Daniel Bernoulli established the principle that the number of modes of vibration in a system is equal to the degrees of freedom. In his method of hanging weights on a string, I assume without any weights the string has one mode. Clearly, a system can have at most one fundamental. But the problem is how can the string be defined by a fundamental frequency while radiating a complex sinusoidal wave?
I am confused, therefore, about how to count the degrees of freedom of movement that exist in a string detained between two point with constant tension, length, mass, and composition. We assume the parameters allow the string to be subject to harmonic motion when deformed. The question seems a simple geometric problem.
I seems to me that the string has only one mode of vibration which is d2y/dt2 = 0 but then there is a second mode, which you can see with the naked eye, where the shape of the string concatenation is not a circle in cross-section so the axis of the string elliptic rotates.
This makes me ask if d2z/dt2 = 0 is also a boundary condition?
If we imagine the string is plucked so the deformation vector is only in the direction of the y-axis, then clearly the concatenary does not remain in the x-y plane. But the string does not necessarily assume a circular cross section either. In fact, I cannot see any reason why the cross-section would be circular at all.
Some say the string has n modes or even an infinite number of modes at the same time, which I think more closely applies to air and electromagnetic fields that have a broad frequency response but not to the detained string defined on the interval 0 to 1 where x is fixed and there must be a fixed point.
I have an intuitive answer for the degrees of freedom of string movement that I hope someone can state formally. There must be two modes of string vibration, a dominant and subdominant string mode. Then we have the string tone and overtones as two sets.
I think the string has only 2 possible modes of vibration, perhaps dy/dt and dz/dt, or it may be a radial and rotational mode, but such that the fundamental has only one mode possible at a time, but there is a secondary mode induced by the fundamental on any mode in the frequency expression that is not isochronus because d2y/dt2 is not zero Perhaps there is one degree of freedom that is perpendicular to the string axis and then there is a rotational degree of freedom perpendicular to dy/dt?
I don't see the equation for the string concatenation form is any where stated in the literature. Am I missing it?
Is there any way that dy/dt and dz/dt could have different frequencies so that the string radiates a complex sinusoidal function that is the sum of two frequencies? I know that doesn't make sense, but there must be a simple explanation for tones and overtones that is easy to understand and that has been so far overlooked.
I am assuming that the deformation and concatenary forms are states of system and not modes.
A mode of vibration in the string means a direction of vibration, e.g. x, y, and how many cycles of vibration there are along the string.
I expect Bernoulli's treatment started with a massless string with tension, with equally spaced point loading with masses (perhaps with half spacing at the ends), with vibration in one dimension. This is a simple and probably good starting example. In this case the number of modes is equal to the number of weights. If you allow vibration in two dimensions this doubles the number of modes. All the elliptical vibrations can be made by adding perpendicular modes. If the string has a very large number of (almost infinitesimal) weights, as a real string has, then the number of modes is very high, and in the end is determined by exactly how many atoms there are in the string and their arrangement. The infinite number of modes is only true for a string made out of continuous material, i.e. not atoms.
To Malcomb White. Thank you!
But there is a problem with dividing the string into smaller and smaller parts without end because clearly there are a finite number of modes.
More importantly, since it is clear the frequency is higher with each successive mode, it is clear the fundamental mode is the lowest energy state in the system.
It is clear that each mode interferes with every other mode so the question of the number possible modes does not affect whether the string can have more that one mode at a time.
Videos of string modes make with a frequency driver and strobe show the string is in one mode or another at any point is time. I think we see the string as a mode with 1 to at most 8 waves and then there is a perturbation of the mode so that the "standing wave" is subject to its own deformation. This allows the overtones to be added to the mode without nodal interference. (I think these experiments depend on careful choice of string parameters and are not generally true expect in a closed and bound range of tension, length, and mass.)
I like your answer up to the part where you show there are 1 or 2 modes, before you suddenly jump to the conclusion that the number of modes is large. That would be true in air, water, electricity but not true for a string under tension.
Here is an very interesting formulation: http://www.johnsankey.ca/string.html
But the author uses a continuous function when he admits the solution is discreet.
Equation 5 in this reference seems to contradict the existence of a large number of modes.
The number of modes on a string can be very large. To a first - and usually good - approximation they do not interact, i.e. they are orthogonal. The shape that the string is bent to before it is released can be described by a combination of modes that can be found by Fourier analysis of the shape, using the sinusoidal modes of the string. If there is a sharp corner in the string then there can be hundreds of modes excited when the string is released. The higher order modes tend to have less amplitude than the lower order modes, and also to have higher losses and so die out earlier.
I don't think equation 5 limits the number of modes - there is an 'n' (the order) that can take any integer value.
The stiffness of the string will make things non-linear (i.e. make the simple equations less accurate) long before the fact that the string is made of atoms limits the number of modes that you can observe.
If you pluck a guitar string 1/10 of the way along while touching it 1/5 of the way along, you can excite a mode with 2.5 cycles along the string. This becomes harder to do as the number of cycles increases, but it should be possible to pluck it 1/20 of the way along and touch it 1/10 of the way along and get 5 cycles along the string. There is nothing stopping modes with 100 cycles along the string, except the difficulty of accurately getting the touch and pick points, and removing the touch fast enough so that the mode isn't damped away too quickly to observe.
If you pluck a guitar string and record the sound it makes, there a programs on the internet to display the spectrum, and you will see that the may be a dozen or more harmonics, corresponding to the modes on the string. These can be excited individually as described in the paragraph above.
The string shape when plucked (before release) is a triangle. I agree the relaxation of perturbation drives overtones but these overtones are not harmonic on the string fundamental as multiples but come from frequency characteristic of the contenation that has precessionary mode. The earth turns at one frequency, but the north pole has another mode. But that is all there is, because the tone and overtone sets are defined as first and second order modes on the string boundary condition d2y/dt2 = 0, which is the position the string returns to when released (not the midline when d2y/dt2 is a maximum).
You cannot assume the string has the shape of the sound wave that is emitted since clearly the string has only 2 modes of vibration (y and z) which are the only harmonic and inharmonic modes.
I think at some point you will realize the string is an integral domain, whereas vibrations in air are not characterized by the identity of 1, which the string has. The string is indivisible, the smallest possible whole that moves about as 1.
You need to do more reading and maths. There are harmonics on the string. It can only resonate, i.e. have persistent vibrations, at its own harmonics, which are its resonant modes. The resonant modes are not just x and y, they are also all the harmonics, ie. with 1, 2, 3, 4 etc (up to infinity for an ideal string but just to a very large number for a real string) half wavelengths along the string. These, to a first order, do not interact with each other and vibrate independently. The string takes up the shape at each position which is the sum of the offset due to each mode. Study the Fourier transform. This breaks up the triangular shape at plucking into all the sinusoidal modes that are necessary to be added together to make this shape. Each vibrates at its own frequency and will radiate this sound into the air so will be visible as a harmonic in a sound recording. Because they all vibrate at their own frequency and die out at different rates they never add up to the triangular shape again. Most of the technology you use depends on this sort of maths being correct, and it works, so you had better believe it, and find out what you don't understand about it.
Ah, but the string is primitive and not technological, so you must prove your assertion string is Fourier and not simpletons in the subspace topology. Maybe you need to read about subspace topology?
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