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Page 00000001 Modelling a Drum by Interfacing 2-D and 3-D Waveguide Meshes Marc Aird* Dept. Of Mathematical Sciences University Of Bath Joel Lairdt Digital Music Research Group University Of Bristol John flitcht Media Technology Research Centre University Of Bath Abstract The Digital Waveguide Mesh is an extension of the 1-D Digital Waveguide technique (Smith, 1992), and offers the ability to model two and three dimensional wave propagation (Van Duyne & Smith, 1993; Van Duyne & Smith, 1996; Fontana & Rocchesso, 1999). So far 2-D triangular meshes have been used to model wave propagation in membranes (Fontana & Rocchesso, 1998; Laird et al., 1998), while 3-D meshes have been proposed in room acoustics simulations (Savioja et al., 1994). In this paper we propose a model of a drum based upon a 2-D mesh for the drum skin and a 3-D mesh to model the interior air cavity. We also attach rimguides (Laird et al., 1998) to each of these structures to model correctly the circular boundary for low frequencies. Due to the differences in wave speed through the two media, the membrane mesh will be far denser than the 3 -D mesh, with non-integer ratios between mesh radii, and hence interfacing the two mesh structures is non-trivial. We consider the use of low-order approximations to pressure fronts in the air and their inverse operations on the membrane to model the contributions from one medium to the other and vice versa. We also propose extensions to the work to include higher order approximations at the interface using interpolation, and the building of an interface between the air and the drum shell. 2 Digital Waveguide Meshes 2.1 The Lossless Scattering Junction We create waveguide meshes by constructing lattices of unit length waveguides interconnected at lossless scattering junctions (Van Duyne & Smith, 1993). By definition, the velocities of each waveguide at the junction should all be equal to the junction velocity, and the forces must sum to zero. For a junction of N waveguides, each with impedance Ri, where i = 1,.., N it can be shown that the junction velocity vj may be written as, _2(Ei=1 R ) v = N J C 1 Ri (1) where v+ is the incoming velocity from the ith waveguide. Furthermore we can write the outgoing v- velocity along the ith waveguide as v- = vj - v+. (2) A special case arises when the impedance of each waveguide is the same in which case we have, N 2 i=1 (3) 1 Introduction Digital waveguide meshes model wave propagation through 2-D surfaces and 3-D spaces. So far 2-D and 3 -D meshes have been implemented separately to model ideal membranes or rooms. However, in both instances accurate modelling at the mesh boundaries is nontrivial. In this paper, by constructing a model of a drum, we also create a framework for the modelling of acoustic spaces, terminated by flexible 2-D boundaries. * firstname.lastname@example.org t email@example.com Sjpff@maths.bath.ac.uk 2.2 Choosing Mesh Structures Waves of any frequency travel through the air at the same speed, and this is also the case in an ideal membrane; however all digital waveguide meshes suffer from dispersion, where the speed of wave travel decreases with frequency (Van Duyne & Smith, 1993; Van Duyne & Smith, 1996; Fontana & Rocchesso, 1998; Fontana & Rocchesso, 1999). It is well known that direction independent dispersion error is desirable for waveguide meshes, and this is achieved in 2-D by using a triangular mesh, but this property is harder to attain using 3-D meshes. The 3-D tetrahedral mesh was introduced in (Van Duyne & Smith, 1996) and offers slightly better dispersion characteristics than the 3-D rectilinear mesh, as well as multiply free implementation. However for this study, we restrict ourselves to rectilinear
Page 00000002 Rm R Rm^~~^^ Rm 'Rm m Ra Wy 1 -'I^ ^-< ' -ay ay = sinmc a~y ax = cosc wx We Figure 1: Membrane mesh scattering junction. structures for the comparative ease afforded in the interfacing procedure. In further studies it is envisaged that a tri-linearly deinterpolated mesh structure will be introduced to replace the rectilinear mesh (Savioja, 1998). This will have little bearing on the interfacing method as the nodes have the same spatial positions. In three dimensions the deinterpolated mesh involves the consideration of a hypothetical mesh of 26-port junctions, where waveguides extending in diagonal directions are spread (deinterpolated) on to their surrounding nodes. Although computationally expensive the deinterpolated 3-D mesh offers near direction independent dispersion error through every plane of view. 3 Building The Drum Model 3.1 Interfacing The Meshes We consider building the drum model using a triangular mesh for the drum skin and a 3-D rectilinear mesh for the air. Due to differences in the mesh geometries the nodes in the membrane mesh will not in general be spatially positioned above nodes in the underlying air mesh. Furthermore, the speed of wave travel in the air will be far greater than that over the membrane, resulting in a greater node density in the membrane mesh. Hence in order to pass contributions from membrane to air and vice versa it would become necessary to increase the complexity of the air mesh by using an interpolation technique, while decreasing the complexity of the membrane mesh by using a deinterpolation method. In order to pass contributions from the membrane to the air and vice versa, where an impedance discontinuity occurs, we use a triangular mesh of 7-port scattering junctions. That is, of the 7 waveguides, 6 carry the membrane impedance Rm and are placed in a triangular geometry in the 2-D plane of the membrane, while the 7th waveguide extends downwards, or upwards, into the air, and has impedance Ra, the impedance of air. This junction structure is described graphically in Figure 1. To create the interface from skin to air we make the assumption that the output pressure at an air mesh node represents a constant output pressure over an element of area A = l, where la is the length of a unit waveguide in the air mesh, and where the element of Figure 2: Calculating Bilinear Interpolation Coefficients. area is centred over the air mesh node. We then assume that it is this pressure value which is passed to the vertical input of the membrane node. This pressure is converted to a velocity by dividing by the air impedance Ra (Hall, 1987). However, this represents the velocity of an air element of area A, and if we are passing this velocity to, say, n membrane nodes, then each membrane nodes air impedance must be scaled by A, /n in order to preserve volume velocity. We may now compute the membrane junction velocities, using (1), as 2(Rm v +1 R+A Va+) 6Rm + A (4) where each vt is the incoming velocity from a neighbouring membrane node, while v+ is the incoming velocity from the air. We now consider the interface from skin to air. Having calculated the reflected velocity from each of the membrane nodes using (2) we convert back to a pressure by multiplying by Ra. We then place each pressure at the input of the relevant air mesh node. This is equivalent to attaching n vertical waveguides to the air mesh node, and thus the junction pressure equation looks like, 5 + (1 p i i= Pi nPm 3 (5) where p+ is the sum of the n inputs from the membrane mesh. Note that each waveguide has equal impedance, so the junction pressure equations takes the form of (3). The node nominally connects six waveguides, and hence the impedance at the vertical input from the membrane must be split between its n inputs so as not to cause an impedance discontinuity. Now, each of the n inputs from the membrane are stored in order to compute, using (2), the reflected pressures back in the direction of the membrane. In this way the process repeats. This interfacing procedure could be easily thought of as an interpolation/deinterpolation process, where in this case the interpolation is zero-order, resulting in representing the incoming pressure front to the membrane by a surface which is not continuous. Possible
Page 00000003 Air Mesh Rimguide Membrane Mesh 0 100 200 300 400 500 600 700 800 900 1000 Frequency(Hz) Figure 4: Spectral plot of Membrane only. Figure 3: Membrane mesh with underlying air mesh. extensions to this model could be to use a higher order interpolation such as bilinearly interpolating (Savioja & Valamaki, 1997) the four air mesh nodes at a membrane node position as shown in figure 2. This would result in an approximate pressure front which was continuous, although with discontinuous derivatives. 3.2 Using Reflection Filters and Rimguides To complete the model we consider terminating each mesh rigidly and model the circular boundary accurately for low frequencies by using rimguides (Laird et al., 1998). This involves extending the mesh from its outermost points by non-integer length waveguides, each consisting of an integer delay and an allpass filter to model the fractional part. The rimguide impedances are adjusted so that they cancel out the impedance mismatch between perimeter nodes and nodes within the mesh. Note also that the rigid termination gives an inverting reflection for the velocity waves on the membrane, but a non-inverting reflection of pressure waves in the air mesh. Figure 3 shows the complete model in plan view, including the rimguides and both mesh structures. Lastly we would add an IIR lowpass filter to model energy losses at the boundary of each mesh. 4 Results Of The Simulation We considered simulating a drum of radius r = 0.2m, with membrane density a = 0.262kg/m2, tension of T = 1850N/m, and at a sample rate of F = 11025Hz. We present a spectral analysis of the results in the lossless case, concentrating our analysis at the first five resonant modes of the membrane. Figure 4 shows the impulse response of the membrane excited at the centre node so as to stimulate only the central modes of vibration. The membrane model is as in (Laird et al., 1998), where the rimguides will help us to accurately model the modes of vibration for low frequencies. 0 100 200 300 400 500 600 700 800 900 1000 Frequency(Hz) Figure 5: Air Mesh only with depth 0.5m. 1.41 1------------1----- ~3 0.8 bC 0.6 0.2 II I.I ~ L~ ~u~ nr. lu 0 100 200 300 400 500 600 700 Frequency(Hz) 800 900 1000 Figure 6: Complete Drum with depth 0.5m. ' 0.5 - bCo 0.4 - 0.3 - 0.2 -0.1-, I i l ~ S.......-. 0 100 200 300 400 500 600 700 Frequency(Hz) 800 900 1000 Figure 7: Complete Drum with depth 1.0m.
Page 00000004 Shown in figure 5 is the impulse response of the cylindrical air column modelled by the air mesh, when excited at the central mode closest to the skin. In this case the membrane has been forced to zero, but the interface procedure still acts and the membrane is employed as a perfect reflector. Upon connecting the membrane to the air column we notice a significant change in the spectrum. Figure 6 shows the output of a membrane attached to an air column of depth 0.5m. Notice how the fundamental frequency is lowered, and also how the resonant modes of the air alone are also prevalent. Finally, figure 7 shows the output of the same membrane, this time attached to an air column of depth 1.0m, and an expected (Fletcher & Rossing, 1991) lowering a reordering of the partials is observed. 5 Conclusions The method developed allowed a realistic model for a drum to be built from quite simple constructs as waveguide meshes and at a low sample rate. The interfacing technique used is not limited to models of drums, but potentially to any acoustics simulation where a 3-D space is terminated by vibrating surfaces. A potential comparison could be made to Finite Element Methods, where adaptive meshes are used to obtain increased accuracy at places such as the boundaries, or at corners. With this model, we are able to model the boundary carefully, while retaining regularly spaced meshes in the interior of the model. 6 Future Work Continued research by the authors is concentrating on manipulating mesh structures which model wave propagation in stiff surfaces. These, together with the improved deinterpolated 3-D mesh structures, could be interfaced to model acoustic spaces terminated by flexible yet stiff surfaces such as glass or wood. Furthermore, the introduction of higher order interpolations could increase the accuracy in the interface. In the context of the current work, improved methods to model the lowpass losses and extensions to the model to include diffusion as presented in (Laird et al., 1999) would increase the accuracy of the model, while the flexibility of the method could easily be extended to model drums of many shapes and sizes, the construction of abstract instruments, or in the area of instrument morphing. References Fletcher, N. H., & Rossing, T. D. 1991. The Physics Of Musical Instruments. Springer-Verlag, New York. Fontana, F., & Rocchesso, D. 1998. Physical Modelling of Membranes for Percussion Instruments. Acustica - Acta Acustica, 84, 529-542. Fontana, F., & Rocchesso, D. 1999. Signal-Theoretic Characterisation of Waveguide Mesh Geometries for Models of Two-Dimensional Wave Propogation in Elastic Media. IEEE Trans. on Speech and Audio. Hall, D. E. 1987. Basic Acoustics. Harper and Row, New York. Laird, J., Masri, P., & Canagarajah, N. 1998 (Nov). Efficient and Accurate Synthesis of Circular Membranes Using Digital Waveguides. In: IEE Colloqium: Audio and Music Technology: The Challenge of Creative DSP. Laird, J., Masri, P., & Canagarajah, N. 1999. Modelling Diffusion at the Boundary of a Digital Waveguide Mesh. In: ICMC Proceedings. Savioja, L. 1998 (Sept). Improving the ThreeDimensional Digital Waveguide Mesh by Interpolation. In: Nordic Acoustical Meeting (NAM 98). Savioja, L., & Valamaki, V. 1997. Improved DiscreteTime Modeling of Multi-Dimensional Wave Propogation using the Interpolated Digital Waveguide Mesh. In: 1997 IEEE International Conference on Acoustics. Savioja, L., Rinne, T., & Takala, T. 1994. Simulation of Room Acoustics with a 3-D Finite Difference Mesh. In: ICMC Proceedings. Smith, J. 0. 1992. Physical Modelling Using Digital Waveguides. Computer Music Journal, 16(4). Van Duyne, S. A., & Smith, J. 0. 1993. Physical Modelling with the 2-D Digital Waveguide Mesh. In: ICMC Proceedings. Van Duyne, S. A., & Smith, J. 0. 1996. The 3-D Tetrahedral Digital Waveguide Mesh with Musical Applications. In: ICMC Proceedings.