Re: Does not have any harmonics

It isn't just a matter of definition. In classical mechanics, most simple sorts of motion (a mass on an ideal spring, the motion of a point on a rotating sphere, and so on) are sinusoidal. Sines and cosines are the eigenfunctions of all sorts of vibrations, including drum heads. (Drum heads go 'thump' instead of 'bonnnnggggg' because their sinusoidal oscillations don't occur at evenly-spaced frequencies.)

Frequency-dependent loss, for instance due to damping, makes essentially all vibrations become more and more like damped sinusoids as they die away.

In optics, a (weak) sinusoidal grating produces only one diffracted order (one on each side of the specular reflection). All other profiles produce multiple orders.

In quantum mechanics, the time evolution of every eigenstate is sinusoidal. Sinusoids are the only functions for which the Planck relation e = h * nu applies.

So it isn't just a matter of definition--exponential functions, including sines and cosines, really are fundamental to a wide range of physical processes. Another way of saying this is that just about all the equations of mathematical physics are second-order differential equations.

Cheers,

Phil Hobbs

It is a definitional matter.

Originally: "Why does a sinusoidal waveform alone does not have any harmonics or distortion?"

The fundamental ("alone") is the "first harmonic." That is, it is self-referential which in ordinary language also comes out to "definitional," "self-evident," "axiomatic" and other such lingo. It is the _basis_.

You can look at it mathematically too. It is energy at a single "line": cos(wt) for example. The single line "is what it is," and as a _fundamental_ it refers to nothing else because that is what it means to be fundamental. Anytime you end up having no choice but to speak in tautology to "explain" something, it is a basis; it is definitional.

That is to say, X = -X'' (or -X double dot).

That's an undamped oscillator equation, which qualifies but isn't the only example...Maxwell's equations, the heat equation, the Schrodinger equation, the harmonic oscillator equation with damping, the wave equation--they're all second order linear DEs. (Laplace's equation is too, but it doesn't have a time dependence.)

Cheers,

Phil Hobbs