Helmholtz Resonance in Speaker Cabinets

Ryan Morris, Christopher O’Brien, Andrew Swisher, Ryan Yount (ryan.morris@pop.belmont.edu)(christopher.obrien@pop.belmont.edu)(andrew.swisher@pop.belmont.edu) (ryyount@gmail.com)

Belmont University, Nashville, TN, United States.

ABSTRACT:

Loudspeakers are one of the most common formats for audio playback in the world,

ideally providing the user with the flat response throughout the frequency spectrum. Speaker

cabinets and enclosures are often tuned to either accentuate or remove specific frequencies

within the audible frequency spectrum known as the speaker’s natural “resonant frequency.”

This paper will explore the effects of cabinet volume on the resonant frequency of a ported

speaker through the expansion of the depth of such a cabinet. First, hypothetical predictions

about the speaker’s resonance will be given. Second, methods of measurement and testing will

be presented. Third, quantitative data will be presented from the outcome of physical testing; this

will include comparisons between expected and measured data. Finally, improvements and

extensions for future experimentation will be suggested.

0. INTRODUCTION

In the case of most loudspeaker manufacturers, the goal is to create an enclosure that uses

its dimensions and design to accentuate or reduce specific frequencies in order to obtain a “flat”

frequency response. In the case of this experiment, a rear­ported style design was chosen as it is

commonly seen in use for studios and home theatres. These ports are used to alter the

low­frequency output of the speaker at a specified or “tuned” frequency. This frequency is

altered by the port, acting as a Helmholtz­style resonator; it acts as an “acoustic impedance

transformer presenting a high impedance to the rear of the loudspeaker cone and a low

impedance to the air” (1). By designing a speaker cabinet capable of “telescoping,” which

effectively alters the length factor in the volume of the cabinet, it was expected that the resonant

frequency of the speaker cabinet would descend the frequency spectrum per increment of

distance moved. No absorption or damping would be added to the inside of the enclosure, as “the

width of this absorption band depends on the friction of the system” (2). First, the expected

results were calculated using a formula for Helmholtz resonance, and then actual measurements

were taken from the physical enclosure.

1. METHODOLOGY: THE SPEAKER BOX

In this section we will review the concept of Helmholtz resonators and outline the design

and construction of our own Helmholtz­resonating speaker cabinet.

1.1 Design and Construction

The walls of the speaker cabinet were constructed from a ¾ inch sheet of medium­density

fibreboard (MDF), cut to the dimensions pre­determined by a three­dimensional planning sketch.

MDF was chosen for its high degree of sound transmission, which provides a “medium Q” factor

to the resonant frequencies, making them more manageable to view and calculate (3). The

speaker cabinet was designed to “telescope,” meaning that there would be two different­sized

halves of the enclosure that would fit together. The loudspeaker driver was three inches in

diameter, removed from a cheap thrift store surround­sound set of speakers. The two sides of the

cabinet were cut so that the internal dimensions measured 11.5in x 9.5in x 7in (height, width and

depth, respectively). Due to the “telescoping” nature of the cabinet, the depth was expandable

from 7 inches to 13 inches. A hole of three inches in diameter was cut in the internal half of the

cabinet to fit the three­inch driver, and a hole of one and one­half inches was cut in the external

half of the enclosure for the port­hole. Images of the actual enclosure can be found in Figure 1.1.

Figure 1.1 - The two halves of the speaker cabinet separated

1.2 Calculation of Resonance

In the formulation below, the resonant frequency of a cabinet is calculated using the

speed of sound (v) in feet per second, the area of the hole (A) in square inches, the volume of the

box (V) in cubic inches, and the length of the hole (thickness of the front panel of the cabinet)

(L) in inches.

(4) fresonance = V2π√ AV ×L

This formula will give the fundamental frequency of resonance for the speaker cabinet of

specified dimensions. In the case of the expandable speaker cabinet, the only changing variable

is the volume of the container, as the length dimension changes incrementally.

1.2.1 Expected Calculations

From the above formulas, measurements were taken of all dimensions of the resonator at

specific points of extension (0, 2, 4, and 6 inch extension of length), which effectively altered the

volume of the container. The following resonant frequencies were found for the following

extension points:

69.9Hzf0 = 1 50.9 Hzf2 = 1 35.5 Hzf4 = 1 24.7 Hzf6 = 1

1.3 Physical Measurements

Actual measurements were taken utilizing the following: Room EQ Wizard software for

sine sweeps and measurement, Focusrite Scarlett USB Interface for I/O, Audix TR40

omnidirectional measurement microphone to record the sweeps, and an isolation booth. Room

EQ Wizard was set to record a sine sweep from 0Hz to 1000Hz, as the goal was to test only the

low­end response of the speaker, where exists the lowest modal density. Measurements were

taken at telescoping extensions of zero, two, four, and six inches of extension. For the

front­of­speaker testing, the microphone was placed exactly eight inches in front of the speaker

cone. The back­of­speaker (porthole) measurements were taken with the capsule just inside the

hole. A total of eight measurements (four distances in front, four distances in back) were taken

using Room EQ Wizard.

2. RESULTS

In this section we will review the outcome of testing the resonance of the speaker cabinet,

sharing how the theoretical results compared to the quantitative outcomes.

2.1 Measurement Results

There was very little discernable resonance observed from the measurements taken eight

inches away from the front side of the speaker. As illustrated in Figure 2.1, there were slight

variations in the low­midrange of the frequency spectrum (50­500 Hz), but there was no

profound resonance found from the front of the speaker cabinet.

Figure 2.1- Level of signal outputted from cabinet in dB (y-axis) by HZ (x-axis) with microphone placed in front of the

cabinet.

Seeing as the two measurements with the cabinet elongated zero inches and six inches were

extremely variable from each other, we found that these measurement did not yield themselves to

interpretation. Furthermore, neither measurement displayed a clear resonant peak. We have

therefore decided to consider the measurement received by placing a microphone on the back of

the cabinet to be more representative of the speaker’s resonance peaks.

The measurements received from the aforementioned procedure were most conclusive

when the microphone was placed on the back of the cabinet as opposed to the front. Figure 2.2

provides a graph that displays the sound intensity level in dB of each frequency in Hz from 15

Hz to 1,000 Hz when the microphone is placed behind the speaker cabinet.

Figure 2.2- Level of signal outputted from cabinet in dB (y-axis) by HZ (x-axis) with microphone placed behind cabinet.

By comparing the line denoting when the speaker cabinet was set to six inches of extension

(purple) with the line denoting zero extension (green), we are able to ascertain that the

loudspeaker enclosure has resonant peaks at 137 Hz and 140 Hz respectively. Surprisingly, the

bandwidth of the resonant peak became much narrower as the length of the box was elongated.

The Q value of the bandwidth at the 137 Hz peak was 2.07, while the Q at 140 Hz peak was

3.34. This was a surprising result, as the expected result was for the frequency of the resonant

peak to drastically change; the bandwidth of the resonant peaks presented the most change in

measurement.

2.2 Theoretical Results

Subsequent to experimentation, the results did not match our calculated expectations, nor

did they follow what would be expected in relation to the knowledge gained from previous

scientific experimentation on the subject. We expected that an increase in the volume of the

speaker cabinet would result in a lower fundamental “room mode” due to the now longer

dimension of the cabinet, but measurement yielded alternate results. The fundamental frequency

of the cabinet decreased only slightly as the cabinet was extended; the Q value is the only

significant change observed through testing.

When the cabinet was closed a peak at roughly 140Hz can be recognized by examining

Figure 2.2. When the cabinet was extended 6” that same peak can was measured to be at 137Hz.

According to prior theoretical calculation of Helmholtz resonance, the speaker box should have

held resonance at 169.9Hz when fully “collapsed,” and resonance at 124.7Hz when the cabinet

was fully extended. A calculated 17.6 percent error was calculated for our measurement of

resonance at the collapsed state, and a 9.9 percent error was calculated for measurement at full

(six inch) extension.

In addition to this error, Room EQ Wizard’s frequency sweep showed a “phantom” lower

harmonic below the resonant frequency centered around 50 Hz. This was either a result of a

room mode within the isolation booth we were using for measurement, or could have been an

indication of poor seal in our speaker enclosure. This was the main cause for surprise when

reading measurements when comparing to calculations.

2.3 Improvements

Some of the ways in which we went about the experiment could have been done more

accurately and efficiently. We could have gone a few steps further had we had more time and

materials. In terms of our actual experiment we could have used a power saw to get straighter

cuts which would in turn make a tighter seal on the box. In addition to the caulking of the box

we could have weather­striped the gaps between the two portions of the box instead of using tape

to create a tighter seal.

In terms of alternatives to our actual experiment, we could have used a larger speaker

driver and larger cabinet and compared that to the smaller cabinet and driver to find differences

in even larger volumes versus the smaller. Different port sizes, depths, and positions would also

be an informative avenue to explore. Due to differences in modes excited at different portions of

the boxes we could place the port in alternate locations. It would also have been interesting to

test a front facing port versus a rear facing port, both being common to most studio monitors

today.

An alternative way of going about our actual experiment would be to calculate the

volumes of the box and resonance of the port first, and build the box based around certain

frequencies we wanted to excite, rather than building a box first and finding its inherent

frequencies. That way we could test our own theoretical values versus the experimental values

instead of finding experimental and then comparing to theoretical.

3. CONCLUSION

Despite the possible faults in experimentation that could account for error in our

calculated and measured values of resonance, one thing holds certain: it is important to

understand the resonant­frequency created by a port in a speaker, acting as a Helmholtz resonator

(or absorber). The pressure differential between the inside and the outside of the box (created by

the movement of the speaker cone) causes massive pressure of air to be pushed through the port,

giving the its enclosure a "resonance” (5). This resonance should be carefully tuned as needed

from calculations of the speaker's pre­existing frequency response in order to help achieve the

desired sonic characteristics of the speaker. From our testing, it was found that ported speaker

cabinets require a larger amount of overall surface area; the enclosure in question should have

been at least double its constructed size, installed with a larger loudspeaker. This could be a

factor of blame for the failed attempts to match the theoretical calculations with the physical

measurements. From a perspective of auditory perception, the expanded speaker cabinet offered

a much slower low­end transient response than when the cabinet was unexpanded. From this

slightly less quantitative assessment, one can conclude that cabinet volume does in fact have a

noticeable, prominent effect on the sonic characteristics of a loudspeaker.

4. REFERENCES

[1] A. Salvatti, A. Devantier, and D. Button, “Maximizing Performance from

Loudspeaker Ports.” J Audio Eng. Soc. Vol 50, No.1/2. (January 2002). 210

[2] F. Everest and K. Pohlmann, “Master Handbook of Acoustics.” McGraw­Hill.

Edition 5. Chapter 12, Page 210. (June 2009).

[3] A. Schweikert, “A Whitepaper: The Audibility Of Cabinet Panel Resonances and

Pat. Pend. Method of Reduction of Audible Coloration.” DaGoGo. (July 2009).

[4] R. Nave, “Cavity Resonance.” HyperPhysics­Sound. Web,

http://hyperphysics.phy­astr.gsu.edu/hbase/waves/cavity.html.

[5] Lucas, A, "Sealed vs. Ported Enclosures." Bass Gear Magazine. (June 2011).

Web, http://www.eminence.com/2011/06/sealed­vs­ported­enclosures/