Spherical Waves, Radiator Groups

Transcription

1 Waves, Radiator Groups ELEC-E5610 Acoustics and the Physics of Sound, Lecture 10 Archontis Politis Department of Signal Processing and Acoustics Aalto University School of Electrical Engineering November 30, 2015

2 1 waves Waves, Radiator Groups 2/22

3 1 Radiation A point P can be given in spherical coordinates using radius r angle θ (from z-axis) angle φ (from x-axis) Relation to cartesian coordinates: z θ P. φ y Dipole x = r sin θ cos φ r = x 2 + y 2 + z 2 y = r sin θ sin φ θ = arccos(z/r) x z = rcosθ φ = arctan(y/x) Waves, Radiator Groups 3/22

4 1 II Radiation Calculations for point sources become simpler when switching from cartesian to spherical coordinates set the origin at the source the sound field is symmetric w.r.t. origin, angles θ and φ become irrelevant Dipole Waves, Radiator Groups 4/22

5 1 Radiation Dipole Some definitions related to spherical sound fields: acoustic center is the center of spherical sound waves radiating outward a sound source 1/r-law illustrates the behavior of the sound field w.r.t. distance from the source. in effect, pressure and velocity amplitudes decrease 6 db when the distance doubles 1/r 2 -law illustrates the behavior of the power w.r.t. distance from the source. in effect, intesity decreases 6 db when the distance doubles. Why? Waves, Radiator Groups 5/22

6 1 of a Wave Radiation Dipole The characteristic impedance of a spherical wave can be given as z s = p ( ) ikr u = ρc (R&F:(6.29), FF:(3.45)) 1 + ikr ( k 2 r 2 ) = ρc 1 + k 2 r + 2 i kr 1 + k 2 r 2 What is the limit value far away from the source? when kr z s ρc in other words, the wavefront starts to resemble a plane wave! Waves, Radiator Groups 6/22

7 1 of a Wave II Radiation Dipole Figure: The characteristic impedance of a spherical wave in real and imaginary parts (right) and its relation to the characteristic impedance of a plane wave (left). Waves, Radiator Groups 7/22

8 1 Radiation Dipole Near field: the relation between p and u differs from the plane-wave case close to the sound source when considering the wavelength dimensions of the source some wave components attenuate quickly when moving away from the source Far field: the wave field acts locally as a plane wave Waves, Radiator Groups 8/22

9 1 Radiation Dipole Consider a sphere with radius R. The sphere pulsates, i.e. periodically increases and decreases its radius so that the radial displacement is ξ the movement is sinusoidal, so that radial velocity v = v 0 e iωt frequency of pulsation is ω If R 0, the source becomes an elementary monopole. R ξ Waves, Radiator Groups 9/22

10 Radiation Dipole 1 Created by an The volume velocity (or source strength) created by the pulsating sphere is q 0 = v 0 da = 4πR 2 v 0 A (i. e. surface area times radial velocity). The pressure wave at a distance r is given as p = iωρq 0 4πr ikr e ik(r R) (FF:6.15b) What is the pressure field created by a point source? p = iωρq 0 e ikr 4πr (FF:6.20,R&F:7.4) Waves, Radiator Groups 10/22

11 1 Intensity Created by an Radiation Dipole Active intensity in the radial direction: Re[I] = 1 2 p 2 Re [1/z r ] e r = 1 2ρc p 2 e r (FF:6.27) Integrating Eq. (FF:6.27) over a spherical surface gives P = ω2 ρ q 0 2 ( ) 1 8πc 1 + kr 2 (FF:6.18) which becomes for a point source P = ω2 ρ q 0 2 8πc (FF:6.19,R&F:7.5) What can you say about the point source as a bass source? Also, power is mainly a property of the source. Waves, Radiator Groups 11/22

12 1 Mechanical Radiation Radiation Dipole The mechanical radiation impedance of a surface gives the ratio between the radial velocity and the resulting force exerted on the surface by the fluid property of the vibrating surface and fluid not a property of the actual vibrating object The radiation impedance of a pulsating sphere is given as ( ) (kr) z mrad = 4πR 2 2 ρc 1 + (kr) + 2 i kr 1 + (kr) 2 (1) Remember k = ω/c. For small ω, Im[z mrad ] Re[z mrad ]. Waves, Radiator Groups 12/22

13 1 Mechanical Radiation II Radiation Dipole By rearranging the imaginary part of Eq. (1), one obtains Im[z mrad ] = iω 4πρR3 1 + (kr) 2 Does this impedance look familiar? It is of the form iωm s, where m s = 4πρR3 1 + (kr) 2 is called attached mass. This attached mass represents the inertial effect of the surrounding fluid, and can be approximated for low frequencies as 3 the mass of the fluid replaced by the sphere. Waves, Radiator Groups 13/22

14 1 Dipole Radiation Dipole Placing two elementary monopoles with opposite phases at a distance d between them creates an elementary dipole. A vibrating sphere may be considered as an elementary dipole, if kr 1. We will study the sound field created by an elementary dipole in what follows. d Waves, Radiator Groups 14/22

15 1 Dipole II Radiation Dipole Consider a case where each monopole is placed on the z-axis. P is the observation point at a distance r from the dipole midpoint θ is the angle between P, dipole midpoint, and z-axis x Note the rotational symmetry around z-axis! d z θ r P. y Waves, Radiator Groups 15/22

16 1 Radiation Dipole The expression for the sound pressure at P is ( p(r, θ) = ω2 ρ ) e ikr µ cos θ 4πcr ikr (R&F:7.7) where µ = q 0 d is the dipole moment. What is the sound pressure at the xy-plane? Zero, since θ = 90. In the far field r, so pressure becomes p(r, θ) = ω2 ρ 4πcr e ikr µ cos θ Equations for the particle velocity, characteristic impedance, and impedance can be found in FF:p.115. Waves, Radiator Groups 16/22

17 1 of an Dipole Radiation Dipole The power that an elementary dipole radiates to the far field is given as P = ω4 ρµ 2 24πc 3 (FF:6.35a) What can you say about the bass response of the elementary dipole? - Extremely poor radiator at low frequencies (acoustic short-circuit)! Waves, Radiator Groups 17/22

18 2 Radiator Groups Waves, Radiator Groups 18/22

19 2 Radiator Groups Radiator Groups Sound Pressure Directivity Patterns Wavefield Synthesis Consider a case where a group of equal-phase monopoles are placed on the z-axis, d meters apart. P is the observation point at a distance r from the group midpoint θ is the angle between P, group midpoint, and z-axis x z d d d d d θ r P. y Waves, Radiator Groups 19/22

20 2 Sound Pressure Radiator Groups Sound Pressure Directivity Patterns Wavefield Synthesis The sound pressure at P caused by N monopoles can be given as ] p(θ, r) ( iωρq0 4πr ) e ikr [sin ( Nπd λ sin ( πd λ cos θ) cos θ) (R&F:(7.17)) which consists of sound pressure of a monopole directivity function (note that book version of R&F:7.17 uses a number of 2N monopoles) Waves, Radiator Groups 20/22

21 2 Directivity Patterns Radiator Groups Sound Pressure Directivity Patterns Wavefield Synthesis Figure: directivity patterns created by 7 point sources (illustrated with red circles) [?] lobe number increases with frequency! Waves, Radiator Groups 21/22

22 2 Wavefield Synthesis Radiator Groups Sound Pressure Directivity Patterns Wavefield Synthesis Generally, the radiation patterns of group sources can be varied by varying the phases, amplitudes, and delays of the individual sources.this enables wavefield synthesis: principle: http: // virtual sources inside the room: The same principle can be used also for microphone arrays! acoustic beamforming by summing the mic signals Waves, Radiator Groups 22/22