1. Passive Acoustic Amplifier Report
Introduction to Engineering Systems II 10112 – Section 4, Group 3
Professor Peter Ivie
Chris Brady, Patrick Corbin, Lauren Czaja, Noah Holmes, Luke Rafferty, Joe Trzaska
April 26, 2016

2. 1
Abstract
In discussing project ideas, the design group sought to create a piece of technology which
could be useful in dorm rooms, which eventually led to the idea of a passive acoustic amplifier
that could be mounted to the base of a smartphone to amplify music for the occupants of the
room without auxiliary cords or Bluetooth. The group decided to optimize an amplifier design in
MATLAB for various parameters and geometries and finally 3-D print the product.
In order to accomplish this task, the group first discerned relevant equations related to the
movement of sound with respect to different conic structures. These equations will be discussed
in greater detail later, but, to summarize, the group modelled volume as a function of distance
from the speaker, and horn geometry (throat and mouth cross-sectional areas) as a function of the
frequency of the audio to be played and the desired length of the horn. Code was written to
tabulate basic matrices using parameters set by the graphical user interface (phone size, speaker
length, music frequency), then write and call further functions to plot the desired conic horn
geometry (linear, hyperbolic, or exponential) again based on the specifications set in the GUI by
the user. Finally, one more layer of functions was used to optimize each speaker for the
maximum ratio of volume to surface area, a quality closely related to the amplification potential
of a passive amplifier.
In the end, the design group modeled and printed an amplifier optimized for an iPhone 6
and calculated its reverberation time, or the time it takes for the sound produced to decay to
essentially zero. The calculated reverberation time was approximately .82 seconds, which is
approximately double the reverberation time of a standard classroom. Thus, the amplifier
produces roughly doubles the acoustic efficiency of a given phone’s speaker.
Needs Assessment and Problem Formulation:

3. 2
The goal of the project was to create a program to model and optimize a passive acoustic
audio amplifier for iPhones, Galaxies, or similar devices and export the model to a printable
format. The modeled amplifier came in three distinct cone-based geometries which could be
chosen by the user: exponential, hyperbolic, and linear. The amplifier was optimized by
modifying the flare of the horn for a maximum time of reverberation (time it takes for a sound to
be reduced 60 dB) given certain parameters (volume, frequency, and range). A 3-D model was
then produced in a GUI in MATLAB, and the files were converted to the stl. format that could be
printed by a 3-D printer to create the final product (see Figure 1 below).
Figure 1. Stereolithographic Export
The cost of the first trial 3D print was 71.00 USD which accounted for the printer’s
material price as well as the price of usage of the machine. This trial run was unsuccessful as the
horn was not hollowed out and thus could not allow sound to travel through. A hollow horn
would have cost less money as it would have required less material and time to print. This

4. 3
adjustment had to have been made on the printer, not the MatLab or stl file. This ability was
unfortunately not made available to the group, and thus it was impossible to notice that the fill
percentage was not what it should have been, leading to the misprint. Though this print was
unsuccessful, the nature and complexity of this design project’s algorithms and plotting to the
GUI make it a still fully-functional computational group.
To design the acoustic amplifier, a number of implicit and explicit constraints needed to
be accounted for. Some of the most apparent were the requirements for the physical build aspect
of the artifact. The designed amplifier needed to be 3-D printed for under one hundred dollars.
Additionally, the amplifier needed to be a reasonable size for the sake of portability and
practicality. To determine the optimal shape of an amplifying cone, it was necessary to find
formulas that were both relevant and reasonably understandable.
Abstraction and Synthesis:
To model and optimize the passive acoustic amplifier in MATLAB, several decisions had
to be made;included among these were what the geometry of the horn would consist of, and what
acoustic dampening
phenomena would
be significant. From equations (3) through (5):
(3) (4) (5)
the optimal surface areas of the throat and and mouth of the amplifier were calculated given a
chosen target frequency and length of the amplifier (see Appendix A for details). Equations (3)
through (5) are standard equations for the frequential optimization of an exponential horn; that
they would optimize for the frequency clarity of the amplifier geometry was assumed.

5. 4
With the surface areas of the throat and the mouth of the amplifier known, the remaining
geometric constraints of the amplifier were the cross-sectional shape and the flare (i.e. the
curvature) of the amplifier. Because the goal of design project was to produce a prototype
amplifier for a smartphone, and the base of a smartphone is most nearly elliptical, the cross-
sectional shape of the design was chosen to be an ellipse that would surround the phone’s base,
in order to minimize the sound lost at the throat of the amplifier. Equations (7) and (8) describe
the geometry of an ellipse.
The flare of the amplifier was modelled in with three different curvatures: hyperbolic,
linear, and exponential curves. The standard equations for for these curvatures are equations (8)
and (9) respectively. Because MatLab surface plotting functions require explicit formulas for the
x-y-z coordinates of the plotted curvatures, parameterizations for the surfaces were developed to
yield the needed explicit equations. These parameterizations for hyperbolic, linear, and
exponential curves are displayed in equations (14) through (16) respectively below (see
Appendix A for details).
(14)
(15)
(16)
With the above parameterizations, surface plots for each of the three curvatures were produced,
and from the x-y-z data from these plots, stereolithographical files (the file format necessary for
3D printing) were produced with a conversion program.
In regard to the acoustic dampening of the amplifier (i.e. how much the amplifier
impedes the travel of sound), factors causing the decay of sound volume were explored. For any
passive amplifier, the two factors that contribute to sound decay are acoustic impedance and

6. 5
sound absorption. Acoustic impedance, which is modelled by equation (2), is air’s resistance to
the travel of sound; it was assumed that the effects of acoustic impedance were to be negligible.
Sound absorption is the material of the amplifier’s tendency to absorb the sound waves, thereby
decaying entering sound. Sound absorption was found to be primarily responsible for the decay
of sound in the amplifier, and therefore this factor was minimized in the optimization of the
amplifier’s shape.
To optimize the amplifier, the group sought to minimize sound absorption, which is
modelled by Sabine’s equation:
(1)
which describes the time of reverberation of an object (i.e. the time it takes for sound to decay
sixty decibels) in terms of the constants ‘c’ and ‘a’ and geometric parameters of volume and
surface area. Reverberation time is inversely related to sound absorption, so, by maximizing
reverberation time, the design group could minimize the sound absorption of the amplifier.
Because volume and surface area are the only variables in the equation (1), reverberation time
maximized was maximized by maximizing the ratio of volume to surface area of the amplifier.
Maximizing the ratio of volume to surface area was done in the following way. For an
exponential curvature described by the equation:
(9)
the parameter ‘b’, which relates to how pronounced the exponential nature of the curve, was
varied over a large range of values. Equation (9) was then revolved around an axis to find the
volume of the exponential amplifier, as described in equation (10). An areametric revolution of
equation (9) was used to find the surface area of the exponential amplifier, as described by
equation (11). Then the different values of ‘b’ were tested in the above equations for volume and

7. 6
surface area to find the value of ‘b’ that minimized the volume/surface area ratio. This ideal
value of ‘b’ was used to create the exponential amplifier in the GUI.
For a hyperbolic curvature which results in a hyperboloid of one sheet described by the
equation:
(8)
the parameter ‘c’, which relates to how quickly the curve approaches a linear asymptote, was
varied of a large range of values. Standard equations (12) and (13) for the surface area and
volume respectively of a elliptical hyperbolic one sheet were used the calculated the surface area
and volume of the hyperbolic amplifier at different values of ‘c’. The value of ‘c’ that minimized
the volume/surface area ratio was deemed ideal and passed to the GUI to create the hyperbolic
amplifier.
Limitations to the project’s geometric choices and optimization techniques include size
limitations, which limited how closely the frequentially optimal surface areas of the throat and
mouth could be matched via equations (3) through (5), and variations in the frequency of music,
which limits the accuracy of the frequential optimization that was calculated for a single median
frequency.
Analysis:
In Figure 2 below, the GUI is pictured modeling an exponential amplifier.

8. 7
Figure 2. GUI Modeling an exponential acoustic amplifier.
The GUI tool, as designed, allows the user to manipulate the key parameters of the
speaker, thereby generating a 3D surface model of the amplifier, which is presented in a figure in
the GUI. This model could then be exported to an stl. file to be printed. The user can alter 4
parameters using sliders: length of speaker, width of phone, depth of phone, and frequency. If the
user wishes to build a speaker for an iPhone 5 or 6 or a Samsung Galaxy, these width and depth
dimensions are programmed in and can be selected from a drop down window. Additionally, the
user is able to select which shape they want the amplifier to be. These options, hyperbolic, linear,
or exponential, are defined in a menu of options. The length is set at a maximum of 50
centimeters, and width and depth both have a maximum value of 10 cm. Lastly, the frequency is
locked between .1 and 1000 Hz. All of these sliders and menu options were synchronized with
the GUI so when the user altered them, they would be altering corresponding values within the
optimization formulas. With the constants of the equations set via the GUI, the figure was plotted
based on information called from the GUI. First, the type of horn selected corresponded to a

9. 8
particular switch which would call the unique code for each kind of cone. From here, the
program built the models by drawing the two-dimensional graphs of the optimized horn at each
z-value ranging from zero to the inputted user length.
Implementation:
Overall, the GUI creates an accessible interface, where the user can see how their data
inputs affect the amplifier that is optimized for them. The biggest challenge was designing an
algorithm that optimizes each speaker type. The linear cone was simple in that the optimization
factor was the length of the cone, and so the “optimal” linear-cone speaker was always the shape
created using the input base dimensions and the full length of the inputted length dimension. The
hyperbolic and exponential cones were more challenging due to the more complex shape of the
created amplifier, and thus the more complex method of minimizing the surface area to volume
ratios. For the exponential cone, equation (9), a planar projection of the equation used to graph
the shape, was determined, and then this curve was rotated along the y axis volumetrically to
determine the volume of the shape, and areametrically to determine the shape’s surface area. As
“a” was the width of the horn’s throat, and “b” was the flare of the horn, b was varied and,
through the optimization code, the value of b which minimized the volume to surface area ratio
was determined. For the hyperbolic cone, equation (8), used in the plotting, was taken and used
to model the volume and surface area by the standard equations for such for elliptical
hyperboloid sheets. With “a” and “b” being the dimensions of the throat and c the constant
related to the growth of change of the hyperboloid, similar analysis was run on “c” for the
hyperbolic cone as was run for “b” for the exponential cone, and thus the optimal c value for the
horn shape, given the user inputs, was able to calculated.

10. 9
The validity of these optimizations was tested by calculating how close the optimized
values of the reverberation time given by the optimization equations were to the actual predicted
reverberation time of .82 seconds (given by research and calculation). Although this was
considered a computational group, in that all the calculations and final products (plotted
graphical models, ability to input data, and ability to export as a stl. file) were able to be done
solely via the GUI, had the prototype printed correctly, it would have been able to be tested as a
build as well. This would have been done by using a sound collection and analysis app to
determine the actual experimental value of the reverberation time, and been able to compare this
to the estimated calculated value. The prediction was that the actual time of reverberation would
have been slightly lower than mathematically predicted, mostly due to the assumption that the
sound interference from the volume of air inside the horn would be negligible. Thus to better the
design, air volume and pressure inside the horn could be modeled and accounted for, and these
interference values could be incorporated into the existing code.
This design project succeeds in that it creates an interactive interface where a user can
model, modify, and actually create something that could be useful to them. The equations written
and linked to compute these parameters and specifications and optimize the amplifiers’ ability to
amplify sound are reliable, as they are adaptations of equations for geometric shapes and the
physical movement and degeneration of sound, and their accuracy in implementation can be
tested via their correspondence to scientifically determined values.
Supporting Documentation
"Acoustic Impedance, Intensity and Power." Acoustic Impedance and Intensity: From Physclips
Waves and Sound. School of Physics Sydney Australia, Web. Mar. 2016.

11. 10
"Dynamic Loudspeaker Principle." HyperPhysics. Georgia State University, Web. Feb. 2016.
"Elliptic Cone." Wolfram MathWorld. Web. Apr. 2016.
"Geometrical Exponential Horn Calculations." Audio Heritage. Computer Systems Design Co.,
Web. Feb. 2016.
"The Inverse Square Law." HyperPhysics. Georgia State University, Web. Feb. 2016.
King, Martin J. "Horn Physics." AccessScience (2008): QuarterWave. Web. Feb. 2016.
Link, Emmanuel. "Acoustic Impedance." University of Wisconsin Physics (2013): 5. 2009. Web.
Mar. 2016.
"One-Sheeted Hyperboloid." Wolfram MathWorld. Web. Apr. 2016.
"Parameterization Tricks." Cornell University Math. Web. Apr. 2016.
"Sabine Equation Example Calculation." Duke University Physics. Web. Apr. 2016.
"Volume, Lateral Area, and Surface Area of an Elliptic Conical Frustum." StackExchange
Mathematics. Web. Apr. 2016.

12. 11
Appendix A