Abstract
This chapter deals with the radiation of vibrating structures in air, with application to stringed and percussive instruments. Basic notions are first presented with the help of an introductory example of a beam coupled to an air column. The important concept of critical frequency is then introduced through the example of an infinite thin plate radiating in air. The radiation models of finite plates and their results can be applied to real musical instruments. Recent methods are then presented for calculating the radiation of unbaffled plates, structural volumes, and nonplanar sources. Finally, the questions relative to the appropriate choice of material, and to the compromise between radiation efficiency and tone duration, are illustrated on several stringed instruments.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Notes
- 1.
In reality, the modes of the coupled system are complex, and we could think of applying the rigorous theory of complex modes presented in Chap. 5. However, since the air coupling can be most often considered as weak in musical acoustics, the method of projection on the in vacuo modes is preferred here, which corresponds to current practice.
- 2.
Notice that this property is no longer valid if there is another structural damping inside the beam.
- 3.
Such an effect can be taken into account as a length correction in tubes.
- 4.
In what follows, the symbols “\(\tilde{\ }\)” will be omitted for clarity.
- 5.
- 6.
Recall that H(t) is the Heaviside function.
- 7.
Here, an interesting link can be made with the linear array of monopoles shown in Fig. 12.7 It was shown in the previous chapter that, as the number N of monopoles tend to infinity, then the direction of radiation tends to \(\varTheta \,=\, 0\), which is equivalent to \(kd\cos \theta \, =\,\varphi\). This can only occur under the condition \(\left \vert \varphi \right \vert \, <\, kd\). In the present example of the plate, the definition of the angle \(\theta\) is modified, so that we must here convert the \(\cos \theta\) of the monopole array in \(\sin \theta\), which yields
$$\displaystyle{ \sin \theta = \frac{\varphi } {kd}. }$$For the plate, the phase shift \(\varphi\) between two consecutive “monopoles” is given here by \(k_{B}d\, =\,\varphi\). As d tends to zero, we find Eq. (13.56) again. This shows that the linear array of monopoles is of the supersonic type. We can further add that if a condition such as \(\left \vert \varphi \right \vert \, >\, kd\) would have been obtained as N tends to infinity, then no radiation would have exist in the far field, because \(\varTheta\) could not be zero. This remark will be useful in Chap. 14 to understand why there is no critical frequency in wind instruments.
- 8.
In this expression, it is assumed that the Poisson’s coefficients are identical in both directions.
- 9.
These values correspond to the maxima of the functions \(\left \vert \frac{(-1)^{m}e^{+jk_{x}L_{x}}-1} {(k_{x}L_{x})^{2}-m^{2}\pi ^{2}} \right \vert \) and \(\left \vert \frac{(-1)^{n}e^{+jk_{y}L_{y}}-1} {(k_{y}L_{y})^{2}-n^{2}\pi ^{2}} \right \vert \). Except for the lowest values of m and n, these maxima are close to m π and n π.
- 10.
The issue of the material choice for wind instruments is a very intricate and controversy matter. Two different materials handled by the same tool do not produce the same geometry, and the nature of the material intervenes also by its porosity, the state of its surface, its heat capacity, etc.
- 11.
By analogy with the electric filters, this cutoff frequency can be defined as the value for which the pressure is reduced by a factor of −3 dB compared to its low-frequency asymptotic value (see Fig. 13.38).
- 12.
Similar results were obtained by Suzuki and Tichy [37], using the theory of spherical harmonics for expanding the pressure (see Chap. 12). These authors report that, due the diffraction effects, an attenuation of the order of −5 dB between ka = 0. 4 and ka = 4 is obtained for the convex caps, whereas an amplification of 4 dB is obtained for the concave case.
References
Abramowitz, M., Stegun, I.A.: Handbook of Mathematical Functions, with Formulas, Graphs, and Mathematical Tables. Dover, New York (1972)
Arcas, K.: Physical model of plate reverberation. In: Proceedings of the 19th International Congress on Acoustics, Madrid (2007)
Ashby, M.F.: Materials Selection in Mechanical Design. Butterworth-Heinemann, Oxford (2004)
Barlow, C.Y.: Materials selection for musical instruments. In: Proceedings of the Institute of Acoustics, ISMA’97, pp. 69–78 (1997)
Boullosa, R.R., Orduna-Bustamante, F., Lôpez, A.P.: Tuning characteristics, radiation efficiency and subjective quality of a set of classical guitars. Appl. Acoust. 59, 183–197 (1999)
Brémaud, I.: Acoustical properties of wood in string instruments soundboards and tuned idiophones: biological and cultural diversity. J. Acoust. Soc. Am. 131(1), 807–818 (2012)
Busch-Vishniac, I.J.: Drive point impedance of an infinite orthotropic plate under tension. J. Acoust. Soc. Am. 71(1), 368–371 (1982)
Chaigne, A.: Structural acoustics and vibrations. In: Springer Handbook of Acoustics, pp. 901–960. Springer, New York (2007)
Conklin, H.A.: Design and tone in the mechanoacoustic piano. Part III. Piano strings and scale design. J. Acoust. Soc. Am. 100, 1286–1298 (1996)
Cummings, A.: Sound radiation from a plate into a porous medium. J. Sound Vib. 247(3), 389–406 (2001)
Dérogis, P.: Analysis of the vibrations and radiation of an upright piano soundboard, and design of a system for reproducing its acoustic field (in French). Ph.D. thesis, Université du Maine, Le Mans (1997)
Elliott, S.J., Johnson, M.E.: Radiation modes and the active control of sound power. J. Acoust. Soc. Am. 94(4), 2194–2204 (1993)
Gautier, F., Tahani, N.: Vibroacoustic behavior of a simplified musical wind instrument. J. Sound Vib. 213(1), 107–125 (1998)
Gautier, F., Nief, G., Gilbert, J., Dalmont, J.: Vibro-acoustics of organ pipes-revisiting the Miller experiment. J. Acoust. Soc. Am. 131, 737–738 (2012)
Gough, C.: Violin plate modes. J. Acoust. Soc. Am. 137(1), 139–153 (2015)
Gough, C.E.: A violin shell model: vibrational modes and acoustics. J. Acoust. Soc. Am. 137(3), 1210–1225 (2015)
Graff, K.F.: Wave Motion in Elastics Solids. Dover, New York (1991)
Griffin, S., Lane, S.A., Clark, R.L.: The application of smart structures towards feedback suppression in amplified acoustic guitars. J. Acoust. Soc. Am. 113(6), 3188–3196 (2003)
Heckl, M.: Vibrations of one- and two- dimensional continuous systems. In: Encyclopaedia of Acoustics, vol. 2, pp. 735–752. Wiley, New York (1997)
Hu, Q., Wu, F.: An alternative formulation for predicting sound radiation from a vibrating object. J. Acoust. Soc. Am. 103(4), 1763–1774 (1998)
Junger, M.C., Feit, D.: Sound, Structures and Their Interaction. Acoustical Society of America, Melville (1993)
Kausel, W., Chatziioannou, V., Moore, T., Gorman, B., Rokni, M.: Axial vibrations of brass wind instrument bells and their acoustical influence: theory and simulations. J. Acoust. Soc. Am. 137, 3149–3162 (2015)
Lambourg, C., Chaigne, A.: Measurements and modeling of the admittance matrix at the bridge in guitars. In: Proceedings of the SMAC 93, pp. 448–453 (1993)
Le Pichon, A.: Method for predicting the acoustic radiation of volumic structures composed of one or several vibrating surfaces. Application to stringed musical instruments (in French). Ph.D. thesis, Université Paris 11 (1996)
Le Pichon, A., Berge, S., Chaigne, A.: Comparison between experimental and predicted radiation of a guitar. Acust. Acta Acust. 84, 136–145 (1998)
Lesueur, C.: Acoustic radiation of structures (in French). Collection de la direction des études et recherches d’Electricité de France. Eyrolles, Paris (1988)
Nederveen, C.J., Dalmont, J.: Pitch and level changes in organ pipes due to wall resonances. J. Sound Vib. 27, 227–239 (2004)
Nightingale, T.R.T., Bosmans, I.: On the drive-point mobility of a periodic rib-stiffened plate. Tech. Rep. NRCC-45609, National Research Council Canada, Ottawa (2006)
Pico, R.: Vibroacoustics of slightly bent cylindrical tubes. influence of the wall vibrations on the oscillations of wind musical instruments (in French). Ph.D. thesis, Universités du Maine (France) et de Valence (Espagne) (2004)
Quaegebeur, N.: Nonlinear vibrations and acoustic radiation of thin loudspeaker-like structures (in French). Ph.D. thesis, ENSTA ParisTech (2007)
Ruzzene, M.: Structural acoustics. Class notes, School of Aerospace Engineering, Georgia Institute of Technology (2003)
Schedin, S., Lambourg, C., Chaigne, A.: Transient sound fields from impacted plates: comparison between numerical simulations and experiments. J. Sound Vib. 221(3), 471–490 (1999)
Skudrzyk, E.: The mean value method of predicting the dynamic response of complex vibrators. J. Acoust. Soc. Am. 67(4), 1105–1135 (1980)
Soedel, W.: Vibrations of Shells and Plates, 3rd edn. Marcel Dekker, New York (2004)
Stepanishen, P.R.: Transient radiation from piston in an infinite baffle. J. Acoust. Soc. Am. 49(5B), 1629–1638 (1971)
Suzuki, H.: Vibration and sound radiation of a piano soundboard. J. Acoust. Soc. Am. 80(6), 1573–1582 (1986)
Suzuki, H., Tichy, J.: Diffraction of sound by a convex or a concave dome in an infinite baffle. J. Acoust. Soc. Am. 70(5), 1480–1487 (1981)
Wallace, C.E.: Radiation resistance of a rectangular panel. J. Acoust. Soc. Am. 51, 946–952 (1972)
Williams, E.G.: Numerical evaluation of the radiation from unbaffled finite plates using the FFT. J. Acoust. Soc. Am. 73, 343–347 (1983)
Williams, E.G.: Fourier Acoustics: Sound Radiation and Nearfield Acoustical Holography. Academic, New York (1999)
Wolfe, J.: Music acoustics web page, the University of New South Wales. http://www.phys.unsw.edu.au/music/ (2007)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
Copyright information
© 2016 Springer-Verlag New York
About this chapter
Cite this chapter
Chaigne, A. (2016). Radiation of Vibrating Structures. In: Acoustics of Musical Instruments. Modern Acoustics and Signal Processing. Springer, New York, NY. https://doi.org/10.1007/978-1-4939-3679-3_13
Download citation
DOI: https://doi.org/10.1007/978-1-4939-3679-3_13
Published:
Publisher Name: Springer, New York, NY
Print ISBN: 978-1-4939-3677-9
Online ISBN: 978-1-4939-3679-3
eBook Packages: Physics and AstronomyPhysics and Astronomy (R0)