Fourier series analysis of string vibration

Question

In case of a plucked string,the amplitudes of successive frequencies fall by 1/n^2.

In case of a string which is struck so that say at x=a only the string has a velocity,say v,initially,then the amplitudes of successive frequencies fall by 1/n which implies that it is more enriched with harmonics than plucked one.

Again,for a string struck at midpoint such a way that the initial velocity at each point varies linearly with its distance from nearest end from zero to v,then the amplitudes fall by 1/n^3.

My question is that,they all can be showed by fourier analysis applying proper boundary conditions.

But what is the reason behind it physically?why a struck string has more enriched vibration with different frequencies,than a plucked one,and why the amplitudes fall at such different rates for different cases?

Answer 1

Sine waves are absolutely continuous, so they are good at approximating functions that are also absolutely continuous. By this I mean that a Fourier series for an absolutely continuous function will generally converge fast.

If the function is continuous but has discontinuities in the gradient, like a triangle wave, the convergence will be slower because it's hard to get the discontinuity in the first derivative using sine waves. If the function is discontinuous, like a square wave, then the convergence will be even slower because it's even harder to get discontinuities in the function using sine waves.

So for example the Fourier coefficients of a triangle wave (discontinuities in $f'$) fall as $1/n^2$ while the coefficients of a square wave (discontinuities in $f$) fall as $1/n$.

The point of all this is that the instant after the string has been displaced (struck, plucked or whatever) it will have some profile and the shape of that profile will determine the convergence of the Fourier series used to describe it. The rate at which the higher harmonics fall away will depend on how jagged the initial shape of the string is.

At this point I have to resort to arm waving because I don't know anything about playing stringed instruments. However plucking the string involves smoothly deforming it sideways then releasing it, while striking the string delivers an impulse at one point on the string. It seems a reasonable supposition that the initial profile of a struck string is more jagged than that of a plucked string so the Fourier coefficients will fall off more slowly.

You indicated you were after a physical rather than a mathematical discussion, which is what I have attempted. If you're interested in the maths then Wikipedia has a nice article on the convergence of Fourier series.

Answer 2

Suppose the string stretches between $z$ co-ordinates $z=0$ and $z=L$. Then the Fourier series for the string's sideways displacement as a function of time contains only components of the form

$$\psi_n(z,\,t) = \sin\left(\frac{\pi\,n\,z}{L}\right)\,\cos\left(\frac{\pi\,n\,c\,t}{L}+\delta_n\right)\tag{1}$$

where the cosinusoidal time variation is valid if the string is released at $t=0$. These are standing waves that automatically fulfill the universal boundary conditions that:

$$\psi(0,\,t)=\psi(L,\,t)=0;\;\forall t\in\mathbb{R}\tag{2}$$

i.e. the string is fixed at both its ends and:

$$\left.\frac{\partial}{\partial\,t} \psi(z,\,t)\right|_{t=0}=v_0(z);\;\forall z\in[0,\,L]\tag{3}$$

i.e. the string's initial velocity at all points is the string's initial velocity profile. The first boundary condition dictates the spatial dependence in (1), the second the time dependence - different amplitudes and phases $\delta_n$ let you match your initial displacement and velocity conditions. The velocity $c$ gives the correct phase relationship between the two variations (i.e. the solutions in (1) are indeed made up of complex exponential functions of the arguments $z\pm c\,t$).

Now you must sum terms of the form in (1) to build your space and time dependence of your string's shape. This you do by using the standard formulas for Fourier co-efficients to match the strings initial shape and initial velocity profile. So you choose profiles that describe the conditions you wish to model and then work out the Fourier co-efficients (including the phases) with the standard formulas. It is these co-efficients that set the spectral content of the vibration.

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The OP further asks:

Yes I also did that mathematically,but i shall be helped if you provide me with a physical insight into this system rather than mathematical,on why amplitudes fall in squares or linearly

The Fourier co-efficients and phases are the description of the initial velocity and shape, so they contain exactly the same information as the sentence "the initial shape and velocity profile are .....". So your question is equivalent to asking why the initial shape and velocity profile are as they are: it's just how the system is set up and that is a function of your hammer / plectrum shape, striking mechanism and so forth.

The physics of the system simply tells us we can model the disturbance as sum of terms in (1). Newton's law gives us the wave equation, which is both linear (which justifies breaking the expression up into a sum) and also admits terms of the form $f(z\pm c\,t)$ or any linear superposition thereof. That's all that the physics tells us.

Fourier's theorem then gives us a way to encode the description of the initial conditions in a way from which it is easy to read off what the linear superposition co-efficients must be and be in accordance with the physics. Uniqueness and completeness then guarantees that this is the only way one can combine solutions linear and be consistent both with the physics and the assumed initial conditions.