Why are standing waves on a guitar string sinusoidal?

[Bill Plates]

Ok I understand the idea that a standing wave can be represented as the sum of two traveling waves going in opposite directions with same stuff but what I don't understand is why the waves on a guitar string are sinusoidal. I mean I know looking at them, they look sinusoidal but could they be represented by some other function that just looks sinusoidal. Could you form any other shape wave?

 

[Khashishi]

resonance. The non-sinusoidal parts of the wave die out fairly quickly, leaving just the sinusoidal part.

 

[Bill Plates]

Could you elaborate please?

 

[M Quack]

They are not sinusoidal, there are plenty of harmonics = multiples of the the fundamental frequency = fractions of the fundamental wave length.

Depending on where you pluck the string you excite the harmonics with higher or lower amplitude, resulting in a harder or softer sound. Think of a Fourier decomposition of the initial triangular shape of the string while being plucked.

In general, however, Khashishi is right that harmonics decay faster such that after a while essentially only the fundamental is left.

 

[Bill Plates]

They are not sinusoidal, there are plenty of harmonics = multiples of the the fundamental frequency = fractions of the fundamental wave length.

Depending on where you pluck the string you excite the harmonics with higher or lower amplitude, resulting in a harder or softer sound. Think of a Fourier decomposition of the initial triangular shape of the string while being plucked.

In general, however, Khashishi is right that harmonics decay faster such that after a while essentially only the fundamental is left.

But the fundamentals and harmonics are always sinusoidal, right?

How can I find out more about harmonics decaying?

 

[M Quack]

Yes, a Fourier series basically writes an arbitrary wave form as superposition of sinusoidals of many frequencies.

In an electric guitar, the bridge pickup is more sensitive to harmonics than the neck pickup. If you took the difference, Peter Green style, you would be quite sensitive to the harmonics but less to the fundamental.

If you have access to a numerical oscilloscope or if you can simply record the sound with a microphone and a sound card in your PC, you should be able to chop the sound into bits (say 0.1 seconds each) and calculate the Fourier transform of each bit. You can then compare the amplitude of the different harmonics from one bit to another. Unfortunately I cannot tell you which program would be able to do that...

 

[Bill Plates]

Okay, thank you very much M Quack and Khashishi. I think I am closer to understanding this topic. Have a good day.

 

[bahamagreen]

Maybe someone more knowledgeable in physics and math can clarify or correct, but I think...

Mechanically, simple harmonic oscillation is going to look sinusoidal because the momentum is subject to a force that varies with excursion from the central position. Least action principle is the minimization of the integral of momentum times distance... it is what determines which of all possible curves is actually made manifest.

Mathematically, it is sinusoidal waves that historically comprise the components of the Fourier because it was sine waves that were chosen as the "perspective". But an arbitrary wave (including a sinusoidal wave itself) can also be decomposed into cosine, or triangle, or square, or ramp, or impulse, or any other arbitrary waveform. The choice of the sine wave perspective for decomposition results in simpler results when the applications are subject to the least action principle.

 

[M Quack]

Mechanically, simple harmonic oscillation is going to look sinusoidal because the momentum is subject to a force that varies with excursion from the central position. Least action principle is the minimization of the integral of momentum times distance... it is what determines which of all possible curves is actually made manifest.

yes

Mathematically, it is sinusoidal waves that historically comprise the components of the Fourier because it was sine waves that were chosen as the "perspective". But an arbitrary wave (including a sinusoidal wave itself) can also be decomposed into cosine, or triangle, or square, or ramp, or impulse, or any other arbitrary waveform. The choice of the sine wave perspective for decomposition results in simpler results when the applications are subject to the least action principle.

You need sine and cosine to decompose arbitrary wave forms.

There are many other sets of complete basis functions, e.g. Chebychev polynomials could be used for a string. http://en.wikipedia.org/wiki/Chebyshev_polynomials

Sine waves are chosen because there is a direct corrrespondance between the temporal frequency of the oscillation and the spatial wave length.