# Does the Fundamental Frequency in a Vibrating String NOT Necessarily Have the Strongest Amplitude?

I am doing some experiments on musical strings (guitar, piano, etc.). After performing a Fourier Transform on the sound recorded from those string vibrations, I find that the fundamental frequency is not absolutely the component with the largest amplitude (or energy).

I learnt from introductory physics courses that fundamental frequency is the "major" component of a vibrating string. So why does it sometimes does not possess the highest amplitude?

I did a bit of discussion on this subject in this thread on Music.SE.

The fundamental doesn't necessarily have the strongest amplitude. As said by Alfred Centauri, it depends on the initial configuration: ideally, the string returns to exactly that configuration after each $\tfrac1{\nu_1}$, and the amplitude of each harmonic in frequency space is proportional to its amplitude in the initial configuration in wavenumber space. Now the initial configuration for plucked instruments happens to be roughly what I depicted in the first figure in the Music.SE answer: something between a triangle and a sawtooth, and as we know both have a monotonically descreasing1 sequence of Fourier coefficients ($\mathcal{O}(\tfrac1{n^2})$ for triangle, $\mathcal{O}(\tfrac1{n})$ for sawtooth), so the fundamental does tend to have the strongest amplitude on the string in the beginning. But this may not hold true for an actual instrument sound for a variety of reasons: on the guitar, if you strike the string quickly with a pick, the initial configuration may be more like a smoothened Dirac peak, for which all of the lower frequencies have the same amplitude. Similarly for a loud piano note. Then, resonances in the instruments' bodies may lead to the fundamental decaying faster than some of the other harmonics, so after a while they are stronger, though instrument builders classically try to prevent this. Similarly, as said by John Rennie, the bodies may be more efficient at conducting some of the harmonics to the air than the fundamental and likewise the microphone at measuring the sound.

Finally, you should assure yourself of what this Fourier spectrum of yours actually shows: in audio production, spectrum analysers do not display something proportional to the amplitude $A(\omega)$ at frequency $\omega$, but in fact to $A(\omega)\cdot\sqrt{\omega}$ so that pink noise appears to have constant amplitude. (For both historic / technical and convenience reasons.) So if in such a spectrum the fundamental turns up slightly weaker than e.g. the second harmonic, it may in actuality still be stronger.

1Actually, the sequence of Fourier coefficients of a triangular wave itself is not monotonic, but alternates between a monotonic sequence (for odd $n$) and zero. Since the fundamental has odd $n=1$, that amounts to the same result.

I would expect that if you pluck a guitar string the mode with the highest amplitude is the fundamental, though this isn't necessarily the mode with the highest energy since the energy of a vibrating string is proportional to the square of the frequency (and the square of the amplitude).

However you're not measuring the amplitude of the string vibration. You're measuring the EMF generated by a microphone that is recording the air vibrations produced by the string. You need to consider how energy is transferred from the string to the air, and from the air to the microphone. In the case of a guitar a lot of the sound is generated by the body of the guitar, so you also need to factor in the efficiency of transfer of energy from the string to the body of the guitar.

All this seems complicated to me, and I don't know of any simple rules for how much of the energy of each mode eventually ends up at the microphone. I did do similar experiments at school, but that was a long time ago and I honestly can't remember whether the fundamental was the biggest mode or not. I remember that the higher harmonics were pretty big, i.e. the waveform looked nothing like a sine wave, so if you tell me the fundamental sometimes isn't the biggest component I can believe it.

It all depends on the initial configuration. If the string's initial configuration (shape of string at $t=0$) includes a node at the center, the string will not vibrate at the fundamental frequency.

This YouTube video is a demonstration of this phenomenon.