# Is it possible there can be a non-Fourier model of string vibration? Is there an exact solution?

I am looking for a model of string vibration that does not assume the string is Fourier. Is there a Hamiltonian?

The equation of motion must be a function of length and tension, not time, and it must explain why the Mersenne equivalence relation is true only when the string tension and length have boundary conditions. That is, sustained action does not occur unless the tension and length are greater than some minimum value. And the equation should give the frequency of vibration as a constant of motion. I believe that requires an integral that is not found in the literature but clearly must exist.

I don’t see how Fourier analysis can determine the frequency of vibration, particularly since strings of different length and tension can have the same frequency.

That is to say I want the Hamiltonian equation. I don’t see it in the literature.

So maybe there is like a Newtonian and non-Newtonian way to understand string mechanics?

It is not coherent action for a string to vibrate in many different modes at the same time. How is this possible if the higher modes have higher energy levels? Won’t the string be required to seek the lowest energy level under the principle of least action?

I don’t accept superposition on the string.

For example, if a string vibrates in the fundamental mode, the midpoint of the string is moving and not critical but in the octave mode, the midpoint is critical and stationary. So, if the fundamental and octave coexist, how can the midpoint be both moving and stationary at the same time? A point cannot be critical and not critical at the same time, it would seem. String modes cannot superimpose!

If we could add the octave and fundamental, the result is the fundamental. The fundamental is inertial and cannot go to the octave mode unless a force acts to stop the critical midpoint. Clearly, the octave has a critical Morse point.

D’alembert’s archaic, pre-Newtonian equation used to derive a partial differential equation of string motion postulates the string has two degrees of freedom in vibration (that is angular momentum is conserved) but isn’t it true there is only one (so it is energy which is conserved instead). Besides, doesn’t the PDE describe motion in a plane when the string is not constrained to move in a plane? Shouldn’t the equation of motion give us the shape operator as a minimum surface of revolution?

I think it is logical that the number of modes cannot be greater than the degrees of freedom. The string reasonably has one or two degrees of freedom. Infinite degrees of freedom is nonsense! If there is only one degree of freedom then there must be an exact solution. If we have an exact solution then we known the shape of the string is a catenoid not a sine wave.

• Please clarify your specific problem or provide additional details to highlight exactly what you need. As it's currently written, it's hard to tell exactly what you're asking.
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Commented Jun 28 at 22:08
• I am asking how many modes of vibration can a string have? More than the degrees of freedom? Commented Jun 28 at 22:31
• It seems to me that you are mistaking the path in phase space for the deflection of the string. The phase space of the string has an infinite number of degrees of freedom. Commented Jun 28 at 23:21
• Comment to the post (v4): What does string is not Fourier mean? That the Fourier series does not converge? Or that the vibration is not monochromatic? Or something else? Commented Jun 29 at 15:47
• "String modes cannot superimpose!" Yet, empirically, they do. Commented Jun 29 at 15:50

A vibrating string can support a very large number of vibrational modes simultaneously. This is because waves on strings superimpose linearly. Note here that an electric guitar string struck hard with a stiff pick puts a transient sharp flexural kink in the string, which can be decomposed into a very large number of modes (including anharmonic ones that get suppressed quickly). So pick strikes carry a lot of high-frequency content.

Assuming linearity, the response of the system can be written as the superposition of its modes.

How many modes does a continuous medium have? An infinite number, in general.

How many modes have non-negligible contribution? Usually, it depends on the forcing of the system. In Nature it's likely to have forcing with limited bandwidth, so it's usually possible to truncate the summation over the infinite number of modes only to a finite number of modes, those which have natural frequency lower than the "cut-off" frequency. Sometimes, truncation is quite a rude way and better ways to model high-frequency modes exist (but this is not the focus of your question, and my answer).

As an example, a string with clamped edges have modes with wave number $$k_n = n\frac{\pi}{L}$$ and frequency $$\omega_n = a \, k_n$$, being $$a$$ the "speed of sound" in the string, usually $$a = \sqrt{\frac{N_0}{m}}$$ where $$N_0$$ is the tension of the string and $$m$$ is the mass linear density. If a forcing $$F(t)$$ acts on the system, with maximum frequency content at $$\omega_c$$ (i.e. the frequency content of $$F(t)$$ is written as its Fourier transform, and all the contributions are negligible for $$\omega > \omega_c$$), it's possible to write the response of the system as a linear combination of the modes of the system with frequency $$\omega_n < \omega_c$$, $$n < \overline{n} = \omega_c \frac{L}{\pi a}$$, i.e.

$$w(x,t) = \sum_{n = 0}^{\overline{n}} a_n(t) \sin(k_n x) \ .$$

• Doesn’t an infinite number of modes violate conservation of energy? Commented Jun 29 at 0:52
• Does the string truncate the modes as part of the equation of motion or is that just because there is no way to form an integral on a series that does not converge? Commented Jun 29 at 0:57
• Answer to the 1st comment: no. Does an infinite number of terms in a summation makes the sum infinite? No, since convergent series exists Commented Jun 29 at 13:29
• Answerr to 2nd comment: truncation is not related to divergence of the series. Instead, you can truncate the series and get a result converging to the dynamics of the system you observe since the series is converging and you can neglect high-frequency terms whose contribution is close to be zero. Commented Jun 29 at 13:32
• What do you mean with "archaic model"? Can I ask you what's your background that makes you do this sentence? Physics? Math? Engineering? Commented Jun 29 at 14:03