jonas stated in his answer to your question.

It is very tempting to think of a wave (e.g., an electromagnetic wave – a photon) as a particle travelling along a wave-like (sinus) curve. However, this is indeed not true. A wave doesn't mean a particle travelling in a wave-like trajectory – instead, there is no necessity to include particles when talking about waves