I am currently trying to understand standing waves. The key to the phenomenon is that: $$A\sin(kx+\omega t) + A\sin(kx-\omega t)=2A\sin(kx)\cos(\omega t)$$ The shape of $2A\sin(kx)\cos(\omega t)$ is constant, as only the amplitude changes with time due to the $\cos(\omega t)$. This creates the familiar "standing wave", as the shape remains the same while the amplitude goes up and down, occasionally hitting $0$. The following post also explains that the wavelength is not actually required to fit $L/0.5n$ to form a standing wave: That is only a requirement if the amplitude on both ends of the string has to be $0$. In most string setups, the ends are fixed though, so practically speaking this is a requirement.
Suppose that I have a string of length $L$ with fixed ends, and that I have somehow managed to make a perfect sin-wave on one side of the string. This wave is traveling to the other side of the string:
As most articles and videos have described it, when a wave hits the end, it will both change direction, and flip/be inversed. But I have great difficulty seeing how this ends up creating an identical wave with identical phase to the first one. There is also a great deal of ambiguity in how one is supposed to visualize this flip and reverse action:
- Is the whole wave flipped at once, or is it flipped "gradually" as each point of it hits the end?
- Does the wave move while it's in the action of being inverted? If so, wouldn't that mean that a lot of parts of it never hit the end, and are thus never reflected properly?
- Does the standing wave phenomena only occur because new waves are constantly created on one end. Is it these waves that the original waves form interference with, or does the wave form interference with "itself"?
The problem isn't that I don't get how the initial wave is created on a string instrument: I get that when plucking a string instrument, a triangle shape is created, and that this shape can be described as a sum of harmonic frequencies of varying amplitudes. The problem is that I don't really get the geometry of this whole inversion process, and how it forms two identical waves.