Page 28 - 2016Summer

Violin Acoustics
The Acoustic Spectrum
Unlike loudspeakers designed to have as flat a frequency re- sponse as possible, the spec- trum of the violin fluctuates wildly, with many strong peaks and troughs reflecting the relatively weakly damped, multiresonant responses of the instrument. This will vary markedly in detail from one instrument to the next, even between different Stradivari and Guarneri violins, giving each instrument its individual sound quality.
Figure 4. RSuperimposed spectra of the radiated sound pressure measured in the bridge plane at 0°, ± 30° and ± 60° in front of the top plate of the Willemotte 1734 Stradivari violin. Red boxes: Frequencies of the open G0 to E0 strings and the first three octaves of the open E-string, E1 to E3. Data courtesy of Curtin, personal communication.
Figure 4 shows the radiated
sound measured by Curtin
in five different directions for
the Willemotte 1734 Stradivari violin investigated in the Strad 3-dimensional project (Zygmuntowicz and Bissinger, 2009). The acoustic response was measured by tapping the bass-side top corner of the bridge in a direction parallel to the plates. This simulates the component of the bowed string force in the same direction. The fast Fourier transform (FFT) of the recorded sound has been normalized to that of the force of the light impact hammer exciting the violin modes. To simplify the acoustic response, the strings were damped, although string resonances can make a significant contribu- tion to the quality of the radiated sound (Gough, 2005).
The observed resonances are those of the independent nor- mal modes of the freely supported instrument, which have individual resonant responses just like a single damped mass-spring oscillator. They describe the coupled compo- nent mode vibrations of the body shell, the air inside the cav- ity, and all attached structures such as the neck, fingerboard, tailpiece, and strings (Gough, 2015b). To avoid potential confusion between the uncoupled normal and coupled com- ponent modes, capital letters will be used for the former (A0, CBR, B1−, B1+,...) and small letters for the latter (f-hole, cbr, breathing, bending, ...).
The “coupled oscillators” text box illustrates how the cou- pling between coupled component modes result in the veer- ing and splitting of the frequencies of the resultant normal
modes describing the in- and out-of-phase vibrations of the component modes.
The radiated sound is the sum of the radiated sound from each of the excited normal modes. For typical Q-values (25-50), the amplitude and width of each resonant peak is damping limited over about a semitone or two of its reso- nant frequency. Because of the logarithmic sensitivity of the ear, each mode still contributes significantly to the perceived sound well away from its resonance, where its response is determined by its springiness and effective mass below and above its resonant frequencies.
The effective mass of the individual shell modes can be de- termined from the measured mobility or admittance (in- duced velocity/applied force) in the direction of the force at the point of excitation. For a given mode, the lighter the plates, the stronger the radiated sound. Curtin (2006) has suggested that one of the reasons for the general decline in quality of violins from around 1750 onward was the use of somewhat heavier plates than those of the Italian masters.
The radiated frequency response can be divided into three overlapping regions.
(1) A signature mode range over the first two oc- taves up to around 1,000 Hz, where there are a rela- tively small number of well-defined resonant modes such as the A0, B1−, and B1+ modes indicated. Their resonant frequencies and intensities provide
26 | Acoustics Today | Summer 2016