Guitar resonance and soundhole geometry – Part 7: HIGH PERIMETER TO AREA RATIO (P:A) SOUNDSLOTS

PART 7: COMPARISON OF HIGH P:A SOUNDSLOTS WITH ROUND SOUNDHOLES

This is the seventh in a series of nine posts summarising the results of an experiment I carried out during 2017 and 2018 to try to increase my understanding of resonance in acoustic guitars, and in particular how the design of soundholes could be improved.

So far this series, Parts 1 to 6 have looked into the function of soundholes in general and the performance of the traditional round guitar soundhole. 

Part 5 shows how Helmholtz resonance is detectable in the response of my rigidly-confined airbody, excited by a signal from the loudspeaker sited in the lower bout. 

For this parlour-sized cavity the Helmholtz resonant frequency was at around 150Hz, and varied with soundhole size as theory predicts. The measured values coincide quite closely with predictions made using the equation provided by Gore & Gilet (Contemporary Acoustic Guitar – Design 2-14)

The strongest low frequency resonant response of an airbody is the Helmholtz mode, pictured as a plug of air in the region of the soundhole pushed and pulled back and forth as the airbody inside the soundbox expands and contracts at its fundamental resonant frequency

The most important experimental findings so far have been:

  • Blocking off the soundhole of a guitar reduces the emission of sound by about 30% depending on the instrument (the other 70% is direct transmission of sound to the listener by the soundboard vibration)
  • The larger a soundhole is the better it is at emitting soundwaves, and the more complex the tone it can produce – it has better acoustic conductance
  • Helmholtz resonance is not detectable except very close to the soundhole, so does not directly contribute to the sound of the guitar
  • The oscillating airflow associated with Helmholtz resonance is strongest at the edge of a soundhole and weakest in the centre (the edge effect)

My interpretation of these results is:

  • While the Helmholtz resonant response can only be detected very close to the soundhole, the three spring model indicates that the airbody in a soundbox acts as an elastic connection between the top and back panels. Helmholtz resonance, although not audible, therefore plays an important role in shaping the overall sound qualities of the instrument (loudness, timbre)
  • Much of the sound coming from the soundhole is produced by complex processes inside the soundbox from soundwaves pumped into the box by the vibrating soundboard
  • These internal processes include standing wave resonances, reflection, interference, and diffraction as the sound finds its way out through the soundhole

From Part 7 onwards, the emphasis shifts from round soundholes and detecting Helmholtz resonance to a direct comparison between round soundholes and soundslots.

The question now becomes: will a longer skinnier soundslot (high perimeter : area ratio) be a more efficient radiator than a shorter fatter one, as the Royal Society paper (see Part 1) asserts?

My experiments have shown that with Helmholtz resonance there is indeed higher activity at the edges of a soundhole than in the centre. The Royal Society paper links this to their finding in violins that the overall efficiency of slots (f-holes) is greater than round holes because they have a greater length of edge (perimeter) for their area and thus more scope for the edge effect to come into play.

In order to see if the same effect measured in violins happens in guitars as well, I repeated the original soundhole resonance experiment using a set of rectangular soundslots, all with the same area but with increasing perimeter to area (P:A) ratio – that is, longer and narrower.

These all had the same area as a round 45.4mm (R45.4) radius soundhole for comparison. [1]

SOUNDSLOT GEOMETRY

The soundslots used in the experiment were the following sizes:

SLOT NUMBERDIMENSIONS (mm)AREA (mm2)P:A RATIO
SLOT 3 (RECTANGLE)150 x 436450 0.0598
SLOT 4 (RECTANGLE)170 x 3864600.0644
SLOT 5 (RECTANGLE)190 x 3464600.0693
SLOT 6 (RECTANGLE)210 x 3165100.0740
SLOT 6A (RECTANGLE)250 x 2665000.0849
SLOT 6B (FLAT C SHAPE) 300 x 21.765100.0959
SLOT 7 (FLAT C SHAPE)341 x 1964110.1122
SLOT 8 (FLAT C SHAPE)386 x 1765620.1228
DOUBLE SLOT 1 (SHALLOW S)[380 x 1764600.1263
 MEAN AREA:6480 +/- 1.2% 

The error margin of +/- 1.2% in the slot areas could be improved, but is reasonable for drawing general conclusions.

The flat C shape for slots 6B to 8 was made to fit across the upper bout of the Parlour size cavity. Each of these C shapes added a curved end to a straight slot.

The S shape for double slot 1 fits around the curve of the upper bout edges and is inset by 15mm from the edge of the cavity.


The graph below shows the overall response of the slots. Again, there was little activity above 1,500Hz, so this shows the plot from 0 to 1,000Hz.

Figure 1: Response of soundslots 

The overall form of this graph is similar to that for round soundholes (see Figure 2below), with a strong response between 150 and 300Hz and a scattering of low-order high frequency responses. [2]

Figure 2: Response of round soundholes – note the overall similarity to Figure 1

Both holes and slots each follow the same spectral pattern no matter their size. For both, greater hole area allows stronger radiated power.

From the data represented in Figure 1, the overall strength of the sound emission from each slot can be calculated. 

(Remember that all data is relative to the closed soundbox, so is unfortunately not an absolute measure that can be used to calculate the performance of holes or slots from scratch.)

Figure 3 belowshows how the radiative power of the slots varies with increasing perimeter to area.

Figure 3: Soundslot relative radiated power by P:A ratio

Figure 4presents the same data, showing how the slots compare to the round soundhole (radius 45.4mm, purple data point):

There is clearly a relationship between the radiating power of a soundslot and its P:A ratio. A linear regression fit gives an R2value of 0.953 – a good result.

Figure 4: Comparison of round soundhole to soundslots of the same area by P:A ratio

Figure 5 below gives a clearer view of the total relative radiated power of the slots.

It’s clear that below a P:A ratio of 0.08 the slots have little advantage over the round hole. However as P:A becomes greater than 0.1 there is a large increase in sound conductivity as found in the Royal Society paper.

Figure 5: Comparison of radiated power 50 to 1,000Hz for soundslots[3]

The high P:A ratio slots 7 and 8 show an improvement in conductivity of 60% over the equivalent round hole (R45.4).

But what about the quality of sound produced by the slots? Dividing the radiated power into octave bands shows a consistent increase in power in the Octave 3 band (165 to 330Hz). Most of the increase in power of the high P:A slots is in this octave covering most of the guitar’s range.

Figure 6: Soundslot radiated power by octave. P:A ratio increases from left to right.

Here is the same data expressed as a percentage of total radiation for each slot, again broken down by octave frequency band:

 Figure 7: Soundslot radiated power by octave band as a percentage of total power

This shows a decrease in bass response as the P:A ratio increases, the difference being made up by a similar increase in the higher octave band (covering the notes E5to C6– guitar string 1 is E4).


HIGH P:A SOUNDSLOT CONCLUSIONS

  • This data provides good support for the assertion in the Royal Society paper that a soundhole with a long perimeter for its area will be a more efficient overall radiator that one with a lower P:A ratio, and not just for air-cavity resonance
  • Higher P:A ratios show a decrease in bass response and an increase in treble response
  • To provide a worthwhile advantage over a round soundhole, the P:A ratio should be greater than 0.1

FOOTNOTES

[1]It turns out that fitting soundslots onto the Parlour sized soundboard without compromising strength means a restriction in their total area to this smaller sized equivalent round hole size

[2]Note also the 430Hz peak in Figure 1, which I suggested in Part 6 is a standing wave resonance set up along the 380cm length of the cavity.

[3]Slot6B was slightly under size

Leave a Reply

Fill in your details below or click an icon to log in:

WordPress.com Logo

You are commenting using your WordPress.com account. Log Out /  Change )

Google photo

You are commenting using your Google account. Log Out /  Change )

Twitter picture

You are commenting using your Twitter account. Log Out /  Change )

Facebook photo

You are commenting using your Facebook account. Log Out /  Change )

Connecting to %s