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Resonance and Resonators Dept of Speech, Music and Hearing ACOUSTICS FOR VIOLIN AND GUITAR MAKERS Erik Jansson Chapter II: Resonance and Resonators Fourth edition 2002 http://www.speech.kth.se/music/acviguit4/part2.pdf Index of chapters Preface/Chapter I Sound and hearing Chapter II Resonance and resonators Chapter III Sound and the room Chapter IV Properties of the violin and guitar string Chapter V Vibration properties of the wood and tuning of violin plates Chapter VI The function, tone, and tonal quality of the guitar Chapter VII The function of the violin Chapter VIII The tone and tonal quality of the violin Chapter IX Sound examples and simple experimental material – under preparation Webpage: http://www.speech.kth.se/music/acviguit4/index.html ACOUSTICS FOR VIOLIN AND GUITAR MAKERS Chapter 2 – Fundamentals of Acoustics RESONANCE AND RESONATORS Part 1: RESONANCE 2.1. Resonance 2.2. Vibration sensitivity 2.3. The mechanical and acoustical measures of the resonator 2.4. Summary 2.5. Key words Part 2: RESONATORS 2.6. The hole-volume resonator 2.7. Complex resonators 2.8. Mesurements of resonances in bars, plates and shells 2.9. Summary 2.10. Key words Jansson: Acoustics for violin and guitar makers 2.2 Chapter 2. FUNDAMENTALS OF ACOUSTICS - RESONANCE AND RESONATORS First part: RESONANCE INTRODUCTION In chapter 1, I presented the fundamental properties of sound and how these properties can be measured. Fundamental hearing sensations were connected to measurable sound properties. In this, the second chapter the concept of RESONANCE and of RESONATORS will be introduced. Resonators are fundamental building blocks of the sound generating systems such as the violin and the guitar. The chapter starts with introducing the resonance and how a resonance is described. Thereafter vibration sensitivity (technical term mobility) of resonances is discussed and finally how mechanical and acoustical measures of a resonator are related. 2.1. RESONANCE A RESONATOR or a system of resonators gives one or several RESONANCES. The blown tone of a bottle is the result of a resonance and the bottle is the resonator. A practical property to measure a resonator is its frequency response (vibration sensitivity, techn term mobility). The response curve gives a picture of each resonance, i.e. its FREQUENCY (the peak frequency), its resonance LEVEL response (the peak height for the selected driving) and its BANDWIDTH (the peak width). These properties are related to the mechanical properties of the resonator; the MASS (weight), STIFFNESS (spring) and FRICTION. Often a resonator consists of a vibrating area, such as a violin top plate. Then the distribution of the vibrations are important. Positions of maximal vibrations are called ANTINODES and the positions of no vibrations are called NODES. Furthermore the decay (the reverberance) of a resonance can be important. The relations between these different measures and properties will be explained in this chapter. Let us first look at the guitar tone displayed in Fig. 1.9. The time history of the tone is smooth except the sharp wiggles soon after the attack. The tone is not made up by a single partial but a spectrum of partials, the levels of which slowly change. The frequency separation between the partials is constant and equals the frequency of the fundamental (the first partial). It is the level of the partials that give the guitar tone its character, the timbre. But why do we obtain a decaying spectrum of partials at these very frequencies? Let us cautiously move towards the answer by looking at a vibrating string. AN EXAMPLE OF RESONANCE: THE VIBRATING STRING A large scale picture of the vibrating string can be presented by means of a long rubber band with one end fixed to a wall. The rubber band is held stretched by a hand at its free end. If the free end is slowly moved up and down, the whole band moves up and down in time with the hand. If the beat tempo, the frequency, is increased the string tends to swing out more between the fixed end and the hand end. At a specific frequency the band will swing up and down in a large bend between wall and hand, but still only requires small hand movements. The vibrations at the bends are maximum. If the beat frequency is increased further the bend will diminish and it is difficult to make the rubber band vibrate visibly. With still further increase of the beat frequency the rubber band will divide itself into two bends. At a specific beat frequency the Jansson: Acoustics for violin and guitar makers 2.3 vibrations at the bends are maximum. With further increased beat frequency the bends diminish and thereafter the rubber band will divide itself into three bends, etc. Let us repeat the same experiment with better control and use a small electrical vibrator attached to a rubber band with its other end fixed to a wall (a similar demonstration can be made using a string of a musical instrument). An electrical tone generator (oscillator) provides the electrical driving signal. The magnitude and the frequency of the driving signal are easy to adjust with high accuracy. The vibrator is replacing the hand. At low frequencies the rubber band again moves in phase with the motion of the vibrator, but with increasing frequency it starts bending out more and more in the middle. At a specific frequency, the first resonant frequency the bend becomes maximum. The position of maximum motion is called an ANTINODE. With further increase of the frequency the bend will first vanish and thereafter at the second resonant frequency, two maxima of motion will come out and the rubber band does not move in the middle. We have now two ANTINODES and in between a NODE. If we continue to increase the frequency the two antinodes vanish and at a higher frequency we will find three bends, antinodes, and between the antinodes, nodes. Further increase of the frequency will show vibrations with four, five, six etc. antinodes with three, four, five etc. nodes. We shall find that the maxima of vibrations are at at fixed frequencies. By measuring the frequencies we will find that the frequencies are 2, 3, 4, 5 times the frequency with maximum vibrations with only one bend. The experiment has shown that certain things happen at specific frequencies (the resonant frequencies) - the rubber band vibrations become maximum at specific positions (antinodes) and are zero at other positions (nodes). THE ORIGIN OF A RESONANCE What is behind these peculiarities and what has that to do with the guitar tone? It is RESONANCES. The vibrating rubber band and also the vibrating string have resonances. But before I explain what is happening to the guitar string, let me explain the origin of a resonance, how its properties are described and are measured. The RESONANCE is an acoustical building brick of fundamental importance. To describe a resonance we need to answer the following questions: At what frequency is a resonance, the resonant frequency? How limited is this resonance in frequency, the bandwidth? How easily are the vibrations to excite, the vibration sensitivity at resonant frequency? How are the vibrations distributed, the modes of vibration (or operating deflection shapes)? A simple resonator is a small ball attached to a string of rubber bands, c.f., Fig. 2.1a. When the finger is slowly moved up and down the ball moves in the same way up and down (in time, in phase with the hand). If the beating frequency of the finger is increased the up and down motion of the ball increases more and more. At a certain beating frequency the ball motion becomes maximum. Further increase of the frequency will make the ball motion smaller and finally at high frequencies only the finger will move. Jansson: Acoustics for violin and guitar makers 2.4 Figure 2.1. A simple resonance: a) a simple resonator - a ball (C) attached to a rubber band (B) and hung on a finger (A), and b) vibration sensitivity curve of the resonance - resonance frequency (RF), peak height (RL) and bandwidth (B measured 3 dB below the peak maximum). The size of the ball vibrations (upand down motion) follow the vibration sensitivity curve plotted in Fig. 2.1b. The curve shows low vibration sensitivity at low and high frequencies. In between there is a peak, a resonance peak giving the resonant frequency (the peak frequency), the peak height giving the vibration sensitivity at the resonance and the peak width giving the bandwidth. The bandwidth of resonances vary much from a fraction of a Hz for the string to tenths of Hz for body resonances. The resonator consists of a mass (weight) - the ball - and a spring - the rubber band. The turbulence in the air surrounding the ball tries to brake its motion and provides friction. If the stiffness of the rubber band is increased the resonant frequency is increased. If the mass (weight) of the ball is increased the resonant frequency is decreased. If the friction (against the air) is increased by a thin plate under the ball, the width of the resonance peak is increased. It is difficult to move a finger up and down maintaining the same beat size and beat frequency. Therefore an arrangement as shown in Fig. 2.2 may be used. A loudspeaker gives a strong tone. By turning a knob on a tone generator the tone frequency is changed without changing the strength of the tone. The plate with attached rubber band is hung over the loudspeaker and the strong loudspeaker tone sets the plate into vibration. The frequency is slowly changed and at the same time the magnitude of the plate vibration is measured.
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