Vibroacoustic Behaviour of Harp Sound Boxes - Daltrop

Vibroacoustic Behaviour of Harp Soundboxes

by

Shira Daltrop

A THESIS SUBMITTED IN PARTIAL FULFILLMENT OFTHE REQUIREMENTS FOR THE DEGREE OF

BACHELOR OF SCIENCE

in

The Faculty of Science

(Physics)

THE UNIVERSITY OF BRITISH COLUMBIA

(Vancouver)

April 2009

c© Shira Daltrop 2009

All rights reserved. This work may not bereproduced in whole or in part, by photocopy

or other means, without permission of the author.

ii

Abstract

This project involves studying the acoustic velocity field around a soundbox of a musical string instrument.The relationship between the low frequency modes of the air in sound holes and the soundboard is importantin determining the quality of sound production of an instrument. Using an impact hammer to apply aforce, an accelerometer to measure the movement of the soundboard, and a velocity probe to measurethe movement of the air, this relationship is examined. This study is focused on the harp, however aguitar was also measured to ensure this method can reproduce previous results. The lowest air and woodmodes were examined and the results agreed with theory. From looking at the phase difference betweenthe soundboard velocity and the acoustic velocity, it is determined when the soundboard and sound holesare in and out of phase at the four lowest resonances.

iii

Table of Contents

Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ii

Table of Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iii

List of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . v

Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

2 Soundboxes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42.1 Sound Radiation From a Point Source . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42.2 Vibration of Plates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42.3 Vibration of a Shell and a Hole . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

3 Previous Work on Harps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

4 Experiment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104.1 The Impact Hammer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104.2 The Velocity Probe . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114.3 Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114.4 Calibration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

4.4.1 Accelerometer and Impact Hammer . . . . . . . . . . . . . . . . . . . . . . . . . . . 124.4.2 Velocity Probe . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134.4.3 Converting From Decibels . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

4.5 Error . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

5 Analysis and Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155.1 HarpIII . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155.2 Aoyama Amphion Harp . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

6 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

Table of Contents iv

Appendices

A Aoyama Ampion Harp: Another Resonance . . . . . . . . . . . . . . . . . . . . . . . . . . 28

B Aoyama Amphion Harp: Holes H2-5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

v

List of Figures

1.1 The Configuration of a Harp . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

2.1 The Amplitude and Power of a Spherical Monopole . . . . . . . . . . . . . . . . . . . . . . . 52.2 Admittance of a Violin Plate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62.3 Admittance of a Guitar . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62.4 Acoustic Velocity Divided by the Force of a Guitar . . . . . . . . . . . . . . . . . . . . . . . 72.5 Phase Difference Between the Plate Velocity and Acoustic Velocity of a Guitar . . . . . . . 72.6 Schematic Diagram of the Guitar Soundbox . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

4.1 The Impact Hammer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104.2 The Velocity Probe in a Sound Hole . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114.3 The Accelerometer on the Soundboard . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124.4 The Inertance From a 200g Mass . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124.5 Phase Difference of Plate Velocity and Acoustic Velocity . . . . . . . . . . . . . . . . . . . . 134.6 Phase Difference of Soundboard Velocity and Force of Aoyama Amphion . . . . . . . . . . . 14

5.1 Conventions for Analysing Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155.2 Acceleration Versus Time of HarpIII . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 165.3 Acoustic Velocity Versus Time of HarpIII . . . . . . . . . . . . . . . . . . . . . . . . . . . . 165.4 Increasing Resonances of a Harp Soundboard . . . . . . . . . . . . . . . . . . . . . . . . . . 175.5 Admittance of HarpIII . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 175.6 Acoustic Velocity Divided by the Force of HarpIII . . . . . . . . . . . . . . . . . . . . . . . 185.7 Phase Difference Between Acoustic Velocity and Soundboard Velocity of HarpIII . . . . . . 185.8 Admittance of Aoyama Amphion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 205.9 Phase Between Soundboard Velocity and Force of Aoyama Amphion . . . . . . . . . . . . . 205.10 Acoustic Velocity Divided by the Force of Hole H1 of Aoyama Amphion . . . . . . . . . . . 215.11 Phase Difference of Acoustic Velocity and Soundboard Velocity of Aoyama Amphion at

297mm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 215.12 Phase Difference of Acoustic Velocity and Force of Hole H1 of Aoyama Amphion . . . . . . 225.13 Phase Difference Between Acoustic Velocity and Soundboard Velocity at 223Hz . . . . . . . 225.14 Phase Difference Between Acoustic Velocity and Soundboard Velocity at 195Hz . . . . . . . 235.15 Phase Difference Between Acoustic Velocity and Soundboard Velocity at 183Hz . . . . . . . 235.16 Phase Difference Between Acoustic Velocity and Soundboard Velocity at 342Hz . . . . . . . 245.17 Phase Difference of Acoustic Velocity and Soundboard Velocity of Aoyama Amphion at

712mm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

List of Figures vi

5.18 Schematic Representation at 223Hz and 180Hz . . . . . . . . . . . . . . . . . . . . . . . . . 255.19 Schematic Representation at 342Hz . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 255.20 Schematic Representation at 458Hz and 500Hz . . . . . . . . . . . . . . . . . . . . . . . . . 25

A.1 Phase Difference of Acoustic Velocity and Soundboard Velocity of Aoyama Amphion at1145mm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

A.2 Phase Difference Between Acoustic Velocity and Soundboard Velocity at 458Hz . . . . . . . 29A.3 Phase Difference Between Acoustic Velocity and Soundboard Velocity at 500Hz . . . . . . . 29A.4 Schematic Representation at 577Hz and 635Hz . . . . . . . . . . . . . . . . . . . . . . . . . 30

B.1 Acoustic Velocity Divided by the Force of Hole H2 of Aoyama Amphion . . . . . . . . . . . 31B.2 Phase Difference of Acoustic Velocity and Force of Hole H2 of Aoyama Amphion . . . . . . 31B.3 Acoustic Velocity Divided by the Force of Hole H3 of Aoyama Amphion . . . . . . . . . . . 32B.4 Phase Difference of Acoustic Velocity and Force of Hole H3 of Aoyama Amphion . . . . . . 32B.5 Acoustic Velocity Divided by the Force of Hole H4 of Aoyama Amphion . . . . . . . . . . . 33B.6 Phase Difference of Acoustic Velocity and Force of Hole H4 of Aoyama Amphion . . . . . . 33B.7 Acoustic Velocity Divided by the Force of Hole H5 of Aoyama Amphion . . . . . . . . . . . 34B.8 Phase Difference of Acoustic Velocity and Force of Hole H5 of Aoyama Amphion . . . . . . 34

1

Acknowledgements

I would like to thank my supervisor, Dr. Waltham, for his guidance and support. I would also like tothank the School of Music for the use of their Aoyama Amphion harp.

2

Chapter 1

Introduction

A harp is made up of a base, a column, a string arm, a soundbox with sound holes in it and strings which areattached to a soundboard (Figure 1.1). What makes harps different from most other string intruments is thefact that the plane of the strings is perpendicular to the soundboard as opposed to parallel. As well, thereis no bridge present in the harp to connect the strings and the soundboard as in other string instruments.Instead the strings connect directly to the soundboard, therefore it needs to be thick enough to supportthe force of the strings. In string instruments, the soundboard is present as a sound radiator, while thestrings are the resonators. To increase the sound production, an acoustic radiator, the soundbox, is added.

Figure 1.1: The configuration of a harp.

In most instruments, this box has dimensions andholes which allow for optimal sound. Indeed, in theharp it has been shown that the power generatedby the sound holes is on average greater than thepower output of the soundboard [1]. However, theholes in the soundbox are placed more out of con-venience for mounting the strings than improvingthe sound [2]. In the guitar, however, the soundhole is designed to excite the Helmholtz mode at acertain frequency, leading to a greater sound pro-duction [1]. Thus it is evident that it is possibleto maximize the sound production by strategicallyplacing the sound holes.

The acoustic motion in the soundbox createstwo modes, corresponding to the wood and airmodes [2]. The combination of these two modesprevents “dead notes” - frequencies for which theradiativity is zero - in the frequency range domi-nated by these modes. In the analysis done on theviolin, it is found that it is very important to havethese modes well spaced in the low to mid rangefrequency of the instrument [3]. These two lowestresonances are key in determining the sound qual-ity of the instrument [4].

In the making of a string instrument, knowledgeof the relationship between these resonances and the force applied is important for the optimal dimensionsof the soundbox and the location of the holes. This relationship has been studied and understood for

Chapter 1. Introduction 3

the violin [5] and the guitar [6]. The soundboard properties of a harp through construction have alsobeen studied [7], however little work has been done on studying the air modes. By gaining a greaterunderstanding of the acoustic modes, a larger and more even sound production will be achievable. In thisproject, the soundboard and air responses are studied for the guitar and harp, but the main focus of thestudy is the harp.

4

Chapter 2

Soundboxes

2.1 Sound Radiation From a Point Source

As a first order approximation one can consider the soundbox to be a small monopole source, at leastat low frequencies. The following equations describe the properties of a point source [8], the limit of apulsating sphere as its radius goes to zero. For a sphere of radius r and pulsating at a frequency ω, theamplitude is given by

Q = 4πr2u(r) (2.1)

where u(r) is the radial velocity at the surface. Since the wave produced is a spherical wave, the pressureand velocity are given by

p(r) =A

re−jkr (2.2)

andu(r) =

A

ρcr

(1− j

kr

)e−jkr (2.3)

where k = ωc . Combining equations (1) and (3) and assuming kr << 1, one can solve for A from equation

(2), which gives

p(r) =jωρ

4πrQe−jkr. (2.4)

The radiated power P is equal to 12p

2/ρc integrated over a spherical surface, giving

P =ω2ρQ2

8πc(2.5)

The amplitude and power are plotted in Figure 2.1. The power drops off very quickly below resonantfrequencies, while the power output above resonant frequency is much greater and depends on the strengthof the resonance. This is a problem for the harp since its lowest vibrational mode has very little soundproduction.

2.2 Vibration of Plates

The soundboard acts like a vibrating plate. In such a system, two consecutive modes have modal constantswith the same sign, therefore there will be an antiresonance at a frequency between the resonances.Figure 2.2 shows the admittance, the acceleration of the soundboard divided by the force, of a violin plate.There are deep minima between resonances where very little sound production, “dead notes”, would occurif there was no soundbox.

2.3. Vibration of a Shell and a Hole 5

Figure 2.1: The amplitude and power of a spherical monopole with r=15cm, f=262Hz and Q=30. Theseparameters were chosen to be typical for a violin.

2.3 Vibration of a Shell and a Hole

At low frequencies, a shell and a hole, like the soundbox and sound holes of a string instrument, will actas follows. As the soundbox volume decreases, the same volume of air exits through the sound holes,therefore the volume of the air plus soundbox remains constant. This is known as the “toothpaste” effect,for its similarity to a tube of toothpaste [5]. At low frequencies, it is known that the sound radiation isdominated by the size of the overall change in volume of the box plus air [5]. Because of this, the staticmotion of the soundbox and sound hole must break down at some point for sound to be produced. Thispoint is assumed to be where the shell has its first resonance. The wood motion and air motion must haveopposite phase, otherwise there would be exact cancellation and there would be no sound radiation.

Figures 2.3 to 2.5 show data from a guitar taken during this project. In Figure 2.3, the admittanceof the soundboard, there is a deep minimum between the well-spaced resonances. Figure 2.4 shows theacoustic velocity divided by the force. Here, no deep minimum is seen between the two modes. This showsthe importance of the soundbox as a radiator to avoid the deep mimima, corresponding to “dead notes”.

Figure 2.5 shows the phase difference between the acoustic velocity and soundboard velocity. The twoare in phase at approximately 100Hz, the first air mode, then becomes out of phase at 200Hz, the firstwood mode. These are determined by a separate set of experiments involving mass-loading the plate.Whether modes are wood or air modes is determined by which of these dominates the movement. Thesetwo modes are well defined and separate in a guitar [6]. Figure 2.6 shows the direction of the air andsoundboard at the air and wood modes.


Figure 2.2: The admittance of a violin plate. There are deep minima between resonances.

Figure 2.3: The driving-point admittance at the center of the bridge of a guitar.


Figure 2.4: The acoustic velocity in the sound hole divided by the force applied to the center of the bridgeof a guitar.

Figure 2.5: The phase difference between the plate velocity at the bridge and the acoustic velocity in thesound hole of a guitar. Note the change of π from approximately 100Hz to 200Hz.


Figure 2.6: Direction of air in the sound hole at 100Hz, when the air and soundboard are out of phase,and 200Hz when they are in phase.

9

Chapter 3

Previous Work on Harps

The first study of the harp was done on the Scottish harp (clarsach) [9]. Modal analysis was performedon the soundboard at different stages of manufacture. Air resonances in a damped soundbox were alsoinvestigated. There was no evidence of the Helmholtz resonance found. The relationship between vibrationmodes of the soundbox and radiation sound was first looked at on a Spanish harp from the baroque period[10]. It was found that interactions between the soundboard vibrations and air cavity movements leadto two coupled modes corresponding to A0 and T1 modes. This was also found on an unstrung SalviOrchestra Concert Harp. The author identifies these modes by measuring the changes of the structuralresponse with closed sound holes. Another conclusion found in this study is that due to the stress on thesoundboard from the strings, the A0 mode in particular is difficult to identify for a strung harp. Anotherstudy performed on the Salvi Orchestra soundbox confirmed that if the wall vibrations are damped, theHelmholtz mode is present in the acoustic response [11].

Le Carrou and Gautier [2] did an experimental modal analysis to show the importance of the A0 andT1 modes. They measured the acoustic velocity in the sound holes of an Atlantide Prestige concert harpusing a velocity probe. The soundboard velocity was measured using a laser vibrometer and the instrumentwas excited by a shaker, connected to the soundboard by a rod, driven by a white-noise source. Theyexamined the low frequency range and looked at when the soundboard and sound holes were in and outof phase.

Waltham and Kotlicki [7] examined a harp’s soundboard through construction. They also measuredadmittances on the soundboard of several different harps. They found a relationship between the patternsof resonant frequencies of the soundbox and fundamental frequencies of the strings.

10

Chapter 4

Experiment

The first harp studied, called HarpIII through this paper, was measured in an anechoic chamber. Thesecond harp measured was an Aoyama Amphion concert harp. Due to the size of the harp, it could not betested in the anechoic chamger. Thus the measurements were taken in the Chan Center on the Universityof British Columbia campus, whose size and quietness made it the best alternative. The strings weredamped by pieces of paper to avoid any influence from the strings. An impact hammer was used to applythe force and a velocity probe was used to measure the sound velocity in the sound holes.

4.1 The Impact Hammer

To apply and measure the force of a tap to the soundboard, an impact hammer was used (Figure 4.1) [12].

Figure 4.1: The impact hammer with amini-XLR connector.

The hammer uses the fact that a force applied in time ∆tcontains Fourier components with frequencies from zero to1

∆t . Testing the response using an accelerometer, a FrequencyResponse Function can be determined:

Hjk(ω) =Xj(ω)Fk(ω)

(4.1)

Here Xj(ω) is the displacement at position j and Fk(ω) is theforce applied at position k. For the use of this hammer in thisproject, only the case where j=k is considered.

This metal hammer is made from a piezo-electric crystal,one of which is found commonly in barbecue lighters. It waschosen specifically with musical instruments in mind, allowinga tap to give a large signal without risk to the instrument.There is continuous shielding from the crystal to the data acquisition system to reduce noise. The dataacquisition system is programmed to take data long enough to record most of the signal but not longenough to take unnecessary noise.

4.2. The Velocity Probe 11

4.2 The Velocity Probe

To determine the velocity of the air, Newton’s Second Law is used in the form

∂p

∂n= −ρ0

∂un

∂t(4.2)

where p is the pressure, n is the distance, ρ0 is the density of the air and un is the velocity of the air [8].To measure the velocity of the air, a velocity probe was used. Figure 4.2 shows the velocity probe set

up in the sound hole of a harp. The velocity probe is made up of two microphones 12mm apart facingeach other.

Figure 4.2: The velocity probe in a soundhole of the Aoyama Amphion Harp.

By looking at the pressure readings of the two microphonesand the distance between them, one can derive the velocityof the air between them. The direct outputs of the velocityprobe through a Norsonics Real-time Analyser type 830 arethe pressure and the velocity, where they are defined as

p =p1 + p2

∆x(4.3)

v =1

ρ0∆x

∫(p1 − p2)dx (4.4)

Here, p1 and p2 are the pressures from the two microphonesand ∆x is the distance between them.

4.3 Method

The soundboard of the Aoyama Amphion harp was tapped235 times, 47 strings for each of the five holes. The stringswere damped by paper for these measurements.

For each sound hole, the velocity probe was positioned inthe center of the hole to measure the acoustic velocity (Fig-ure 4.2). The accelerometer was attached to the soundboardnext to one of the 47 strings and measured the acceleration,and therefore the velocity of the soundboard (Figure 4.3). Theimpact hammer was used to tap the soundboard next to the accelerometer to measure the force, thereforegiving the driving-point admittance. The voltage outputs from the accelerometer, impact hammer andReal-Time Analyser were read by a Measurement Computing USB-1208FS analog to digital converter.It was connected to a standard computer by USB where it was analyzed by a visual basic data acquisi-tion system. A Fast Fourier Transform of the time dependent data was taken and gave both frequencydependent data and the phase of each component measured.

4.4. Calibration 12

Figure 4.3: The accelerometer on the soundboard of the Aoyama Amphion Harp with the impact hammerand mounting wax shown.

4.4 Calibration

4.4.1 Accelerometer and Impact Hammer

To calibrate the accelerometer and impact hammer, an object of known mass was hung by a light string andallowed to swing freely. The accelerometer was placed with mounting wax on the mass at approximatelythe same height as the center of mass. The impact hammer tapped the mass on the opposite side asthe accelerometer. The resulting force divided by the acceleration of this tap, called the inertance, isequal to the mass of the object. This measurement was done five times on a 200g test mass, where theresulting average inertance divided by 200 gives a calibration constant. Figure 4.4 shows the inertanceafter calibration.

Figure 4.4: The average inertance after five taps of a 200g mass after calibration.

4.5. Error 13

4.4.2 Velocity Probe

The phase difference between the plate velocity and acoustic velocity was examined for a speaker at givenfrequencies. The speaker had no reflex enclosure so that the movement of the speaker and the air above itwas known to be in phase. The phase difference was plotted against the distance to the source multipliedby k, where k = 2π/λ at a distance of 49cm (Figure 4.5). Theory states that the phase should go to zeroas kr goes to zero. Thus a phase correction in the data was implemented.

Figure 4.5: The phase difference between the plate velocity and the acoustic velocity at 49cm beforecalibration.

4.4.3 Converting From Decibels

The velocity probe was calibrated using a sound source of known sound level, given in decibels. To convertfrom dB to an acoustic velocity in meters per second, the following equation was used [8]:

β = 10 log10

I

I0. (4.5)

where β is the sound level, I is the intensity and I0 is a constant equal to 10−12 W/m2. The intensityrelates to the velocity by

I = v2ρc (4.6)

where v is the velocity, ρ is the density of air and c is the speed of sound.

4.5 Error

The uncertainty in phase measurements was established as follows. The phase difference between thesoundboard velocity and force versus frequency was measured for all five sound holes. By taking thestandard deviation of these five plots the error was determined (Figure 4.6). The error is much greater

4.5. Error 14

at lower frequencies, where the velocity probe is less accurate. Lower frequencies correspond to longerwavelengths, thus the distance between the two microphones is a smaller fraction of the wavelength.

The error in hole H5, the top hole, is much higher. The hole is further away from most of the strings,thus there is a greater fraction of the wavelength between the driving point and the velocity probe.

Figure 4.6: The phase difference between the soundboard velocity and the force on the Aoyama Amphionharp, showing the error for each frequency. This is from a position of 297mm from base of soundboard,corresponding to string D2.

15

Chapter 5

Analysis and Results

Figure 5.1: For the following analysis, these are the conventions used for “in” and “out of” phase and thelabelling of the sound holes.

5.1 HarpIII

The first harp looked at was a small harp constructed by Dr. Waltham (Figure 1.1), a copy of an 1820Morley harp. It has four sound holes and 36 strings. The soundboard is 983 mm in length, 329 mm wideat the base and 66 mm wide at the top. The thickness ranges from 4.7 mm at the base to 2.4 mm at thetop [7]. Figures 5.2 and 5.3 show the time dependent acceleration and acoustic velocity, the raw data thatis then Fourier Transformed. Figure 5.4 shows the increasing resonances of a harp soundboard, created byDr. Waltham in SolidWorks/COSMOS. Figure 5.5 shows the driving-point admittance of the soundboardof HarpIII, where resonances of the soundboard are seen. Figure 5.6 shows the acoustic velocity dividedby the force of hole H1 of the harp. The first air and wood modes are seen clearly and are well spaced atapproximately 175 and 225 Hz. In Figure 5.7 there is a clear phase change of π between 175 and 225 Hz.

5.1. HarpIII 16

Figure 5.2: The acceleration versus time of HarpIII.

Figure 5.3: The acoustic velocity versus time of HarpIII

5.1. HarpIII 17

Figure 5.4: Increasing resonances of a harp soundboard, created by Dr. Waltham in SolidWorks/COSMOS.

Figure 5.5: The driving-point admittance plot of the soundboard.

5.1. HarpIII 18

Figure 5.6: The acoustic velocity in hole H1, divided by the force applied at each string-attachmentposition on the soundboard, measured in m/s/N.

Figure 5.7: The phase difference between acoustic velocity and soundboard velocity in hole H1 of HarpIII.

5.2. Aoyama Amphion Harp 19

5.2 Aoyama Amphion Harp

The second harp measured was the Aoyama Amphion harp. This soundboard is approximately 1.4 m inlength, .5 m wide at the base and .1 m wide at the top. The thickness ranges from 11-12 mm at thebottom to 2-2.5 mm at the top [7]. Figure 5.8 shows the driving-point admittance of the soundboard.Figure 5.9 shows the phase difference between soundboard velocity and the force, where the phase staysbetween −π/2 and π/2. This is what theory suggests for the phase of the driving point mobility for thevibrations of a plate [8].

Figure 5.10 shows the acoustic velocity in hole H1 divided by the force applied to the soundboard. Incomparison to HarpIII, there are not two distinct peaks around 200Hz. Here there is one large peak at223Hz, then the phase difference decreases slowly as the frequency decreases. This leads to a more gradualphase change from in phase to out of phase, as seen in Figure 5.11, data taken at string D2. Figure 5.12shows the phase difference between the acoustic velocity and force.

The lack of two clearly separated peaks in the Aoyama Amphion harp compared to HarpIII is due tothe relative size of the two soundboxes. The mass of the Aoyama Amphion’s soundboard is approximatelysix times greater than the mass of HarpIII’s, and there is approximately three times as much air in theAoyama’s soundbox than in HarpIII’s. Therefore the soundboard to air ratio is approximately two timeslarger in the Aoyama Amphion than in HarpIII. The smaller soundboard to air ratio means the air cannotsplit the soundboard resonance as easily.

Looking at 223Hz more closely (Figure 5.13), the soundholes are out of phase with the lower half ofthe soundboard and in phase with the upper half. Each hole switches phase in order of position: hole H5changes first at approximately 550Hz, then hole H4 at 600Hz and so on until hole H1 at 800Hz. Figures5.14 and 5.15 show that as the frequency decreases, the holes start to change from out of phase towards inphase. At low frequencies, below 150Hz, the error from the velocity probe becomes greater (see Figure 4.6).Thus the frequency in which the acoustic velocity and soundboard velocity are in phase is not seen in thisexperiment.

The next large peak on the admittance plot occurs at 342Hz (Figure 5.16). This is when there arethree antinodes in the movement of the soundboard. Here, hole H1 is out of phase with the rest of theholes and the lower part of the soundboard. Looking at Figure 5.17 which shows a cross section of thephase difference between acoustic and soundboard velocities at a fixed position instead of frequency, thereis a clear difference of π between hole H1 and the rest at 342Hz.


Figure 5.8: The average driving-point admittance plot of the soundboard.

Figure 5.9: The phase difference between the soundboard velocity and the force on the Aoyama Amphionharp. This is from a position of 297mm from base of soundboard, corresponding to string D2.


Figure 5.10: The acoustic velocity in hole H1, divided by the force applied at each string-attachmentposition on the soundboard, measured in m/s/N

Figure 5.11: The phase difference between the acoustic velocity and soundboard velocity of the five soundholes. Note the gradual phase change from zero to −π between 180Hz and 223Hz. This is from a positionof 297mm, corresponding to string D2.


Figure 5.12: The phase difference between the acoustic velocity and the force of hole H1.

Figure 5.13: Phase difference between the acoustic velocity and soundboard velocity at 223Hz.



Figure 5.17: The phase difference between the acoustic velocity and soundboard velocity of the five soundholes. This is from a position of 712mm, corresponding to string D4.


Figure 5.18: A figure which shows when the air in the sound holes is in and out of phase with thesoundboard. The figure to the right is at 223Hz, the figure to the left is below that.

Figure 5.19: The direction of air and wood movement at 342Hz.

Figure 5.20: A figure which shows when the air in the sound holes is in and out of phase with thesoundboard. The figure to the left is at 458Hz, the figure to the right is at 500Hz.

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Chapter 6

Conclusion

The relationship between the wood and air modes of a harp was studied on a small harp and an AoyamaAmphion harp. To begin a guitar was studied, since it is a simple, well studied system, to allow for abetter understanding of the data aquisition system used. The data matched previous results. On bothharps the expected resonances of the soundboard were seen. The lowest air and wood modes were wellseparated on the smaller harp and results corresponding to theory were seen. On the larger concert harp,these modes were too close together to see expected results.

From here, this data can now be used to investigate improving the sound production of a harp. This canbe done by moving the sound holes to more optimal locations on the soundbox to correspond to resonancefrequencies. The technique can also be used to examine different harps or other musical instruments.

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Bibliography

[1] F. Gautier and N. Dauchez. Acoustic intensity measurement of the sound field radiated by a concertharp. Applied Acoustics, 65:1221, 2004.

[2] F. Gauthier J.L. Le Carrou and E. Foltete. Experimental study of a0 and t1 modes of the concertharp. J. Acoust. Soc. Am., 121:559–567, 2006.

[3] G. Weinreich. Sound hole sum rule and the dipole moment of the violin. J. Acoust. Soc. Am.,77:710–718, 1985.

[4] C. M. Hutchins. The physics of violins. Sci. Am., 207:78–93, 1962.

[5] G. Weinreich. Klopsteg memorial lecture (august, 1992): What science knows about violins-and whatit does not know. Am. J. Phys., 61:1067–1077, 1993.

[6] O. Christensen. Simple model for low-frequency guitar function. J. Acoust. Soc. Am., 68:758–766,1980.

[7] C. Waltham and A. Kotlicki. Vibrational characteristics of harp soundboards. J. Acoust. Soc. Am.,124:1774–1780, 2008.

[8] Thomas D. Rossing and Neville H. Fletcher. Principles of Vibration and Sound. Springer-Verlag, NewYork, 1994.

[9] I. M. Firth. On the acoustics of the harp. Acoustica, 37:148–154, 1977.

[10] I. M. Firth. Harps of the baroque period. J. Catgut Acoust. Soc., 1(3):52–61, 1989.

[11] A. J. Bell. The helmholtz resonance and higher air modes of the harp soundbox. J. Catgut Acoust.Soc., 3(3):2–8, 1997.

[12] C. Waltham and A. Kotlicki. Construction and calibration of an impact hammer. preprint (2008).

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Appendix A

Aoyama Ampion Harp: Another

Resonance

Figure A.1: The phase difference between the acoustic velocity and soundboard velocity of the five soundholes. This is from a position of 1145mm, corresponding to string G6.

Appendix A. Aoyama Ampion Harp: Another Resonance 29

Figure A.2: Phase difference between acoustic velocity and soundboard velocity at 458Hz.

Figure A.3: Phase difference between acoustic velocity and soundboard velocity at 500Hz.

Appendix A. Aoyama Ampion Harp: Another Resonance 30

Figure A.4: A figure which shows when the air in the sound holes is in and out of phase with the soundboard.The figure to the left is at 577Hz, the figure to the right is at 635Hz. Note the phase phase of all five holesbetween the two. See Figures A.2 and A.3 at these two frequencies.

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Appendix B

Aoyama Amphion Harp: Holes H2-5

Figure B.1: The acoustic velocity in hole H2, divided by the force applied at each string-attachmentposition on the soundboard, measured in m/s/N

Figure B.2: The phase difference between the acoustic velocity and the force of hole H2.

Appendix B. Aoyama Amphion Harp: Holes H2-5 32



Vibroacoustic Behaviour of Harp Sound Boxes - Daltrop

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