Vibration of strings [duplicate]

Question

What happens if we clamp a string at both it's ends and pluck it at different distances from one of the ends? Will the frequency of oscillation be the same or will it differ based on the place from where it was plucked?

Answer 1

The natural harmonics are solutions to the wave equation with boundary conditions applied (a.k.a. modes). In this case that the string is "clamped" at the ends. The "plucking" of a string at different positions along the strings creates different initial shape of the string, i.e. different initial conditions. By describing the initial shape as an infinite series over the natural modes the time evolution can be easily determined. Since the pluck is momentary, once released only the natural modes will be present in the resulting vibration. As many of the comments have stated you will usually detect (hear) the lowest natural frequency as the note being played and the collection of audible harmonics will contribute to the "tone" of the pitch or note. This description assumes a musical interpretation of the question.

It should be noted that any harmonic that has a node where you pluck the string will be missing from the spectrum and that it is possible to attack a string (impulsively or by some other mechanism) so that the fundamental is missing or very weak.

If the string is driven the frequency of the driver will be present along with a collection of harmonics. Also, the nature of the harmonics is different for a very thin string under tension (like a tight rope) and a "stiff" string (something more like a beam). There are several types of boundary conditions for stiff beams. For a very thin cross section they probably all go to the string spectrum in the limit.

Answer 2

In an idealised mathematical it will. If you assume the string obeys the wave equation, $$\frac{\partial^2 u}{\partial t^2}=v^2\frac{\partial^2u}{\partial x^2},$$ where $u$ is the displacement of the string and $v$ the wave propagation velocity, then any solution will repeat itself after the fundamental period. The fundamental period is given by $T=\frac{2L}{v}$ and is also the time required for a wave to travel from one end of the string to the other end and back.

A simple model for a plucked string is the string starting out as a triangle, where the ends are connected to the point that gets plucked. In this animation you can see how this would evolve under the wave equation:

By adding multiple strings being plucked at different positions you see that they repeat after exactly the same time.

Note that any initial wave shape would repeat after the fundamental period.

So does this also apply in practice? In general, no. The oscillation seen in the animation can also be seen as many sine waves added together, each oscillating with their own frequency. In reality the oscillation will be damped and the wave equation will only be followed approximately. Each frequency will be damped by a different amount so the overall shape of the wave changes over time. This means you won't get a simultanuous repetition like in the second animation.

I assumed in your question you meant the frequency of the entire wave repeating. If you meant audible frequency you have to look at @ggcg's answer.