Resonant Guitar Modes

1. Mode Description

2. You may already have some knowledge of the harmonic modes of a stretched string or sound waves in a pipe (see strings and standing waves for some background theory). Various parts of the guitar display these harmonic resonant modes, which give rise to certain tonal characteristics (viz. spectral response) and how the the instrument emits acoustical radiation.   The guitar's soundboard is by far the most important element in sound production and it displays resonant modes, similar to those in strings/pipes.   However there are two important differences:
1. The soundboard can be considered 2-dimensional (rather than 1-dimensional in the case of a string or pipe )and hence is slightly harder to visualise; and
2. The frequency of the resonant modes generally do not follow a simple harmonic progression. For example, the frequency of the first overtone for a simple vibrating string is twice that of the fundamental, whereas for a soundboard, the (1,0) mode (1st longitudinal overtone) is not twice the frequency of the (0,0) mode (fundamental).

See Chladni Patterns and Modes. Although this is written principally for the violin, it applies equally to guitars. Also see guitar Chladni patterns.

A label in the form of (m,n) denotes the mth harmonic in the longitudinal and nth in the lateral direction.

{Scan in hand-sketch}

Example of how the soundboard vibrates in (1,0) ("dipole") mode.

3. Mode Detection

We can observe the resonant modes of the soundboard in various ways:
1. Optically: Chladni Patterns, holographic interferometry, laser velocimetry - a laser beam is reflected off various points on the soundboard.
2. Acoustically: An array of microphones or a single microphone scanning an area detects various peaks in sound intensity while the guitar is excited.
3. Electrically: Examine the capacitance between a charged plate on the vibrating soundboard and another charged plate held steady.
4. Mechanically: Accelerometer/velocity transducer. The mechanical vibrations are measured quite directly, such as with a phonograph cartridge or impedance head.

Adapted from Fletcher & Rossing: "The Physics of Musical Instruments"*

4. Modal Contributions to Sound Radiation

Because of the interaction among the various components of the guitar, the radiation patterns are produced in an interesting manner, (in a similar way to radio aerials---this is because the fundamental wave equations governing both are very similar.)
The (0,0) fundamental mode usually radiates the highest amplitude sound intensity, and the wave fronts radiate outwards in a roughly spherical manner. On the other hand, the (1,0) dipole radiates a volume with two large, diametrically opposing lobes, with two smaller lobes extending perpendicularly:

{Scan in Hand-Sketch, (1,0) Mode Radiation}

Example of Acoustic Radiation Field Around Guitar in (1,0) mode

These radiation patterns are determined by how the instrument moves air in each mode. For example, the air-flow around a guitar in the (0,0) mode looks like:

{Scan in hand-sketch of airflow, (0,0) mode (?)}

Airflow Surrounding Guitar, (0,0) Mode

5. Modal Contributions to Spectral Response

Resonant guitar modes create large vibrations and hence radiate sound efficiently. This has a direct effect on the acoustic spectral response. Most guitars tend to have three strong resonances in the 100-200 Hz region (due to top/back (0,0) coupling and the A0 Helmholtz mode by virtue of the soundhole)   The higher frequency modes don't show as strong resonances (ie. are less efficient radiators) yet still add spectral features ('colour') to the higher frequencies.   One might naively expect that people would prefer an instrument that had a relatively uniform spectral response (ie. constant intensity over a range of pitches, no 'dead notes') but it appears that this sound is taken by most to be rather simple and it seems that most people prefer a guitar that has interesting spectral features (a 'voice'.)

*Fletcher, N. and Rossing, T. "The Physics of Musical Instruments" (2nd ed.) ©1998, Springer-Verlag New York Inc.

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