CMPT 889: Lecture 8 Digital Waveguides

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1 CMPT 889: Lecture 8 Digital Waveguides Tamara Smyth, tamaras@cs.sfu.ca School of Computing Science, Simon Fraser University February 10,

2 Motion for a Wave For the string, we are interested in the transverse displacement of the wave. Consider the small section of a string shown below θ 2 T y θ 1 x T Figure 1: A short section of a string. The net vertical force on this section is given by the difference between the y components of the tension T 1 : F n = T sinθ 2 T sinθ 1. The mass m of the section is given by m = µ x where µ is the mass per unit length. 1 Tension is the magnitude of the force due to stretching the string CMPT 889: Computational Modelling for Sound Synthesis: Lecture 8 2

3 The wave equation The slope of the string at each end of the section is given by m 1 = y 1 x = tanθ 1 m 2 = y 2 x = tanθ 2 If the displacement of the string from equilibrium is small, then angles θ 1 and θ 2 are small, and the following approximation may be made: sinθ 1 tanθ 1 and sinθ 2 tanθ 2 The net vertical force can therefore be written as F n = T sinθ 2 T sinθ 1 = T (m 2 m 1 ) = T m. By Newton s second law, F = ma, we obtain ( 2 ) y T m = (µ x) t 2 T m x = µ 2 y t, 2 where acceleration is given by a = 2 y/ t 2. CMPT 889: Computational Modelling for Sound Synthesis: Lecture 8 3

4 Taking the limit as x 0 m lim x 0 x = m x = 2 y x 2, we obtain the one-dimensional wave equation given by 2 y t 2 = c2 2 y x 2, where c = T/µ is the speed of wave propagation. CMPT 889: Computational Modelling for Sound Synthesis: Lecture 8 4

5 Travelling Wave Solution The general class of solutions to the lossless, one-dimensional, second-order wave equation can be expressed as where y(t,x) = y r ( t x c ( )+y l t+ x ). c y r (t x/c) right-going traveling waves y l (t+x/c) left-going traveling waves and where y r and y l are assumed twice-differentiable. This traveling-wave solution of the wave equation was first published by d Alembert in Notice the traveling-wave solution of the 1D wave equation has replaced a function of two variables y(t,x), by two functions of a single variable in time units 2, greatly reducing computational complexity. 2 If x is in meters and sound velocity c is in meters per second, x/c is in seconds and the spatial variable x cancels out. CMPT 889: Computational Modelling for Sound Synthesis: Lecture 8 5

6 Sampled Travelling Waves Formally, sampling is carried out by the change of variables x x m = mx t t n = nt where X ct, is the spatial sampling interval. Substituting into the traveling-wave solution of the wave equation gives y(t n,x m ) = y r (t n x m /c)+y l (t n +x m /c) = y r (nt mx/c)+y l (nt +mx/c) = y r [(n m)t]+y l [(n+m)t] Since T multiplies all arguments it is typically suppressed by defining y + (n) y r (nt) y (n) y l (nt) This notation is also used to model pressure in acoustic tubes. CMPT 889: Computational Modelling for Sound Synthesis: Lecture 8 6

7 Traveling Waves A traveling wave is any kind of wave which propagates in a single direction with negligible change in shape. In acoustic wave propagation, pure delays (delay lines), can be used to simulate traveling waves propagating in one direction with a fixed waveshape. Transverse and longitudinal waves 3 in a vibrating string are nearly perfect traveling waves. Plane waves are a class of traveling wave which dominate in cylindrical bores, such as the bore of a clarinet or the straight tube segments of a trumpet. The vocal tract may also be simulated using plane waves, though less accurately. Spherical waves take the place of plane waves in conical bore sections. However, they still travel like plane waves, and we can still use a delay line to simulate their propagation. 3 Recall, transverse and longitudinal waves are waves in which the partical displacement is perpendicular and parallel, respectively, to the direction of the traveling wave. CMPT 889: Computational Modelling for Sound Synthesis: Lecture 8 7

8 Damped Travelling Waves When travelling waves are damped, their losses are distributed along the length of the system. Rather than applying losses at each time-step, they may be accumulated, or lumped, at discrete points along the delay line. If the losses are the same for each frequency, they may be simulated using a simple scaling of the delay line input or output. x(n) z M g M y(n) If losses are frequency dependent they are implemented using a digital filter G(z) with the corresponding frequency response. x(n) z M G M (z) y(n) The input-output simulation is exact, while the signal samples inside the delay line have a slight gain error. If the internal signals are needed later, they can be tapped out using correcting gains relative to the tapping location. CMPT 889: Computational Modelling for Sound Synthesis: Lecture 8 8

9 Losses due to spherical spreading In a spherical pressure wave of radius r, the energy of the wavefront is spread out over the spherical surface area 4πr 2. Since energy is a conserved quantity, the energy per unit area (intensity) of an expanding spherical pressure wave decreases as 1/r 2. This is called spherical spreading loss, which exemplifies an inverse square law. Sound pressure amplitude of a traveling wave is proportional to the square-root of its energy/intensity, leading to an amplitude proportional to 1/r for spherical traveling waves. x(n) z M 1/r y(n) Though delay lines may be used for spherical waves of radius r, there is a gain of 1/r applied to the output/input. Waves propoagation a distance r 0 from the cone apex to a disance r 1 from the cone apex, will experience a pressure scaling of r 0 /r 1. CMPT 889: Computational Modelling for Sound Synthesis: Lecture 8 9

10 Converting Propagation Distance to Delay Length We may regard the delay-line memory itself as the fixed air which propagates sound samples at a fixed speed c (c = 345 meters per second at 22 degrees Celsius). If the listening point is d meters away from the source, then the delay line length M needs to be M = d X = d ct samples, where T denotes the discrete-time sampling interval, and X = ct is the spatial sampling interval. In other words, the number of samples delay is the propagation distance d divided by X = ct, the distance sound propagates in one sampling interval. CMPT 889: Computational Modelling for Sound Synthesis: Lecture 8 10

11 Reflection of Spherical or Plane Waves When a wave reaches a wall or other obstacle, it is either reflected or scattered. A wavefront is reflected when it impinges on a surface which is flat (for plane waves) or curved with the appropriate radius (for spherical waves) over at least a few wavelengths in each direction. A wave is scattered when it encounters a surface which has variations on the scale of the spatial wavelength. Absorption In air, there is always significant additional loss caused by air absorption. Air absorption varies with frequency, with high frequencies usually being more attenuated than low frequencies. Wave propagation in vibrating strings undergoes an analogous absorption loss, as does the propagation of nearly every other kind of wave in the physical world. CMPT 889: Computational Modelling for Sound Synthesis: Lecture 8 11

12 Reflection on a String Reflection at a fixed end If a string is fixed at x = 0, it s displacement y at x = 0 is zero, yielding y = y r (t 0/c)+y l (t+0/c) = 0 y r (t) = y l (t), which says that The reflected wave is equal but opposite to the right traveling wave. Reflection at a free end If a string is free at x = 0, there is no transverse force. Since the net transverse force is proportional to the slope y/ x, y x = x y r(t x/c)+ x y l(t+x/c) = 1 c [y l(t+0/c) y r (t 0/c)] = 0 y r (t) = y l (t). which says that The reflected wave is equal to the incident wave with no change of sign. CMPT 889: Computational Modelling for Sound Synthesis: Lecture 8 12

13 Wave Impedance The wave impedance is a characteristic of the medium in which a wave propagates. It represents the medium s resistance to distortion in the presence of an exterior force. The wave impedance is given by: Z = F v = p U, where v is the particle velocity, p = F/A is acoustic pressure, and U = v/a is the volume velocity (also called airflow). The latter representation is more appropriate for gases such as air. For solids, the characteristic (wave) impedance is also given by Z = ρc, where ρ is the density of the material. For gases in cylindrical tubes with a cross sectional area of A, the characteristic impedance is given by Z 0 = ρc A. CMPT 889: Computational Modelling for Sound Synthesis: Lecture 8 13

14 Reflection due to Change of Impedance It is possible to have a boundary which is neither completely fixed nor completely free. When a wave traveling in a medium with a given characteristic impedance Z 1 confronts a new medium with a different impedance Z 2, the incident wave will be partially reflected with an amplitude and polarity dependent on the impedance of the two mediums. What is not reflected at the boundary is transmitted through to the new medium. medium 1 (Z 1 ) incident wave medium 2 (Z 2 ) transmitted wave x = 0 reflected wave x = J CMPT 889: Computational Modelling for Sound Synthesis: Lecture 8 14

15 Boundary Conditions The pressure and velocity at the junction x = J of both mediums must be equal since they are effectively shared by both mediums (continuity). The pressure p(j) at the junction is given by p(j) = p i (J)+p r (J) where p i (J) is the incident wave at the junction and p r (J) is the reflected wave at the junction. The incident and reflected pressure waves correspond to incident and reflected airflow U by p i (J) = Z 1 U i (J) and p r (J) = Z 1 U r (J), where the negative sign accounts for the fact that airflow is a directional quantity and moves in the direction in which it generates pressure. Volume velocity at the junction U(J) is similarly the sum of the incident and reflected velocity waves and, by incorporating the result for the incident pressure wave, is given by U(J) = U i (J)+U r (J) = 1 Z 1 (p i (J) p r (J)). CMPT 889: Computational Modelling for Sound Synthesis: Lecture 8 15

16 Calculating the reflection coefficient for pressure Recall that the impedance is the ratio of pressure to velocity. At the junction, there is a new medium with an impedance Z 2, which is given by Z 2 = p(j) U(J) = p i (J)+p r (J) (p i (J) p r (J))/Z 1 = Z 1 ( ) pi (J)+p r (J) p i (J) p r (J) The reflection coefficient (the ratio of the reflected pressure wave to the incident wave) is given by p i +p r Z 1 = Z 2 p i p r Z 1 (p i +p r ) = Z 2 (p i p r ) p r (Z 2 +Z 1 ) = p i (Z 2 Z 1 ) p r = Z 2 Z 1. p i Z 2 +Z 1 Therefore, if we know the amplitude of the incident wave p i, and the characteristic impedance of both mediums Z 1 and Z 2, we can determine the amplitude of the reflected wave p r. CMPT 889: Computational Modelling for Sound Synthesis: Lecture 8 16

17 Calculating the reflection coefficient for velocity Similarly, the reflection coefficient for volume velocity is calculated starting with the fact that the new impedance Z 2 is the ratio of the pressure and velocity at the junction. Z 2 = p(j) U(J) = Z U i (J) U r (J) 1 U i (J)+U r (J) We then proceed in calculating the reflection coefficient as follows: U i U r Z 1 = Z 2 U i +U r Z 1 (U i U r ) = Z 2 (U i +U r ) U i (Z 1 Z 2 ) = U r (Z 1 +Z 2 ) U r = Z 1 Z 2. U i Z 1 +Z 2 CMPT 889: Computational Modelling for Sound Synthesis: Lecture 8 17

18 Applying to Reflections on String On a string free at one end, Z 2 = 0. Using displacement velocity as our wave variable, we obtain v r = Z 1 Z 2 = Z 1 = 1, v i Z 1 +Z 2 Z 1 which again says that the reflected velocity wave is equal to the incident velocity wave. The total velocity at the junction is therefore given by v = v i +v r = v i +v i = 2v i. If the boundary of the string is fixed, it s impedance approaches infinity, and Z 2 Z 1, yielding v r = Z 1 Z 2 Z 2 = 1, v i Z 1 +Z 2 Z 2 where Z 1 is considered to be negligible. The velocity at the boundary is therefore v = v i +v r = v i v i = 0. Intuitively, this result is to be expected because the displacement at the fixed end is zero, and therefore the velocity must also be zero. CMPT 889: Computational Modelling for Sound Synthesis: Lecture 8 18

19 Applying to Reflection in a Tube Recall that impedance for air in a confined space is inversely proportional to it s area, that is, the smaller the area, the larger the impedance. For a tube open at one end, we can assume that the wave impedance of open air is negligible to that of the tube cylindrical section. The reflection coefficient for pressure at an open end is given by p r p i = Z 2 Z 1 Z 2 +Z 1 = Z 1 Z 1 = 1. Similarly, for a tube that is closed at one end, we can assume an infinitely small radius and a corresponding wave impedance Z 2 that is very large relative to Z 1, yielding the result p r p i = Z 2 Z 1 Z 2 +Z 1 = Z 2 Z 2 = 1. CMPT 889: Computational Modelling for Sound Synthesis: Lecture 8 19

20 Wave Variables Displacement, velocity and acceleration waves all reflect with the same polarity. That is, if a displacement wave changes sign when reflecting at a boundary, so will velocity and acceleration waves. r x = Z 1 Z 2 Z 1 +Z 2 Force and pressure waves however, reflect with a polarity opposite to displacement, velocity and acceleration. This can be shown using the ratio of the reflected force wave F r to the incident force wave F i. r p = Z 2 Z 1 Z 2 +Z 1 CMPT 889: Computational Modelling for Sound Synthesis: Lecture 8 20

21 The Plucked String When a string is plucked, the finger chooses a point on the string and then displaces it a certain distance. Plucking a string therefore, introduces an initial energy displacement (potential energy). The shape of the string before its release defines which harmonics will be present in the resulting motion. A string plucked at 1/n th the distance from one end will not have energy at multiples of the n th harmonic. The strength of excitation of the m th vibrational mode is inversely proportional to the square of the mode number. A simple example is demonstrated by plucking the string exactly midway between its endpoints. In so doing, we have created a sort of triangle wave, with its corresponding harmonics. CMPT 889: Computational Modelling for Sound Synthesis: Lecture 8 21

22 Figure 2: The motion of a string plucked one-half of the distance from one end. CMPT 889: Computational Modelling for Sound Synthesis: Lecture 8 22

23 String Spectrum Figure 3: Spectrum of a string plucked one-half of the distance from one end. CMPT 889: Computational Modelling for Sound Synthesis: Lecture 8 23

24 Pluck position Take for example the motion of a string plucked one-fifth of the distance from the end. Figure 4: The motion of a string plucked one-fifth of the distance from one end. The motion can be thought of as two pulses moving in opposite directions (see the dashed line). The resulting motion consists of two bends, one moving clockwise and the other counterclockwise around a parallelogram. CMPT 889: Computational Modelling for Sound Synthesis: Lecture 8 24

25 Resulting Plucked String Spectrum The resulting spectrum of the string plucked one-fifth of the distance from one end is given below. Figure 5: Spectrum of a string plucked one-fifth of the distance from one end. CMPT 889: Computational Modelling for Sound Synthesis: Lecture 8 25

26 Digital Waveguides A digital waveguide is a sampled traveling-wave simulation for waves in ideal strings or acoustic tubes. A (lossless) digital waveguide is defined as a bidirectional delay line at some wave impedance R. z N R z N As before, each delay line contains a sampled acoustic traveling wave. However, since we now have a bidirectional delay line, we have two traveling waves, one to the left and one to the right. While a single delay line can model an acoustic plane wave, a digital waveguide can model any one-dimensional linear acoustic system such as a violin string or a clarinet bore. In real acoustic strings and bores, the 1D waveguides exhibit some loss and dispersion so some filtering will be needed in the waveguide to obtain an accurate physical model of such systems. CMPT 889: Computational Modelling for Sound Synthesis: Lecture 8 26

27 Physical Outputs Physical variables (force, pressure, velocity,...) are obtained by summing traveling-wave components. z N Physical Signal z N To determine the value at any physical point, extract a physical signal from a digital waveguide using delay-line taps. The physical wave vibration is obtained by summing the left- and right-going traveling waves. The two traveling waves in a digital waveguide are now components of a more general acoustic vibration. A traveling wave by itself in one of the delay lines is no longer regarded as physical unless the signal in the opposite-going delay line is zero. CMPT 889: Computational Modelling for Sound Synthesis: Lecture 8 27

28 Plucked String Model In this string simulator, there is a loop of delay containing N = 2L/X = f s /f 1 samples where f 1 is the desired pitch of the string and L is the physical length of the string. y(n) + y(n N/2) + z N/2 Bridge -1 (rigid termination) Physical Signal -1 Nut (rigid termination) G(e jω ) Low-pass y(n) z N/2 y(n N/2) What is the impulse response of this structure? Define an input x(n) and an output y(n). CMPT 889: Computational Modelling for Sound Synthesis: Lecture 8 28

Music 206: Digital Waveguides

Music 206: Digital Waveguides Tamara Smyth, trsmyth@ucsd.edu Department of Music, University of California, San Diego (UCSD) January 22, 2016 1 Motion for a Wave The 1-dimensional digital waveguide model