Re: Does not have any harmonics

• posted

> Hi,

> > Why does a sinusoidal waveform alone does not have any harmonics or > > distortion ? > > > For example, (Reference ->
> > Sawtooth wave of constant period contains odd and even harmonics Square > > wave of constant period contains odd harmonics Triangle wave, (an > > integral of square wave) contains odd harmonics > > > But, How is it possible that sinusoidal wave alone does not have any > > harmonics or distortion ? > > I searched the internet,but i did not find any link/pdf that talks in > > detail about these . > > Any ideas ? > > > Thx in advans, > > Karthik Balaguru > > Reading all the responses to this has been like hearing of all the wise, > blind men who are describing an elephant. > > So, IMHO, an elephant is like a rope. And by the way, why is it suddenly > raining fertilizer? > > At any rate, a sine wave doesn't have any because that's how harmonics > are defined. Harmonics are defined in the context of the Fourier series, > and the Fourier series for a pure sine wave is just -- that sine wave. > That's it, no more. Only a periodic wave that deviates from a perfect > sine wave can have harmonics, and (thanks to Fourier) we know that we can > express that periodic wave, if we so choose, as a sum of sine waves at > the fundamental frequency and all of it's multiples.

Thanks. I had similar thoughts to your's.

I had to giggle when Bartoli correctly said, as you did, that it basically a definitional matter, and Allison jumped to the moon.

• posted

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

• posted

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.

• posted

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

• posted

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

ElectronDepot website is not affiliated with any of the manufacturers or service providers discussed here. All logos and trade names are the property of their respective owners.