Doesn’t an infinite number of modes violate conservation of energy?

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?

If the string has one degree of freedom, then it has an exact solution. It is a standing wave and does not depend on time. Your equation assumes the string is a sine wave and assumes superposition is like a fourth Newtonian law. Does anyone have a Hamiltonian equation?

Answer to the 1st comment: no. Does an infinite number of terms in a summation makes the sum infinite? No, since convergent series exists

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.

A string has an infinite number of degrees of freedom, since it can be modeled as a continuous medium. If you manage to force only the first harmonics, the dynamics of the system only involves the first harmonic and it's a standing wave: this solution does depend on time, being $w(x,t) = a_1(t) sin (k_1 x)$ (time dependence in the amplitude of the sine). No 4th Newton's law. I didn't get the question about Hamilton equation

A monochord has maybe 6, 7 modes. If the string has a million modes, I cannot believe their action is sustained in an inertial way like the fundamental. This archaic model of string action is incoherent.

What do you mean with "archaic model"? Can I ask you what's your background that makes you do this sentence? Physics? Math? Engineering?

The ridiculous idea that two waves slam together to make a standing wave is couple hundred years old. You have no curvature, no manifold, no orbit, no integral, no perturbation theory, no way to calculate frequency. You don’t know how many degrees of freedom the string has and can’t explain string shape operator. You postulate the string is a sine wave with superposition but cannot define union, intersection, and complementation. What do you claim is modern here?

You postulate nothing here. You have continuum mechanics here. You have PDEs under the assumption of continuum only. You have exact solutions in simple problems, you have numerical methods approximating and solving the exact eqhations. And trust me: this is the branch of physics used in many engineering fields, from mechanical, to civil, to aerospace engineering

My postulate is string curvature is an inertial constant that cannot change unless the string is acted upon by an external force. This is Newton’s first law and it is true at rest as well as during sustained motion. Everything else follows this assumption.

Sound waves and electromagnetic waves can superimpose but they do not have fixed end points nor a boundary condition imposed by tension. You start by assuming the string is a sine wave but it is not! It is a minimum surface of revolution formed by rotating the catenoid around the string axis.

The string has to be Hamiltonian and I think that by itself implies string shape is a minimal surface of revolution. PDE’s are confined to a plane and they cannot make a manifold.