The Effect
of Finishes on the Vibration Properties of Spruce Guitar Soundboard Wood 
Abstract This paper presents a study of the effect of a sealer and
four finishes on the vibration properties of spruce for guitar
soundboards. Two of the finishes,
dewaxed shellac and nitrocellulose instrument lacquer, are evaporative
finishes, traditionally used for guitars. The third and fourth are reactive
shellacbased finishes. The study measured the fundamental vibrational
frequency, f_{0}, and damping
quality factor, Q, of Sitka spruce test bars machined in alonggrain
and crossgrain orientations, and coated with sealer and the finishes. The sealer alone produced significant changes
in both f_{0} and Q for the two grain orientations: the alonggrain f_{0}
decreased; the crossgrain f_{0} increased, and Q for both grain
orientations decreased. The finish top coats affected f_{0 }for only the alonggrain bars, which decreased
with top coat application. Compared to Q for the sealer coating, all of the top
coats cured for seven weeks increased Q for the alonggrain bars, but did not
affect Q for the crossgrain bars. Statistical
analyses showed that all of the top coat finishes cured for seven weeks were equivalent with respect to their effect on the
vibrational properties of the spruce bars.
I. Introduction
Finishes serve to protect and enhance the beauty of musical instruments. Also, it is wellknown that finishes can modify the acoustics of an instrument. This study reports the results of tests of the effect of a sealer and four top coat finishes on the vibration properties of spruce guitar soundboard wood. Two of the top coat finishes, dewaxed shellac and nitrocellulose instrument lacquer, are evaporative finishes. The third and fourth are reactive shellacbased finishes, that after evaporation of the solvent, continue to cure by chemically crosslinking to form top coats that are more durable than evaporative finishes.
Many of the physical and chemical characteristics of evaporative and reactive finishes are different [1]. This presents an important question for the luthier: do they also affect the vibration properties of the wood differently? The purpose of this study is to compare the vibration properties of spruce soundboard wood finished with sealer, followed by top coats of two evaporative finishes (dewaxed shellac and nitrocellulose instrument lacquer), and two reactive modified shellacbased finishes.
Two methods were used to examine the vibration properties of the bare wood, and the wood with sealer and finishes:
1) Measurement of the fundamental vibrational frequency, f_{0} (resonant frequency), of wood sample strips (thin bars) with free ends; and
2) Measurement of the damping quality factor, Q, at the fundamental resonant frequency by the logarithmic decrement method.
A significant part of this investigation was the measurement and control of sampletosample variation of finish thickness. Previous studies of the impact of finishes (primarily varnish) on f_{0} and Q (or the logarithmic decrement) had noted the consequences of variation in finish thickness for interpreting measurement results, but had not reported finish thickness nor variation in thickness for the samples of the measurements.
Schelleng [2] discussed how varnish thickness affects the vibrational properties of wood, noting deviations occur depending on manner of application and thickness did not differ radically from sample to sample. Schleske [3], in his study of violin varnish, concluded that most of the uncertainty [in the properties measured] is caused by differences in the consistency of application rather than by measurement uncertainties
A focus of this study was to limit measurement errors due to the variability of finish thickness, and to quantify the precision of measurement of finish thickness, and the vibration properties f_{0} and Q so that statistical analyses could be used to evaluate the results. This study included the control and measurement of sampletosample variation of areal density and thickness, a component not included in previous studies.
Details of the measurements and results are presented in the following sections of this paper.
II. Experimental
A. Materials
Photo 1 ─ Location of test bars cut from one of the
book matched panels. The panel at
the top of the photo is 3B. The test
bars were cut from book matched panel 3A at the bottom. Labels give the test bar identification
numbers. L1 through L5 refer to
alonggrain bars 1 through 5. C1
through C5 refer to crossgrain bars 1 through 5. Note:
a faint pencil outline of a guitar top, drawn by the supplier, can
be seen on the panels. Photo 1. Location of test bars cut from one of the book matched panels. The panel at the top of the photo is 3B. The test bars are cut from book matched panel 3A. Labels give the test bar identification numbers. L1 through L5 refer to alonggrain bars 1 through 5. C1 through C5 refer to acrossgrain bars 1 through 5. Note: a faint pencil outline of a guitar top, drawn by the supplier, can be seen on the panels. 
After the panel edges were trimmed to ensure alignment of the wood grain with the long edge, ten bars were machined from the 3A spruce panel. Five of the bars were machined with the grain running parallel to the long edge (the alonggrain bars). Another five of the bars were machined with the grain perpendicular to the long edge (the crossgrain bars). The location of each bar cut from the panel was documented photographically (see Photo 1). The two sets of five bars were stacked and trimmed together to ensure the lengths and widths within each set were the same.
Following careful sizing of the length and width of the bars, they were sanded with a Luthiers Friend[3] sander with a 120 grit drum, to a thickness of 3 mm (which is within the thickness range of 2.7 to 3.2 mm for a large body steel string guitar) [4]. This was followed by sanding, using a block, with 220, then 320 grit paper.
After sanding, bar dimensions and weights were measured, and the densities were calculated. Lengths and widths were measured to the nearest 0.01 cm using a precise ruler, and digital calipers. Thicknesses were measured with a micrometer graduated to 0.001 inch. The bars were weighed with a digital scale precise to 0.1 gram. Length and thickness were sized to produce an approximate fundamental frequency that was above the lower range (approximately 60 Hz) of the sine wave sound generating equipment. The target for the crossgrain bars was ~100 Hz. The target was ~170 Hz for the alonggrain bars.
Alonggrain bars Crossgrain
bars
Table 1
─ Averages and standard deviations for dimensions, weights and
densities of the two sets of the unfinished test bars. 
Table 1 gives the average and standard deviation of the dimensions, weights and densities of the bars prior to application of the finish. As can be seen from Table 1, the dimensions and weights of the bars within both the alonggrain and crossgrain sets were uniform, as were the densities of all the bars.
Uniformity of the annual growth ring (grain) pattern of the crossgrain bars is shown in Photo 2, an edgeon view of the stack of the five crossgrain bars. Alignment of identical features of the grain of each bar demonstrates that the bars were well matched. From Photo 2 it is also seen that panel 3A was well quarter sawn: the annual growth rings, from one end of each bar to the other end, were all perpendicular to the surface of the bar.
Photo 2 ─ Edgeon view of the
crossgrain bars stacked to show alignment of the annual growth ring
pattern. The top bar is labeled
3AC1, and the bottom 3AC5. The
arrows point to distinctive annual growth ring pattern features that are the
same for all the bars. 
1) Sealer:
SealLac,[4]
comprised of dewaxed super blonde shellac and natural resin additives
(comparable to a 2 lb cut[5]
of shellac);
2) Dewaxed shellac: dewaxed super blonde shellac flakes^{4} dissolved in anhydrous 200 proof denatured alcohol to form a 2 lb cut;
3) Modified
dewaxed garnet shellac: RoyalLac Garnet^{4}, formulated
from dewaxed garnet shellac dissolved in anhydrous 200 proof denatured alcohol
and modified with synthetic and natural resins to form a reactive finish;
4) Guitar
lacquer aerosol: ColorTone
Clear Gloss No. 3881 Nitrocellulose Lacquer[6],
in an aerosol spray can;
5) Modified
dewaxed shellac aerosol: RoyalLac
Clear Coat^{4}, formulated from dewaxed super blond shellac dissolved
in anhydrous 200 proof denatured alcohol and modified with
synthetic and natural resins to form a reactive finish, and provided in an
aerosol spray can.
For this finish study, the test bars were paired into five sets of two bars, each set consisting of one alonggrain and one crossgrain bar. The sets received the top coat finish treatments shown in Table 2.
Test bars 3AL3 and 3AC3 were left unfinished (bare wood) to serve as controls during the course of vibration measurements. Measurements made on these control test bars aided in determining the repeatability and precision of the results.
B. Procedure
Application of Finishes Because of the small size of the test bars, spraying was found to be the most effective way to evenly apply the sealer coats, and all of the finish top coats. A Preval portable sprayer[7] was used to apply the sealer, dewaxed shellac, and the modified dewaxed garnet shellac. The guitar lacquer aerosol and modified dewaxed shellac aerosol were applied using the aerosol spray cans.
Determination of Finish Thickness Because finish thickness affects the resonant frequency and damping of the test bars, comparison of the effect of finishes on these vibration properties calls for a uniform thickness of finish film on each bar, and bartobar uniformity of thickness. Film thickness and uniformity are controlled by application technique, monitored by determination of film thickness.
Direct measurement of the thickness of films on the order of a hundred microns, μ (1μ = 10^{6} m) requires special measurement tools. For wood substrates, a dry film ultrasonic thickness gage, not available for this study, can be used. However, disadvantages of the ultrasonic gage are the inability to distinguish film layers of similar density, and the use of a gel to couple the probe to the surface of the film. The gel can contaminate the surface of the film, interfering with adhesion of subsequent coats of sealer or finish.
An alternative to direct film thickness measurement, average areal density, was used in this study to evaluate and monitor film thickness. Average areal density is determined by weighing a bar to measure film mass, then dividing the mass by the area of the surface of the bar. Average film thickness is calculated by dividing the average areal density by a reported value of the dry film volumetric density. To avoid confusion with the term average, meaning arithmetic average, in the following sections of this article the terms areal density and thickness are used to denote average areal density and average thickness as defined above.
Measurement of the areal density to evaluate finish thickness requires only precise rulers, calipers, micrometers, and a weighing scale. However, it is an average value of thickness that is determined; uniformity of film thickness on a particular bar is not evaluated. Undetected variations in the uniformity of finish thickness on a bar will contribute to imprecision in the measured values of the vibrational properties of the coated bars. Accordingly, careful attention was paid to finish application technique (use of spray application, proper thinning, building of thickness with light coats, level sanding between coats, and final level sanding) to reduce film thickness variations. Also, significant effort was spent to evaluate the bartobar precision of the areal density and thickness of the sealer and top coat finishes.
There are several sources that report useful information to serve as a guide for the appropriate amount of finish on instruments, and hence the amount for the test bars used in this study:
1) Michelman [5], in his study to recreate violin varnishes of the old Italian Masters, reported using a thinness of combined subvarnish (i.e. sealer) and varnish coats ranging from 0.0040 inch (101 μ) to 0.0052 inch (132 μ), with a subvarnish thinness of 0.0015 inch (38 μ).
2) Data reported by Schelleng [6] indicates that he used coatings of 0.013 g/cm^{2}, about 0.005 inch (127 μ), on his test bars for vibration property studies.
3) According to Gore and Gilet [7], for their guitar finishes: We frequently use shellac (French Polish) as the base for our nitrocellulose finishes on soundboards and keep the total finish thickness on tops very low, never more than 100 microns.
4) In a YouTube video[8] of a tour of the Taylor Guitars factory in El Cajon, CA, the Taylor Guitars guide and narrator stated that it is important to keep the finish absolutely as thin as you possibly can. He indicated that the thickness of the Taylor uvcured polyester finish is 0.006 inch (152 μ) for most of their guitars. For the Taylor 800 series guitars he indicated the finish thickness was 3 mils (89 μ).
Using this information, it was decided to keep the combined sealer and top coat thickness to less than 100 μ (0.004 inch). This thickness is within the norm of lutherie practice, as discussed above.
The areal density ρ_{a} (mg/cm^{2}) of the film was used to monitor the amount of finish on the surface. Thickness, T, can be calculated from the areal density and reported values of the volume density, ρ_{v} (g/cm^{3}), or specific gravity of the coating materials:
T = ρ_{a} / ρ_{v} (1)
The mass of the coatings (m_{c}) was determined, by weighing the test bars with a digital scale, as the difference between the mass before (m_{1}) and after (m_{2}) a step of the finish process:
m_{c}_{ }= m_{2} m_{1 }(2)
Coating areal density ρ_{a} is determined from the calculated mass of the coating m_{c} divided by the surface area A of the coating:
ρ_{a}_{ }= m_{c }/ A (3)
The surface areas of the coatings were calculated from the average length, width and thickness of the bars, given in Table 1. The area of the edges of the bars (less than 10% of the total area) were included where noted in the following sections.
Bartobar uniformity of coatings, areal densities and thicknesses of the test bars was achieved by calculating, prior to final levelsanding, the mass m_{c} for the desired coating areal density. During final levelsanding, the bars were weighed frequently to monitor the approach to this mass. Final coating areal density and thickness were calculated as noted in equations (3) and (1).
Measurement Precision and Confidence Intervals Comparison of coating amounts to judge the similarity of values, requires an estimate of the precision of the areal densities and thicknesses. For this study, the resolution, or least count of the digital scale, 0.1 g, limits the precision for determination of areal density and thickness. The reading error, 0.05 g, equal to half of the least count, can be taken as an estimate of the standard deviation of the masses of the sample bars.
The areal densities and thicknesses were derived from the difference of two masses, each with a standard deviation of 0.05 g. From the formula for compounding subtraction errors, the estimate of the standard deviation of the mass of the coatings, s_{m}, is √2 (0.05 g), or 0.07 g. This estimate of s_{m} is used as the basis for calculation of the margin of error which reflects the amount of random measurement error.
For the coating areal density, the margin of error, ξ_{A} is:
ξ_{A} = k (s_{m}/√n)/A (4)
where k is the twotailed Student tstatistic for the chosen level of significance for sample size n. For the thickness, the margin of error is:
ξ_{T} = ξ_{A} /ρ_{v} (5)
A confidence interval[9] about a population mean for the areal density ρ_{p}, is constructed from the margin of error and the mean of the areal density measurements for the coatings (the sample mean), ρ_{s}:
ρ_{s}_{ } ξ_{A}_{ }≤ ρ_{p}_{ }≤ ρ_{s}_{ }+ ξ_{A} (6)
As an example of calculating the margin of error and confidence interval for the areal density, consider the data in Table 3 for the alonggrain test bars with sealer applied to both sides. The surface area (both sides and edges) is 290 cm^{2}. (The edge surface area was included because overspray also coated the edges and contributed to the weight of the sealer.) The areal densities were calculated using equation (3). The estimate of the standard deviation for the areal density of the coatings, s_{m}/A is 0.3 mg/cm^{2}. The sample size n (number of bars) is 4 and the sample mean for the areal densities from Table 3 is 5.3 mg/cm^{2}. At a level of significance of 0.05 (95% confidence level), the twotailed Student tstatistic, k, is 3.18. (Note that use of the Student tdistribution for estimating confidence intervals was designed to treat small sample sizes, typically less than 15.)
The margin of error, according to equation (4), is 0.5 mg/cm^{2}, yielding a 95% confidence interval for the population mean of the areal densities, ρ_{p}, of 4.8 ≤ ρ_{p} ≤ 5.8. As all of the areal densities measured for the alonggrain samples fall within this confidence interval, the values can be considered equivalent, with differences due to random measurement error.
Because the same finish procedure was used for all of the bars, the bartobar margin of error can also be used as an estimate of the uniformity of finish coating areal density and thickness for a single bar.
Table 3
─ Sealer mass, areal density and thickness after application to both
sides of the test bars.
Confidence intervals for areal densities (mg/cm^{2})alonggrain
set: 4.8 ≤ ρ_{p} ≤ 5.8;
crossgrain set: 4.6 ≤ ρ_{p}
≤ 5.8. Confidence intervals
for thicknesses (in microns)alonggrain set: 44 ≤ T_{p} ≤ 54; crossgrain set: 42 ≤ T_{p}
≤ 54. 
Table 3 presents the masses of the sealer coatings applied to both sides, and the calculated areal densities and thicknesses. Thicknesses of the sealer coatings were calculated from the areal densities and a value of 1.1 for the specific gravity of shellac, taken from a published range of values^{[11]} (1.02 to 1.12). The area, A, for calculation of the areal density of the bars included the surface of both sides (equal to 268 cm^{2} for the alonggrain bars; 175 cm^{2} for the crossgrain bars), and the edges (equal to 22 cm^{2} for the alonggrain bars; 16 cm^{2} for the crossgrain bars). It can be seen from the confidence intervals in the caption for Table 3 that the areal densities and thicknesses of the coatings of the bars within each set were equivalent.
Table 4 ─ Sealer mass, areal
density and thickness applied to the top side of the test bars. Confidence
intervals for areal densities (mg/cm^{2})alonggrain set: 4.2
≤ ρ_{p} ≤ 5.8;
crossgrain set: 3.7 ≤ ρ_{p}
≤ 5.9. Confidence intervals
for thicknesses (in microns)alonggrain set: 38 ≤ T_{p} ≤ 52; crossgrain set: 34 ≤ T_{p}
≤ 54. 
Removal of the sealer was accomplished by sanding, using a progression of 180 to 220 grit paper to MicroMesh 1500. Progress of sealer removal was judged both visually and by weighing the bars frequently during sanding. The test bar side with the sealer coating remaining was designated as the top side, to be coated with the top coat finish, as listed in Table 2.
Table 4 presents the mass of the sealer coating applied to the top side of each bar, and the calculated areal density and thickness. From comparison of the sealer coating mass values for the bars in Table 3 to those in Table 4, it can be seen that half of the total mass of sealer was removed from the bars coated on both sides. This indicates that, within the margin of error for determining mass, half of the sealer was removed. However, any unmeasurable amount of sealer remaining, or undetected small amount of wood removed during the sanding process, would contribute to measurement imprecision of the vibrational properties.
The area, A, for calculation of the areal density of the bars coated on only the top side included the surface of the top (equal to 134 cm^{2} for the alonggrain bars; 88 cm^{2} for the crossgrain bars), and the edges (22 cm^{2} and 16 cm^{2}, respectively) as previously discussed. Because of the fixed weighing error of 0.07g, and a coated surface area about half of that of the test bars coated on both sides, the calculated margin of error values for the areal densities and thicknesses are larger.
Again, within the margins of error, the areal densities and thicknesses of the sealer coatings of the test bars within a set were equivalent. Additionally, comparison of the means and margins of error for the areal densities and coating thicknesses given in Table 3, to those in Table 4, indicates that the coating thicknesses on each side of the bars, when originally coated, were equivalent.
Top Coat Application As with the sealer, finish top coats were applied by spraying. One pound cuts of dewaxed super blonde shellac and modified dewaxed garnet shellac were applied with the Preval sprayer. The guitar lacquer aerosol and modified dewaxed garnet shellac aerosol were applied according to the instructions on the aerosol cans.
Table 5 ─ Mass, areal density
and thickness of top coat finishes for test bars. Confidence intervals for areal densities
(mg/cm^{2})alonggrain set: 3.7 ≤ ρ_{p}
≤ 5.3; crossgrain set: 3.5 ≤ ρ_{p}
≤ 5.7. Confidence intervals
for thicknesses (in microns)alonggrain set: 32 ≤ T_{p} ≤ 46; crossgrain set: 29 ≤ T_{p}
≤ 49. 
Table 5 presents the masses of the finish coatings applied to the top sides of the test bars, and the calculated areal densities and thicknesses. The area, A, for calculation of the areal density included the area of the top and the edges, as previously given. Thicknesses of the dewaxed shellac, modified dewaxed garnet shellac, and modified dewaxed shellac aerosol films were calculated from the previously given value of 1.1 for the specific gravity of shellac. A value of 1.35 for the dry film density of the nitrocellulose lacquer, determined from solids data for a lacquer formulation by Chemcraft,^{[12]} was used to calculate the thickness of the nitrocellulose lacquer top coat.
Table 6
─ Summary of means and margins of error for the areal densities and
thicknesses of the sealer and top coats, and the sum of these, for the
sample bars. 
Table 6 provides a summary of the areal densities and thicknesses of the finish coatings on the test bars. These are presented as the means of the values from Tables 4 and 5, along with the margins of error (ξ_{A} or ξ_{T}) calculated at the 95% confidence level. The total areal densities and thicknesses were calculated as the sum of the mean values for the sealer in Table 4 and mean values of the top coats in Table 5. The margins of error for the totals of the sealer and top coats were calculated from the compounding of error formula for the sum of two quantities.
As can be seen from Table 6 the
thickness of the sealer is equivalent for the alonggrain and crossgrain bar
sets, as is the thickness of the top coats.
The total thickness of the of the finish (sealer plus top
coat) for the alonggrain bars was determined to be 84 10 μ and that for
the crossgrain bars is 83 14 μ.
Thus the goal of bartobar uniform coatings less than 100 μ was
achieved.
C. Vibration Measurements and Calculations
Photo 3 ─
Setup for measuring the vibration properties of the test bars. The components (A G) are identified as
follows: A) PCbased sine wave tone generator (NCH Software, Inc., http://www.nchsoftware.com/) and
Audacity sound editing software (http://audacityteam.org/); B)
40watt
PA amplifier (RadioShack); C) Four inch audio speaker (Altec
Lansing); D) Dynamic mic
(ElectroVoice N/D367s); E) Digital audio recorder (Marantz PDM 660); F) Test bar support; and G) Test bar. 
Fundamental Resonant Frequency The Chladni method [1113] was used as the primary method to measure the frequencies of fundamental mode vibrations for the bars. Frequency spectrum analyses were used to check and verify the fundamental frequencies.
Figure 1 ─
FFT frequency domain plot of the damping signal displayed in Figure 2 for
bare wood alonggrain bar 3AL3. FFT
specifications: Hann window; 
The resonant frequency measured by spectral analysis used the Audacity software Fast Fourier Transform (FFT) of the audio signal emanating from the vibrating bar during decay of the bar resonance (see the following damping section). Figure 1 shows the FFT frequency domain plot of the damping signal displayed in Figure 2 for bare wood alonggrain bar 3AL3. A Hann window, 44.1 kHz sample rate, and a sample size of 65536 were used to obtain the data in the Figure 1 plot. For determinations of f_{0} using the FFT, sample sizes of 8192 or greater were used. Even spectrum peaks for lower resolution window sizes, used for the bars with lower Q values, agreed within one Hz with the Chladni measurements. For sample 3AL3, the resonant frequency determined by both the Chladni method and spectral analysis was 172 Hz.
The alonggrain and crossgrain Youngs moduli, E_{L }and E_{C} of the bare wood bars, were calculated from the measurement of f_{0} for the alonggrain and crossgrain bar samples [14]:
Figure 2 ─ Computer screen capture
photo of a decay measurement for alonggrain bar
3AL3. This bar was unfinished, and
used as a control throughout the vibration measurements. The resonant frequency f_{0} was
172 Hz. 
Where E is Youngs modulus (GPa), f_{0} the resonant frequency (Hz), L the length (m), T, the thickness (m), and ρ the density (kg/m^{3}). Use of this equation assumes the bars are isotropic in the direction of bar length. While this assumption is valid for the spruce bars without finish, it is approximate for bars with finish on the surface, as the latter are composite structures. Therefore E_{L }and E_{C} were calculated for only the bare wood bars to confirm that the moduli for the spruce chosen for the tests were within accepted norms for guitar top plate wood.
Rearranging equation (7) shows the relationship [15] between f_{0} and the unfinished bar parameters:
f_{0} = 1.028 T/L^{2} (E/ρ)^{1/2 }(8)
For the bars with a finish coating, the resonant frequency f_{0}, though only approximately represented by equation (8), is still an important measure of vibration properties, just as tap tones are for violin plates [16]. Within either an alonggrain or crossgrain unfinished set of bars, the length, thickness and density were the same (see Table 1), and the finish thicknesses for a finish step were uniform (see Table 6). Thus a change in f_{0} reflects a change in the bar stiffness and density resulting from the finish.
Damping Damping, often expressed as the quality factor, Q, was measured by the logarithmic decrement method at the resonant frequency of the bars. A detailed discussion of the measurement of Q is presented by Gore and Gilet [10].
To measure Q, a microphone was positioned above the bar (see Photo 3) to record the audio output at the resonant frequency. The bar, supported at its nodes, was set into vibration at the resonant frequency with a sinusoidal audio signal, the digital recorder was turned on, then the driving audio signal was turned off. After the driving audio signal was turned off, the microphone and digital recorder captured (recording format PCM44.1 kHz) the decay of the amplitude of the resonant frequency audio signal emanating from the bar, as depicted in Figure 2. This audio signal was stored as a WAV (.wav) file to be analyzed by the Audacity sound editing program.
Figure 3 ─ Plot of ln A(t) vs. t for the decay curve shown in Figure 2. 

The amplitude A(t) of the peaks of the decaying signal were measured as a function of the decay time, t. By expanding the amplitude and time scales, amplitude values could typically be read to 0.002 unit and time to one millisecond.
Five to seven peak A(t) values, were taken from a portion of the decay curve. The decay rate equation,
A(t) = A_{0} e^{t/Ԏ} (9)
where Ԏ is the decay time constant, is linearized by taking the logarithm of both sides of the exponential equation, to yield:
ln A(t) = ln A_{0}  t/Ԏ (10)
Figure 3 shows a plot of ln A(t) vs. t for the decay curve shown in Figure 2. A linear least squares fit of the decay curve (ln(A), t) shown in Figure 2 was performed using the LINEST function of Microsoft Excel 2013. Ԏ was determined from the slope of the line (1/Ԏ = 3.604 sec^{1}) to be 0.277 sec. The standard error in Ԏ, 0.015 sec, was calculated using the statistical functions for LINEST.
Q is calculated from f_{0 }and Ԏ:
Q = π f_{0} Ԏ (11)
Damping is also often reported as the logarithmic decrement, δ:
δ = 1/(f_{0} Ԏ) = π/Q (12)
For the example illustrated by Figures 2 and 3, Q is calculated from equation (11) to be 150 2, and δ, from equation (12), to be 0.0209 0.0006. In general the standard error of Q for data from a single measurement was less than 3 for all of the determinations of Q from the least squares fits.
III. Results and Discussion
A. Youngs Moduli of Test Bars
Alonggrain Bars
Crossgrain Bars
Table 7 ─ Youngs Moduli of unfinished (bare
wood) test bars. 
B. Impact of Finishes on f_{0}
Table 8 presents the results of the measurement of f_{0} of the test bars after each finish treatment. The impact of the finish steps on f_{0} for the alonggrain and crossgrain bars is examined both graphically and statistically.
Line graphs (Figures 4 and 5) of the changes in resonant frequency, Δf_{0}, resulting from finish treatments visually depict the trend for each bar and treatment. The value for Δf_{0} of a bar is calculated as f_{0} after a treatment, less f_{0} of the same bar without finish, i.e. bare wood. The value of Δf_{0}, rather than f_{0}, is used to depict the trends because of the bartobar variation in f_{0 }for the unfinished alonggrain bars, as shown in Table 8.
Figure 4 summarizes the trends in Δf_{0} with finish step for the alonggrain bars and Figure 5 the trends for the crossgrain bars. Each line represents a different bar, identified by its designation number and corresponding color given in the key in Table 9. The finish treatments on the horizontal axis of the figures are designated by a finish step number and brief description of the treatment in the key in Table 9. For the detailed description of the finish treatments see the sections titled Sealer Application and Top Coat Application. Also note that bars 3AL3 and 3AC3 remained unfinished throughout the finish treatment steps to serve as an indicator of the repeatability and precision of the resonant frequency and damping measurements.
Table 8
─ Fundamental resonant frequency, f_{0}, of test bars after
each finish treatment.
The mean f_{0} is given for each set of test bars (except
for the controls) and finish treatment for the purpose of performing
significance tests. *Indicates the unfinished control test bars and the f_{0}
values. 
1) For both the alonggrain and crossgrain unfinished control bars (3AL3 and 3AC3), the spread in resonant frequencies was only one Hz for measurements taken over a period of nine weeks. This confirms that the precision and repeatability of the frequency measurement is within 1 Hz. It also indicates stability of the equipment for frequency measurement and the constancy of shop conditions such as temperature and relative humidity that could affect the measurements.
2) Compared to the resonant frequencies of the unfinished bars (finish Step 1), the frequencies of the alonggrain bars with the finish cured for seven weeks (finish Step 5) are lower (Δf_{0} = 7 to 11 Hz; see Table 8 and Figure 4). In contrast, the frequencies of the cured finish at seven weeks for the crossgrain bars (Table 8 and Figure 5) are higher by 5 to 7 Hz (Δf_{0 }= +5 to +7). Effects similar to these have been observed and reported by previous investigators [1821].[13] Hains [19] and Schelleng [22] attribute this effect to the value of the Youngs modulus for the finish (in this case varnish, equal to about 2 GPa [19]) being between E_{C }(1 GPa) and E_{L} (17 GPa) for the spruce (see Table 7). Thus the stiffness of the crossgrain bars is increased and that for the alonggrain bars is decreased.
Figure 5 ─
Line chart of the trend in changes in resonant frequency, Δf_{0},
with finish treatment for the crossgrain bars. See Table 9 for key. 
Figure 4 ─
Line chart of the trend in changes in resonant frequency, Δf_{0},
with finish treatment for the alonggrain bars. See Table 9 for key. 
3)
Finish treatments
(horizontal axis): 1) Unfinished sample
bars; 2) Sealer both sides; 3) Sealer top only; 4) Finish cured 4
days; and 5) Finish cured 7 weeks. Sample bar
designations and top coat finishes: 3AL1 and 3AC1  Dewaxed Shellac 3AL2 and 3AC2  Modified Dewaxed Garnet
Shellac 3AL3 and 3AC3  Bare Wood Control 3AL4 and 3AC4  Guitar Lacquer Aerosol 3AL5 and 3AC5  Modified Dewaxed Shellac Aerosol Table 9
─ Key to line graphs in Figures 4, 5, 6, 7. Each line represents a different sample
bar, identified by its designated number, type of top coat finish, and corresponding
line and data point color.

4) From the comparison of the values for the bars after Step 2 (sealer both sides) and Step 3 (sealer top only) to Step 5 (finishes at seven weeks), it is seen that the sealer contributes as much to Δf_{0}, as do the top coats. This indicates that as much care must be taken with application of the sealer coat as with the finish top coats.
C. Statistical Analysis of the Impact of Finishes on f_{0}
Examination of Figures 4 and 5 suggested the trends in Δf_{0} with treatment step, previously discussed. In a few cases the spread in values of Δf_{0} for the bars of a treatment step, made it difficult to distinguish whether a treatment step had a significant effect on Δf_{0} or not. Paired sample Students tTests at a level of significance of 0.05 (95% confidence) were used to make stepbystep statistical comparisons of the means of f_{0} for samples of each finish treatment step. Additionally, comparison of the mean f_{0} of Step 1 (no sealer) to the mean of Step 3 (sealer on top side) was included to evaluate the statistical difference between a onesided sealer, twosided sealer and no sealer.
Table 10
─ Results of Students tTest for finish treatments
on the alonggrain bars: comparison
of the means of f_{0} for the treatment steps. 
A twotailed tdistribution was used. The tTest returns a probability, P. If P is less than or equal to the level of significance (P ≤ 0.05) the difference in the means for the finish treatments is considered statistically significant (i.e. the finish treatment led to a statistically significant change in the mean of f_{0}).
The analyses were performed with Microsoft Excel statistical analysis tools.[15] The natural pairing of the samples for the tTest, that is, comparing f_{0} for each bar before and after a finish treatment step, discounts differences in f_{0} due to, for example, bartobar variation of Youngs moduli for the alonggrain bars.
Table 10 presents the results of paired sample tTests for the alonggrain bars and Table 11 the results for the crossgrain bars. For each table: the first column describes the finish step; the second column gives the mean of f_{0} for the finish step taken from Table 8; the third column lists P for the results of the tTest for comparing the means of the two finish steps (e.g. P = 0.005 for the comparison of finish Step 3 to Step 4 in Table 10); and the fourth column states whether the difference in the means is significant (YES) or not (NO).
The top sections of Tables 10 and 11 provide a stepbystep comparison of the means. The lower sections provide a comparison of Step 1 (bare wood) to Step 3 (sealer on top side) to allow comparison of the statistical significance of the sealer on both sides (comparing Step 1 to Step 2) to the sealer on one side (Step 1 to Step 3).
From Table 10 it can be seen that a significant decrease in the resonant frequencies for the alonggrain bars occurred at treatment Step 2 (application of the sealer coat on both sides of the bars). Removal of the sealer from one side (Step 3) had an insignificant effect on f_{0}. Another significant decrease in f_{0} occurs at Step 4 (application of the finish top coat). Curing for seven weeks (Step 5) had an insignificant effect on f_{0}. Note that presence of the sealer on both sides (comparison of Step 1 to 2) or only the top side (comparison of Step 1 to 3) produced a statistically significant decrease in f_{0}.
Table 11
─ Results of Students tTest for finish treatments
on the crossgrain bars: comparison
of the means of f_{0} for the treatment steps. 
From the tTest analyses, it is clear that the sealer and the top coats play different roles in modifying f_{0}. For the alonggrain bars, the sealer and top coats both contribute to a reduction of f_{0}. For the crossgrain bars, the sealer increases f_{0}, while the top coats produce no additional change in f_{0}. Note that for both the alonggrain and crossgrain bars, curing for seven weeks produced no significant change in f_{0} as compared to the initial fourday cure.
D. Testing for Differences in the Effect of Top Coats on Δf_{0}
With respect to the effect of the finish treatments on Δf_{0}, the question remains: are there significant differences due to the different top coats? To address this question the uncertainty in the Δf_{0 }values was estimated by calculating the margins of error, and constructing a confidence interval about the mean of Δf_{0} for the top coats cured for seven weeks. The procedure is similar to that previously used to estimate uncertainties in the coating areal densities and thicknesses.
The limits of the confidence intervals were calculated from the margins of error, ξ_{Δf} , and the mean of Δf_{0 }for treatment Step 5. Calculation of ξ_{Δf} requires an estimate of the standard deviation, s_{Δf}, of the mean, Δf_{0mean}, and the value for the twotailed tstatistic, k, at a significance level of 0.05, and sample size n:
ξ_{Δf}_{ }= k (s_{Δf}/√n) (13)
The confidence interval is given by:
Δf_{0mean } ξ_{Δf} ≤ Δf_{0mean} ≤ Δf_{0mean }+ ξ_{Δf} (14)
Table 12 ─ Δf_{0}
values following finish treatment Step 5 (seven weeks cure time) for alonggrain and crossgrain bars, along with the means,
margins of error, and confidence intervals at the 95% confidence level. 
As can be seen from Table 12 the values of Δf_{0} for the alonggrain and crossgrain bar top coats fit within the 95% confidence intervals. Thus, it is concluded that within the limits of measurement precision, all of the top coats are equivalent with respect to the effect on the changes in the fundamental resonant frequencies.
E. Impact of Finishes on the Damping Q Factor
As with the analyses of the impact of the finishes on f_{0} of the bars, similar analyses were performed to quantify the impact of the finishes on the damping, represented by the Q factor. A lower Q represents a higher damping. Table 13 gives the results of the measurement of Q of the test bars after each of the finish treatments.
Figure 7
─ Line chart of the trend in changes in the damping Q factor,
ΔQ, with finish treatment for the crossgrain bars. See
Table 9 for key. 
Figure 6
─ Line chart of the trend in changes in the damping Q factor, ΔQ,
with finish treatment for the alonggrain bars. See Table 9 for key. 
Figure 6 summarizes the trends in ΔQ for the alonggrain bars and Figure 7 the trends for the crossgrain bars. As in the previous charts for Δf_{0}, each line represents a different bar, identified by its number and corresponding color given in the key in Table 9. The finish treatments on the horizontal axis of the figures are designated by the finish step number and brief description of the treatment in the key in Table 9.
Table 13
─ Damping Q factor of test bar samples after each finish treatment. For the purpose of performing
significance tests, the mean Q is given for each set of samples (except for
the controls) and finish treatment. *Indicates the unfinished control samples. 
1) The means and standard deviations for measurements of Q for the unfinished control bars taken over a period of nine weeks were: 160 7 for the alonggrain bars, and 50 5 for the crossgrain bars. This indicates the precision of the measurement of Q and the repeatability over seven weeks.
2) Compared to Q of the unfinished bars, all of the finish steps resulted in lower values of Q. This is to be expected because of the greater vibrational energy loss due to the plasticlike finish coatings. ΔQ values for the alonggrain and crossbars for the treatment steps are similar. However, because of the much lower Q values for the unfinished crossgrain bars (e.g. 44 for bar 3AC4) compared to the alonggrain bars (e.g. 143 for bar 3AL4), the finishes have a greater impact on decreasing Q of the crossgrain bars.
Table 14
─ Results of Students tTest for finish treatments on the alonggrain
bars ─ comparison of the means of Q for the treatment steps. 
3) Applying the sealer to both sides of the bars (finish Step 2) had the greatest effect for decreasing the value of Q. After removing the sealer from the back sides of the bars, Q increased for the crossgrain bars, but not for the alonggrain bars. Curing the finish seven weeks led to larger values of Q for the alonggrain bars but not for the crossgrain bars.
F. Statistical Analysis of the Impact of Finishes on Q
The impact of finishes on the damping Q factor was examined by the same statistical techniques used to examine the impact of finishes on f_{0}. A paired sample Students tTest at a level of significance of 0.05 was used to compare the means of the Q values for the treatment steps. Table 14 presents the results of the paired sample tTest for the alonggrain bars, and Table 15 the results for the crossgrain bars.
For each table: the first column describes the finish step; the second column gives the mean of Q for the finish step taken from Table 8; the third column provides P for the results of the tTest for comparing the means of the two finish steps (e.g. P = 0.005 for the comparison of finish Step 2 to Step 3 in Table 15); and the fourth column states whether the difference in the means is significant (YES) or not (NO).
The top section of each table provides a stepbystep comparison of the means. The lower section provides a comparison of Step 1 (bare wood) to Step 3 (sealer on top side) to allow comparison of the statistical significance of the sealer on both sides (comparing Step 1 to Step 2) to the sealer on one side (Step 1 to Step 3).
Table 16 ─ ΔQ values
following finish treatment Step 5 (seven weeks cure time) for alonggrain and crossgrain bars, along with their
mean, margin of error, and
confidence intervals at the 95% confidence level. 
From Table 14 it can be seen that a significant change in Q for the alonggrain bars occurred for treatment Step 2 (coating with the sealer coat on both sides of the bars). Removal of the sealer from one side (Step 3) had an insignificant effect on Q, as did application of the top coat with a cure time of 4 days (Step 4). However, curing of the top coats for seven weeks was found to significantly increase the value of Q for the alonggrain bars. Note that the effect of the sealer on both sides (comparison of Step 1 to 2) and only the top side (comparison of Step 1 to 3) were equivalent, producing a statistically significant decrease in Q.
Table 15 shows that Q for the crossgrain bars significantly decreases with application of sealer to both sides (finish Step 2), and significantly increases with the removal of the sealer from the back side (finish Step 3). At the 95% confidence level there is no further statistically significant change in Q due to finish Steps 4 (P = 0.068) and 5 (P = 0.061) for the chosen level of significance of P=0.05). However, it can be seen that presence of the sealer on both sides (comparison of Step 1 to 2) or only the top side (comparison of Step 1 to 3 in the lower section of Table 15) leads to a statistically significant decrease in Q when compared to the value of Q for the unfinished bars.
G. Testing for the Differences in the Effect of Top Coats on Q
Table 15 ─ Results of Students tTest for
finish treatments on the crossgrain bars ─ comparison of the
means of Q for the treatment steps. 
For calculation of the margins of error, the standard deviation s_{ΔQ} was estimated from the data for ΔQ (Q for Step 3 less Q for Step1) in Table 13. The estimated value of s_{ΔQ} for the alonggrain bars was 5.2 and that for the crossgrain bars was 7.0. The margins of error, calculated according to equation (13), are 8.3 for the alonggrain and 11.2 for the crossgrain bars.
Values of ΔQ for the bars, along with the means, margins of error and confidence intervals, following finish application and curing for seven weeks are given in Table 16.
As can be seen from Table 16 the values of ΔQ for the alonggrain and crossgrain bar top coats fit within the 95% confidence intervals. Thus, it is concluded that within the limits of measurement precision all of the top coats are equivalent with respect to the effect on the change in the damping quality factor Q.
IV. Conclusion
A key finding of this study is that all of the top coat finishes, both the evaporative finishes (dewaxed shellac and guitar lacquer) and reactive finishes (modified dewaxed shellac and modified dewaxed shellac aerosol), applied over the same sealer, produce equivalent changes in the properties of fundamental vibrational frequency, f_{0}, and damping, Q, of the spruce test bars. Specific effects of the finishes on f_{0} and Q, supported by statistical analyses, show that both the sealer and top coats affect f_{0} and Q, but in different ways:
1) The sealer alone produces significant changes in both f_{0} and Q for the two grain orientations of the spruce test bars: the alonggrain f_{0} decreases; the crossgrain f_{0} increases, and Q for both grain orientations decreases.
2) The finish top coats affected f_{0} for only the alonggrain bars. The alonggrain f_{0 }decreases with top coat application. At the 95% confidence level, application of the top coats have no significant effect on the crossgrain f_{0}.
3) Compared to Q for the sealer coating, all of the top coats cured for seven weeks showed an increase in Q for the alonggrain bars. Q for the crossgrain bars cured for seven weeks showed no significant increase.
V. Bibliography
 Dresdner, M. (1999). The New Wood Finishing Book, Newtown, Ct, The Taunton Press, 35.
 Schelleng, J. C. (1968). Acoustical Effects of Violin Varnish, Journal of the Acoustical Society of America, Vol. 44, No. 5, 11781179.
 Schleske, M. (1998). On the Acoustical Properties of Violin Varnish, Catgut Acoustic Society Journal, Vol. 3 (Series II), 28.*
 Gore, T. and Gilet, G. (2011). Contemporary Acoustic Guitar Design and Build: Volume 2 Build, Terrey Hills, NSW, Australia: Trevor Gore, 627.
 Michelman, J. (1946). Violin Varnish: A Plausible Recreation of the Varnish Used by the Italian Violin Makers Between the Years 1550 and 1750, A.D., published by the author, Cincinnati, OH, 110. Available online at URL: http://www.oosterhofonline.net/pdf/Michelmann's%20violinvarnish.pdf
 Schelleng, J. C. (1968). Acoustical Effects of Violin Varnish, Journal of the Acoustical Society of America, Vol. 44, No. 5, 11771178.
 Gore, T. and Gilet, G. (2011). Contemporary Acoustic Guitar Design and Build: Volume 2 Build, Terrey Hills, NSW, Australia: Trevor Gore, 192.
 Wilson, E. B. (1952), An Introduction to Scientific Research, New York: McGrawHill, Dover Edition (1990), 239 240.
 Fletcher, N.H. and Rossing, T. D. (1998). The Physics of Musical Instruments, 2^{nd} Edition, New York, NY, Springer, 60 64.
 Gore, T. and Gilet, G. (2011) Contemporary Acoustic Guitar Design and Build: Volume 1 Design, Terrey Hills, NSW, Australia: Trevor Gore, 423 to 429.
 Hutchins, C. M. (1981). The Acoustics of Violin Plates, Scientific American, October, 170 186.
 Hutchins, M. A. (1981). Acoustical Parameters for Violin and Viola Back Wood, Catgut Acoustical Society Newsletter, 36, 29 31.*
 Hutchins, M. A. (1991) Effects on Spruce Test Strips of Fouryear Application on Four Different Sealers Plus Oil Varnish, Catgut Acoustical Society Journal 1, No. 7, 11 16.*
 Haines, D.W. (1980). On Musical Instrument Wood Part II Surface Finishes, Plywood, Light and Water Exposure, Catgut Acoustical Society Newsletter, No. 33, 23.*
 Gore, T. and Gilet, G. (2011). Contemporary Acoustic Guitar Design and Build: Volume 1 Design, Terrey Hills, NSW, Australia: Trevor Gore, 171.
 Curtin, J. (2006). Tap Routine, The Strad, October, 48 54. URL: http://www.josephcurtinstudios.com/article/taproutine/
 Haines, D.W. (2000). The Essential Mechanical Properties of Wood Prepared for Musical Instruments, Catgut Acoustical Society Journal, 4, No. 2, 24.*
 Hutchins, M. A. (1991). Effects on Spruce Test Strips of Fouryear Application on Four Different Sealers Plus Oil Varnish, Catgut Acoustical Society Journal, 1, No. 7, 11 12.*
 Haines, D.W. (1980). On Musical Instrument Wood Part II Surface Finishes, Plywood, Light and Water Exposure, Catgut Acoustical Society Newsletter, No. 33, 22.*
 Schleske, M. (1998). On the Acoustical Properties of Violin Varnish, Catgut Acoustic Society Journal, Vol. 3 (Series II), 30.*
 Haines, D.W. (2000). The Essential Mechanical Properties of Wood Prepared for Musical Instruments, Catgut Acoustical Society Newsletter, 4, No. 2, 30.*
 Schelleng, J. C. (1968) Acoustical Effects of Violin Varnish, Journal of the Acoustical Society of America, 44, No. 5, 1179.
 Schleske, M. (1998). On the Acoustical Properties of Violin Varnish, Catgut Acoustic Society Journal, Vol. 3 (Series II), 29.*
*Issues of the Catgut Acoustic Society Newsletter and Journal are available online from an indexed, searchable collection maintained by the Stanford University Libraries, Stanford, California 943046064 at URL: http://www.oac.cdlib.org/findaid/ark:/13030/kt6h4nf6qc/
[1]Albuquerque,
New Mexico.
This research was funded solely by the author.
Manuscript received October 2, 2015
[2]
Luthiers Mercantile International, Inc., 7975 Cameron Drive, Bldg. 1600,
Windsor, CA 95492; http://www.lmii.com/
[3] Ken Picou Design, 5508 Montview, Austin, TX 78756
[4] ShellacFinishes, 7740 Goldfish Way, San Diego, Ca 92129; http://www.shellacfinishes.com
[5] Dry shellac is mixed with denatured alcohol in a particular ratio called a cut, which refers to the amount of shellac in pounds dissolved in a gallon of alcohol. A 2lb. cut of shellac is 2 lb. of shellac resin dissolved in a gallon of alcohol.
[6] StewartMacDonald, 21 N. Shafer Street, Box 900, Athens, OH 45701, http://www.stewmac.com
[7] Chicago Aerosol, 1300 E. North St., Coal City, IL 60416
[8] https://www.youtube.com/watch?v=08vFIhdfXIk , segment time32:30 to 33:40 minutes, R. Christopher, May 12, 2014.
[9] For a discussion of this method of constructing a confidence interval see Wilson [8].
[10] MicroSurface Finishing Products Inc., 1217 West 3^{rd} Street, PO Box 70, Wilton, Iowa 52778, http://microsurface.com/
[11] Kremer Pigmente, Safety Data Sheet, March 1996, pg. 1
[12] Chemcraft data sheet for Chemseal Amber NC Clear Sealer 5465005, February 2010, Akzo Nobel Coatings, Inc., 1431 Progress Ave., High Point, NC 27261.
[13] Hutchins (1991) [18] describes the effect in terms of the change in frequency, whereas Haines (1980) [19], Schleske (1998) [20], and Haines (2000) [21] describe the effect in terms of change in apparent stiffness or Youngs modulus, related to frequency by equation (7).
[14] For example, the mean of f_{0} values from Table 8 (155, 161, 168, 169) for treatment Step 5 for the alonggrain bars is 163 Hz.
[15] For more information on the tTest as performed by Microsoft Excel data analysis tools, and interpretation of results of the test, refer to Microsoft Excel Help for the tTest.