Duct Acoustics. Chap.4 Duct Acoustics. Plane wave

Transcription

1 Chap.4 Dut Aoustis Dut Aoustis Plane wave A sound propagation in pipes with different ross-setional area f the wavelength of sound is large in omparison with the diameter of the pipe the sound propagates as an one-dimensional wave ( λ>>d 1-d wave) Area A Area A1 T R x x x p ei t x / R e i t x / Te i t x / Aeroaoustis 8 in x in x -1- : 입사, R : 반사, T : 투과

2 Transmission & Refletion of Plane Waves Dut Aoustis The mass flux into the juntion must equal the mass flux out A1u1 Au The veloity must equal at both sides of the juntion A1 R A T Energy flux)in = Energy flux)out A1 p1' u1 A p' u The pressure of both sides of juntion is ontinuous R T Aeroaoustis 8 --

3 Transmission & Refletion of Plane Waves Dut Aoustis The amplitudes of other wave, R and T,are an be solve from above the relations R A A 1 1 A A, T A 1 A1 A The transmission loss, L T is symmetri in A 1 and A L T 1log 1log 1 1 inident power transmitted power A1 AT 1log 1 A1 A 4A A 1 Aeroaoustis 8-3 -

4 Transmission & Refletion of Plane Waves Dut Aoustis A single expansion-hamber silener The simple muffler that is a used in ar silener onsists of inlet and outlet pipes with ross-setional area A1, and expansion hamber between them of ross-setional area A and length l l Area A1 Area A1 B R C x x Aeroaoustis 8 T Area A x l -4-

5 Transmission & Refletion of Plane Waves Dut Aoustis The first area hange ours at x= and the seond ours at x=l. p ei t x / R ei t x / in x Bei t x / Ce i t x / in x l Te i t x / in l x The ondition of ontinuity of mass flux, A1 R A B C A1Te i l / A Be i l / Ce i l / at x at x l The ondition of ontinuity of pressure R B C at x Te i l / Be i l / Ce i l / at x l Aeroaoustis 8-5-

6 Transmission & Refletion of Plane Waves Dut Aoustis The algebrai equation when solved for R and T A1 A l i sin A A1 R A A l os l / i 1 sin A A1 ei l / T A A l os l / i 1 sin A A1 However, the simple silener does not redue the total energy of sound in the system. R T Reduing the aousti energy of transmitted wave nreasing in the refleted wave Sound absorbing material redue the aousti energy by onverting it into heat or vibration Aeroaoustis 8-6-

7 Transmission & Refletion of Plane Waves Dut Aoustis The transmission loss, LT is 1log 1 T The transmission loss is maximum at frequenies for whih sin l / 1 i.e. L T L T 1log A A 3 5,,...et l l l 1 A A 1 l sin The effet of expansion ratio m A A 1 m inrease L T (db) frequeny Aeroaoustis 8-7 -

8 Transmission & Refletion of Plane Waves Dut Aoustis Note Tuning for dominant frequenies of noise Theory work for only λ d Low frequeny wave only High frequeny waves behave like 3-D Also, the geometrial shape of the dut is not important (provided the area hange ours in a distane short in omparison with the wavelength Aeroaoustis 8-8-

9 Transmission & Refletion of Plane Waves Dut Aoustis Ref. Theoretial and experimental investigation of mufflers with omments on engine-exhaust muffler design, Davis et al. NACA 119(1954) Effet of expansion hamber ratio Effet of expansion hamber shape Aeroaoustis 8-9 -

10 Transmission & Refletion of Plane Waves Higher order modes a x a x1 x3 As an illustration, the sound of frequeny ω in a rigid walled dut of square ross-setion with sides of length a is onsidered p x, t f x1 g x h x3 ei t With substitution for p into the wave equation, f g h f g h Aeroaoustis 8-1 -

11 Transmission & Refletion of Plane Waves Higher order modes Sine a wall boundary ondition is applied, funtion f is derived like this m x1 f x1 A1 os, a for some integer m Similarly funtion g is derived like this n x1 g x A os, a for some integer n Finally, funtion h is derived to the propagation form h x3 Amne ik mn x3 Bmne ikmn x3 km n a m n The axial phase speed, p=ω/kmn is now a funtion of the mode number and the propagation of a group of waves will ause them to disperse. Aeroaoustis

12 Transmission & Refletion of Plane Waves Higher order modes The pressure perturbation in the (m,n) mode has the form m x1 n x ik x ik x i t p x, t os os Am ne mn 3 Bm ne mn 3 e a a When kmn is real, the pressure perturbation equation represents that waves are propagating down the x3 axis with phase speed. When kmn is purely imaginary, i.e. exeeds the ut-off frequeny, the strength of mode varies exponentially with distane along the pipe. Suh disturbanes are evanesent Aeroaoustis 8-1 -

13 Transmission & Refletion of Plane Waves Pipes of varying ross-setion Wave equation A x x f the pipe diameter is small in omparison with both the aousti wavelength and the length sale over whih the ross-setional area hange, most partile motions are longitudinal. Conservation of mass A ua t x Linearized momentum equation is Modified wave equation Aeroaoustis 8 u p t x A p p A t x x

14 Transmission & Refletion of Plane Waves Pipes of varying ross-setion Appliation to the exponential horn Evaluation of the ase of exponential horn whih ross-setional area defined as, A(x)=Aeαx For suh an area variation of wave equation simplifies to 1 p p p t x x The pressure perturbation in sound waves of frequeny ω then has the form x i t kx i t kx p x, t e Ae Be Disturbane with ω > α/ propagates and the pressure but not the energy flux attenuates during propagation, while lower frequeny modes are ut-off Aeroaoustis

15 Transmission & Refletion of Plane Waves Normal transmission Physis at the interfae When a sound wave rosses an interfae between two different fluids some of the aousti energy is usually refleted. e i t x Te i t x 1 R e i t x, interfae x 1, 1 There are two boundary onditions The pressure on the two sides of the boundary must be equal The partile veloities normal to the interfae must be equal T Aeroaoustis 8 f

16 Transmission & Refletion of Plane Waves Normal transmission The pressure must be equal at the interfae : +R=T The partile veloities normal to the interfae must be equal R T 11 The result pressure oeffiients, R and T, are determined with R T 11 Veloity Transmission Coeffiient : T / 11 / 11 The energy flux of the inident wave per unit ross setional area is equal to that of the refleted and transmitted waves 11 R T Aeroaoustis

17 Transmission & Refletion of Plane Waves Normal transmission Refletion from a high and low impedane fluid A typial example is aerial sound waves inident onto a water surfae. (ρ ρ11 ) R, T Veloity transmission oeffiient 11 so, the transmission wave arries negligible energy Aeroaoustis

18 Transmission & Refletion of Plane Waves Normal transmission Refletion from a high and low impedane fluid n the opposite ase, for sound in water inident onto a free surfae with air, the refleted and transmitted waves are R, T The aousti energy is totally refleted Aeroaoustis

19 Transmission & Refletion of Plane Waves Sound propagation through walls Effet of a wall in transmission A sound wave normally inident on a plane material layer partitioning a fluid whih has uniform aousti properties, ρ Some sound will be refleted from the layer and some will be transmitted through the wall Aeroaoustis

20 Transmission & Refletion of Plane Waves Sound propagation through walls There are two boundary onditions that must be satisfied at all times and points The veloity of the wall must be equal to wave of eah side A pressure differene aross the wall in order to provide the fore neessary to aelerate unit area of the surfae of material By ontinuity the veloity of wall, e i t T i t u R e The pressure differene is the net fore of mass per unit area of the wall p1' R e i t p' Te i t Aeroaoustis 8 R T ei t m u m i T ei t t - -

21 Aeroaoustis Transmission & Refletion of Plane Waves Sound propagation through walls The result pressure oeffiients, R and T, are determined with Surfae mpedane Energy transmitted m i m i R m i T m i T R u p m i m i R R u p ' ' m T

22 Transmission & Refletion of Plane Waves Sound propagation through walls The transmission loss is dependent on the frequeny ω. 4 LT 1 log 1 4 m 1 For high frequeny(ωm ρ), the sound waves mostly refleted For low frequeny(ωm ρ), the sound waves mostly travels through the wall with very little attenuation Low frequeny waves get through a massive wall easily, while high frequeny waves are effetively stopped Aeroaoustis 8 - -

23 Transmission & Refletion of Plane Waves Sound propagation through walls Example) Attenuation by a wall m 5 kg m 41 m kg se Transmission loss L T 1log 1 T L T 1log m 1 for f 1Hz for f 1kHz L T 1dB L T 5dB Aeroaoustis 8-3 -