I want to re-think the shape of a vibrating string to find a source for frequencies that are not emitted by the fundamental mode but do not require points that are fixed and moving at the same time, which would be the case if the fundamental and octave co-exist. If this occurs, then the octave would degenerate to the fundamental (which does not occur).
Do you know the parametric equation for a catenoid (minimal surface of revolution) with ends that closes on endpoints (-1,0, 0) and (1, 0, 0) so the surface is convex?
Wikipedia and Wolfram MathWorld show equations for a concave catenoid with open ends but I need the equation for a catenoid with endpoints. I know the equation for the catenary (the a cosh x/a function for a rope hanging between two points). I just need to know how to make it rotate around the x-axis so the cross section is a circle.
Since the “rope hanging between two points” catenary is an average of exponential functions, can the catenoid equation be written as a time-average of the family of curves passing through the string axis at the same time (particularly the subset of curves with minimal surface area). Is it possible the catenoid surface area is a time-average which is subject to harmonic oscillation? I am thinking perhaps since the product of the y and z vectors is 0, perhaps they can exchange potential and kinetic energy, or somehow make the surface itself have a mode of vibration.
I looked at the previous answer but I am not satisfied using tension and forces tangential to the string. The equation has to have the form of a curve-lifting function where each point on the string maps 1-1 on a point on the string boundary. The points on the string slide along lines orthogonal to the string that are time-averaged so the shape of the string is constant during decay of harmonic motion. There is a problem if the string open to an infinite series of ever smaller modes because then there is no algebraic formula for the string that is regular, coherent.
Instead, I think the string behavior is a retraction deformation on the string at rest with a cylinder map. See Hatcher Algebraic Topology. pp1-2
When you look at the string vibration,it seems to have a rotational wobble, like the space is not round in cross section and the elliptical axis rotates around the string axis.
Is there no experimental evidence regarding whether the string is a parabola or catenoid? I find no documentation in the literature two or more modes coexist when the string is not driven. Parabola and catenoid are apparently very close but arise in different ways. I notice there are many different string equations in the literature (Feller, Atkinson, Krein, Larsen etc). Usually sign of a weak paradigm. I don't get how the second derivative of a standing wave is not zero with respect to time. I don't see how Mersenne's Law uses twice the string length as a wavelength. The string is defined by a point beyond its boundary? This question has been unanswered since Euler made the mistake of assuming the series of infinite modes on the string converge, later proven incorrect by Cantor and Dirichlet. Yet I still see Euler's string equation in the literature, as if it is a natural law.
This is a serious question with a 300 year old answer that does not make sense. The string vibration is coherent but the theory of the string is not.
Fig 1 Hanging Rope Catenary Fig 2 Fundamental and Octave cannot co-exist Fig 3 Schematic showing string catenoid as average of exp functions (I moved the exp points of origin in the diagram).