Abstract
The aim of this introductory chapter is to summarize the main mechanical models which describe the physics of musical instruments and of their constitutive parts. These models derive from the general principles of the mechanics of continuous media (solids and fluids). In this framework, the phenomena are described at a scale of the so-called particle, or element, whose dimensions are infinitesimal in the sense of differential calculus. Particular emphasis is given to the bending of structures and to the equations of acoustic waves in air, because of their relevance in musical acoustics. One section is devoted to the excitation mechanisms of musical instruments. Analogies between vibrations of solids (such as strings) and fluids (in pipes) are underlined. Elementary considerations on the numerical formulation of the models are also given. This chapter should be considered as a summary which contains reference results to help in reading the rest of the book. It focuses on the origin of the equations and on their underlying assumptions, living aside the complete demonstrations.
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Notes
- 1.
Equation (1.9), written here in Cartesian coordinates, can be generalized to other coordinate systems.
- 2.
Here, we do not treat the cylindrical shells theory, which naturally applies to wind instruments, because it requires significant developments that are beyond the scope of this book. Nevertheless, we provide valuable references in Chap. 13 which deals with sound–structure interaction.
- 3.
n is number of moles, and M the molar mass.
- 4.
This difference in notation for pressure and density, although it is apparently illogical, is convenient for the following of the statement.
- 5.
We could also add a source to Eq. (1.97): it would be a heat source, varying in time, which does not occur in musical instruments.
- 6.
Exponentially time-increasing or time-decreasing solutions may also exist if the constant is positive (but this is rare). The case of complex values is treated in the following section. Furthermore we can continue this operation by separating the variables of space. This will happen several times in this book.
- 7.
The average power \(\mathcal{P}_{m}\) is also called active power . A “reactive” power is also defined by
$$\displaystyle{ \mathcal{P}_{r} = \frac{1} {2}\mathfrak{I}m(pv^{{\ast}}) = -\frac{1} {2}\left \vert p\right \vert ^{2}\mathfrak{I}m(Y ) = \frac{1} {2}\left \vert v\right \vert ^{2}\mathfrak{I}m(Z). }$$(1.130)We choose arbitrarily its sign, which will be positive or negative depending on whether the system is dominated by stiffness or mass (we do not go further here, because the chosen quantities are formal).
- 8.
Assuming a given function of time for the displacement, the source term in Eq. (1.111) is entirely known. A “realistic” simple function is, for example, hH(t), where h is the amplitude and H(t) the unit step function (or Heaviside step function). In practice, this means that the membrane is suddenly moved, then blocked. In this case, the source term in (1.111) becomes \(-\rho hS_{m}\delta (\mathbf{r - a)}\partial \left [\delta (t)\right ]/\partial t\), since δ(t) is the derivative of H(t). In the next chapters, a particular case of elementary source called Green’s function , where the source is written \(\delta (\mathbf{r - a)}\delta (t - t_{0})\), will be examined in details. To achieve it in our “thought experiment,” the velocity should be a step function, and therefore, the displacement should increase indefinitely, which is not realistic! Another way to obtain this Green’s function is to write the acoustic wave equation in terms of velocity potential [see Eq. (1.105)]. In this latter case, the source term becomes hS m δ(r − a) δ(t).
- 9.
A wave equation or a boundary condition including a source is called heterogeneous. It can be shown that it is always possible to transform a heterogeneous boundary condition into a homogeneous one by changing the wave equation.
- 10.
The electroacoustic analogy called “acoustic impedance” associates acoustic pressure and velocity to electric voltage and current, respectively. Ohm’s law states that between two points the difference in one of the quantities is proportional to the other quantity.
- 11.
We will encounter several times this concept of lumped-element systems. Notice that the finite difference calculation (Sect. 1.7.1) is based upon the division of a continuous system into lumped elements.
- 12.
Sometimes we will also use an acoustic impedance called specific defined by the ratio pressure/velocity (see Sect. 1.2.4). This choice is convenient for some problems of unbounded media, or for energy transmission between two media with different sound speed or density.
- 13.
This table has some specificity with regard to the dimensions. The quantity f ext, for example, is a force per unit length, whereas the quantity F in Eq. (1.110) is a force per unit mass. In addition, the equation of vibrating strings is written in terms of velocity: this is rather unusual, but it allows to easily highlight some analogies. Finally, the wave equations are written here for a homogeneous medium, although we will have to deal with heterogeneous strings and horns, for which the analogies remain valid.
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Chaigne, A., Kergomard, J. (2016). Continuous Models. In: Acoustics of Musical Instruments. Modern Acoustics and Signal Processing. Springer, New York, NY. https://doi.org/10.1007/978-1-4939-3679-3_1
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