Abstract
Wave analysis of the acoustic pressure field is shown to be a useful complement to modal analysis. In this chapter, many intuitive results are directly obtained in the time domain, by considering an impulse source and successive wave reflections at boundaries, for a one-dimensional medium. In the frequency domain, it is shown that the concept of input impedance (or admittance), of current use in musical acoustics, can be viewed as a generalized frequency response to a sinusoidal source. Furthermore, it is shown how infinite series of modes (resp. waves) can be avoided by using a closed-form of the responses, both in the time and frequency domain. For the sake of simplicity, 2D and 3D media are not considered, but the direct relationship between modes and waves is established for the particular case of a 1D medium. The chapter is mainly based on the example of a cylindrical tube, but all the results can be transposed to the case of a homogeneous string.
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Notes
- 1.
The mathematically accurate notation would be \(f^{+}(t - x/c) = \left [f^{+} {\ast}\delta _{x/c}\right ](t)\), but for numerous successive reflections, it would be very heavy. Let us highlight also that the FT of this expression is \(F^{+}(\omega )\exp (-j\omega x/c).\)
- 2.
This means that radiation impedance is ignored. The equivalent is a string fixed on a perfectly rigid support, acoustic pressure and string velocity being analogous.
- 3.
This impulsive outgoing wave cannot be realized with a small lateral piston, as the flow pulse divides into two: an outgoing wave (or more generally right wave) and an incoming one (or more generally left one). Thus the response to the outgoing wave must include the incoming wave coming back from x = ℓ, but also the incoming wave directly from the source (about this matter, see [1]).
- 4.
The Fourier transform of Eq. (4.1) is calculated by doing the following variable change in the integral: \(t^{{\prime}} = t \mp x/c\).
- 5.
At the open end of a pipe, the impedance is set by the fact that the external medium is passive. It is called radiation impedance. We will come back to this quantity in the fourth part of the book, but for now it is supposed to be zero for the sake of simplicity: the external medium is so huge compared to the pipe that the pressure does not vary much there.
- 6.
Actually, Chap. 9 will explain more precisely that there are frequencies canceling the imaginary part of admittance.
- 7.
At x = x s , it is written:
$$\displaystyle{ P(x_{s},\omega ) = A^{+}\left [1 + B^{+}\right ] = A^{-}\left [B^{-} + 1\right ]\; Z_{c}U_{s}^{+}(\omega ) = A^{+}\left [1 - B^{+}\right ]; Z_{c}U_{s}^{-}(\omega ) = A^{-}\left [B^{-}- 1\right ]. }$$The total admittance at the source point is the sum of the upstream admittances on both sides:
$$\displaystyle{ Z_{c}\frac{U_{S}} {P_{S}} = \frac{1 - B^{+}} {1 + B^{+}} -\frac{B^{-}- 1} {B^{-} + 1}. }$$This leads to the value of \(P_{s} = A^{+}\left [1 + B^{+}\right ]\) as a function of U S . This yields A +, and the final result thanks to (4.36).
- 8.
This condition assumes that one of the ends is absorbing: thus, either \(\left \vert R_{0}\right \vert\) or \(\left \vert R_{\ell}\right \vert\) is smaller than unity. Equation (4.38) remains valid when damping occurs during propagation (This is studied in details in the next chapter), i.e., when jk is replaced by jk +α, where α is positive. This is a third case, where \(\left \vert g_{\mathrm{RT}}(\omega )\right \vert <1\). If there is an energy source at one extremity, or during propagation, this condition implies that globally the round trip must be dissipative, damping prevailing over energy supply.
- 9.
This calculation is equivalent to write the solution at both the right and left side of the source, together with the continuity of the solution at x = x s , and the jump of the first spatial derivative due to the function δ(x − x s ), as it has been done for the time variable of the single-degree-of-freedom oscillator (Chap. 2 Sect. 2.2.2).
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Chaigne, A., Kergomard, J. (2016). Waves. In: Acoustics of Musical Instruments. Modern Acoustics and Signal Processing. Springer, New York, NY. https://doi.org/10.1007/978-1-4939-3679-3_4
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