According to my notes, energy and momentum. So how do you relate that to the way surface waves propagate in water? Or a steel guitar string?
Well, sort of. But that's a discussion for another thread. Really what propagates is a disturbance in the medium. The speed of wave propagation in both cases is determined by some properties of the medium - typically an inertial property and an elastic property. For instance, on the guitar string, the wave speed depends on the mass per unit length of the string (inertial property) and the tension (elastic property).
Ok. But what that disturbance is, corresponds to a transport of "something", right? The medium gets displaced, but not in the direction of propagation.
Yes. There's a transport of "something", but not "stuff". No substance flows along with a wave, and in particular there is no mysterious glowing substance that is somehow attached to a wave. We appear to be in agreement on half of this statement. I don't know why you're still stuck when it comes to the other half.
In fact, the travelling wave displaces the medium transversely to the direction of propagation. There is a displacement wave you might say.
. . . and so what propagates is a dynamical "situation" or state of the medium, in which the displacement corresponds to momentum in the vertical direction, and so there are molecules of water with extra kinetic energy, for say a continuous wavetrain. So one can say with some certainty there is an energy density, in Joules per unit volume. Of course you can't see kinetic energy, but you can say energy is being transported from place to place, by or "in" a wave which carries it.
But this energy "in" the wave is relative to the medium, or to those parts in it which are not displaced, right? And the kinetic energy flows through an abstract area, transverse to the propagation, in a unit time, this is otherwise called the intensity and is proportional to the square of the amplitude.
This may illustrate; Please Register or Log in to view the hidden image! And this looks like a wonderful link. Please Register or Log in to view the hidden image! A pebble thrown into a pond will produce concentric circular ripples which move outward from the point of impact. If a fishing float is in the water, the float will bob up and down as the wave moves by. This is a characteristic of transverse waves. Such waves obey the wave relationship. Please Register or Log in to view the hidden image! http://hyperphysics.phy-astr.gsu.edu/hbase/Sound/wavplt.html
Not necessarily. Often the parts that have zero displacement have a lot of kinetic energy, for instance, whereas parts that have maximum displacement might have no kinetic energy instantaneously, but lots of potential energy of one kind or another. I agree that there is some reference level for energy. That has to be defined, and it's usually arbitrary - which, by the way, is another thing that tells you that energy is not "stuff". Yes. I agree.
But for waves propagating in an elastic medium, it's straightforward to define where a wave has zero amplitude. And if the medium is stationary relative to this wave (often the case, as in a fixed steel string), then the velocity of propagation is also straightforward. So the velocity of propagation can be defined as relative to the medium, generally. Once you have the amplitude defined, and the velocity, the energy is also defined. In an oscillator that has mass m and amplitude A, the total energy is \( \frac {1} {2} m \omega^2 A^2\). In a wave with a defined energy density \( \rho\), it's \(\frac {1} {2} \rho \omega^2 A^2\).
Please explain how you arrive at this result. What is \(\omega\)? What precisely is "energy density"?
This may not be helpful but... It seems both I and arfa brane have a background in electrical engineering (very dim and distant in my case). I've been waiting for the Poynting vector to come up both in this thread and an earlier one. From https://en.wikipedia.org/wiki/Poynting_vector For Poynting vector S S=E x H I'm fairly sure (don't ask me how) something similar could be derived for any travelling wave. From memory S is the instantaneous value and it would be usual to take the rms average value of the peak (possibly). Please ignore this post if not helpful. Edit.. ω is the frequency in radians/second (always)
The result, at least for an object with mass, is found in any textbook that covers SHM. Wikipedia does an ok job here Angular frequency is conventionally given the symbol \(\omega\). As for energy density; this is a physics thing, you say there are some Joules in a volume because it helps solve a certain kind of problem, in this case oscillations near the surface of liquid water. You can define this energy as the motion of particles, or less controversially as motion of small volume elements of the medium. I will be getting to oscillators in electronics at some point; I want to be sure we're all on the same page with material substances, like a volume of gas, or a steel string or the "surface" of a volume of water, before going near the electromagnetic field. But we can see already, for a fixed oscillator like a pendulum in SHM, there is some kind of connection to a "free" propagating wave; both systems constitute a response by a physical system to an input (of "something" we might as well call energy).
There are many kinds of oscillating systems in physics. The three presented so far, a pendulum, a steel string and water are obviously quite different. The steel string produces sound waves because of the atmosphere around it. Generally pendulums and water waves don't; but a Newton's cradle as an example does make a sound, or series of them, water waves can make sound when they get big enough and meet a boundary. So in each case it's about momentum and momentum transfer. Energy and momentum are the things that get carried around, except energy isn't "really' a thing. Mathematically it most definitely exists and you can prove it.
Can we say that energy is the potential (essence) of a dynamic object? A dynamical latency which becomes expressed when obstructed?
Sorry, looks like I did get a bit confused there. The energy density is used to define the intensity for any wave, including the freely propagating kind on a surface. But the symbol \( \rho\) used is actually for mass density, kg per unit volume not Joules. Otherwise, the intensity is still the average energy per second through a unit area transverse to the propagation. In water, there is no mass flow through this area, all the energy is in vertical momentum; there's a gravitational field too, in the frame. For a fixed steel string, the oscillations have no real interaction with gravity, same for say, a metal pipe.