Part 8. Acoustic Radiators

Transkript

1 Pat 8 Acoustic Radiatos

2 Sphical Wavs Oscillating sphical cavity, a va adius of th oscillating sphical cavity vlocity amplitud of th cavity oscillation a) oscillating cavity b) point souc ( a << λ, >> λ ) v a a Pssu and vlocity distibutions: p () po i( k ωt) v() v o i( k ωt) Balanc of momntum: p ρ v t

3 i( k ωt) p 1 ( ik ) po v t ω i v o i( k ωt) Spcific acoustic impdanc of a sphical wav: Zsph p () p o v() v o iωρ 1 ik lim Z sph ρ c lim 0 Z sph iωρ Bounday conditions: i( ka ωt) v( a) vo v a a iωt ika vo vaa v aa i( ka ωt) pa ( ) po Zsph( a) v( a) iωρav a a ω i t ika po iωρa v a po ωρ i a v a sinc ka << 1

4 Volum vlocity of th souc: So 4 πa v a po ωρ So 4 πi Divgncy: p ( ) 1 ik( 1), wh > 1 p ( 1) ( ) p 1 p ( 1)

5 Bipols and High-Od Radiatos Radiation fild of a sphical adiato: (a) in f fild 0 (b) on th sufac of a igid mio (baffl) 90 So So mio Radiation fild of a bipol: ik ( ωt) ik ( ωt) 1 p () po + p o 1 At a lag distanc fom th dipol ( >> d), + sin θd/ and sin θd/ 1 θ dnots th pola angl d siz of dipol ik ( ωt) o θ p () p D() iksin d iksin d θ / + θ / D( θ ) cos ( ksinθ d/ )

6 Dictivity Pattn of a Bipol d / λ 0. o o 0 o 15 o 30 o d d / λ 0.5 o o 0 o 15 o 30 o d / λ 1 o o 0 o 15 o 30 o

7 Radiation fild of a tipol: ik ( ωt) p () 3 po D() θ D( θ ) iksinθ d/ iksinθ d/ D( θ ) [cos ( ksinθ d/ ) + 1] 3 d full siz of th tipol (th spaation btwn nighboing lmnts is d/)

8 Dictivity Pattn of a Tipol d / λ o 15 o 0 o 15 o 30 o d d / λ o 15 o 0 o 15 o 30 o d / λ 1 30 o 15 o 0 o 15 o 30 o

9 Rayligh Intgal Huygns' Pincipl: Evy point on a wav sufac in tun acts as a simpl souc. Small adiato in font of a baffl: p () ωρso πi i( k ωt) v o( x, y ) souc vlocity distibution is A souc aptu dp() i( k' t) o da ω ωρ v πi ' y da x ' P(, θ ) A θ z a ik' ωρ p () dxdy ' ' v o( x', y') πi ' A

10 (without th i t ω tm) ' ( x x') + ( y y') + z x + y + z Appoximations: Clos to th tansduc: na-fild o Fsnl gion Fa fom th tansduc: fa-fild o Faunhof gion Piston Souc: ik' ωρv () o p dxdy ' ' πi A ' Th acoustic fild of a cicula piston adiato at a/λ 10 z 0a z 10a N z a N a / λ

11 Bam Pofil of a Cicula Piston Radiato at Fou Diffnt Distancs a/λ 10 z 001. N 01. a z 01. N a z N 10a z N 0a ξ /a

12 Fa-Fild, Faunhof Appoximation ik' ωρ p () dxdy ' ' v o( x', y') πi ' A x y ' ( x x') + ( y y') + z x' y' ik x y ωρ ik( x' + y' ) p () dxdy ' ' v ( ', ') o x y πi A ik x y p () ωρ i F(, ) Two-dimnsional Foui tansfom: x y x y 1 ik( x' + y' ) F(, ) dx' dy' v ( ', ') o x y π A 1 x F{ vo( x', y')} v o( kx, ky) dx' dy' v o( x', y') + π A i( k x' k y') y F( x, y ) v o( k x, k y ) In cylindical coodinat systm: x + y ξ x' + y' ξ '

13 ξ / sin θ ' ξ'sinθcosα Hankl tansfom: 1 a π vo( ) ' ' v o( ') π 0 0 k dξ ξ ξ dα ik ξ'cos( α) a vo( k) dξξ ' ' v o( ξ') J0 ( kξ') 0 J0 zoth-od Bssl function of th fist kind ζ dζ ζ J0( ζ ) ζ J1( ζ) 0 J 1 fist-od Bssl function of th fist kind J ( ) 1 ( k a v ) o k a vo k a Th total pssu at a givn point,θ: ik p (, θ ) po D() θ po ½iωρa v o D( θ) J1 ( aksin θ) aksin θ

14 D( ζ ) D( aksin θ ) D( ζ) J1 ( ζ) ζ ak πd/ λ d a piston diamt J 1 ( ζ ) ζ ζ Fist zo: ζ 3.83

15 Dictivity Function of a Cicula Piston Radiato o o 0 o 15 o 30 o d / λ 0.5 o o 0 o 15 o 30 o d / λ 1. o o 0 o 15 o 30 o d / λ 3

16 Half-angl of divgnc θd: Fom fist zo 0 aksin θd 3.83 Fom th half-intnsity (-3dB) point of th bam 3dB aksin θd 1.6 Typical immsion typ ultasonic tansduc: d 1/" f 10 MHz c 1.5 mm/µs ak 50 sin θ d θ d 3dB 1.6 θd ak 0.6 λ a 3dB θd 0.35

17 Na-Fild, Fsnl Appoximation Along th axis ( z): ' ' z ξ + a pz ( ) iωρv o dξ' ξ' 0 ik' ' Chang of vaiabl ' ' z ' ξ + ξ ξ' dξ ' ' d' a + z ik' pz ( ) iωρv o d' z plan wav appoximation ikz ik( a + z z) pz ( ) p [1 ] p p p ρcv o p( z) p cos[ k( a + z z)] p α cosα sin p( z) p sin[½ k( a + z z)] p

18 Maxima and Minima Along th Axis z max 4 a λ (n 1) ( n 1,,...) 4 λ(n 1) a λ n zmin ( n 1,,...) λ n pz ( ) p p 4sin [½ k( a + z z)] a/λ 10 and N a / λ 10aN 4 Nomalizd Intnsity 3 1 1/ z / N Fa-fild distibution along th axis:

19 a lim p ( ) ppk lim ( a + ) ppk N p () pp π Last maximum (n 1), na-fild/fa-fild tansition: 4a λ a zmax ( n 1) N fo a >> λ 4 λ λ Th lagst path diffnc: N maks th tansition btwn th na-fild (Fsnl) and fa-fild (Faunhof) gions Δ' + a a Dstuctiv intfnc stats whn: Δϕ Δ k ' >π a λ > 1 < N d 1/" f 10 MHz c 1.5 mm/µs N 70 mm

20 Simplifid Modl of a Cicula Piston Radiato sachlight modl 1 "sachlight" modl -10 db contou simplifid modl a ξ 0-1 θ -10 db z N

21 Radiation Impdanc Total action foc: F dxdy p( x, y,0) A Radiation impdanc: ik' F ωρ Z dxdy dx' dy' v πi ' o A A Z ρca π [ R ( ka ) ix ( ka )] 1( ) 1( ) ( ) 1 J ka H Rka and X( ka) ka ka ka J 1 H 1 fist-od Bssl function of th fist kind fist-od Stuv function of th fist kind At high fquncis: lim R Z ρca π ak At vy low fquncis: ( ak) 8ak lim Z Z [ i ] ak 0 3π 8 3 lim Im{ Z} iω ρ a aω 0 3

22 1. 1 R(ak) R(ak), X(ak) ak X(ak)

23 Radiatd Pow Fom th fa-fild distibution: ik lim p (, θ ) po D( θ ) po ½iωρa v o D( θ) J1 ( aksin θ) aksin θ Total adiatd pow: π/ P π dθ sin θ I( θ) 0 p I( θ ) o D ( θ) ρc π p π/ o P dθsin θ D ( θ) ρc 0 4 ρca k πv π/ o P dθsin θ D ( θ) a k π/ P ρcπa v o dθsin θ D ( θ) 0

24 D( θ) J1 ( aksin θ) aksin θ ξ J1( sin ) J1( ) R( ) π / d sin ξ θ ξ ξ θ θ 1 0 ξsinθ ξ ξ ak P 1 Rvo

25 Diffaction Coction (two idntical tansducs in a pitch-catch mod) Total foc on th civ: F ( z) dxdy p( x, y, z) A z total distanc ik' ωρ p() dx ' dy ' v o( x ', y ') πi ' D L A ( z) F ( z) F ( z 0) Fo a cicula piston (Lomml diffaction coction): i π/ s D () s 1 [ J ( π s) ij ( π s)] L 0 / 1 / s z/n nomalizd distanc Diffaction Coction s lim DL( s) iπ/ s fa-fild asymptot z / N

26 Point Souc in an Infinit Solid Mdium Stok's Poblm F O P u d u d u do ik d only axial (longitudinal) displacmnt O F P us u s u so ik s only tansvs (sha) displacmnt

27 Abitay Ointation: u s θ u d P O F u (, θ ) u cosθ d do ikd u (, θ ) u sin θ s so iks u do F 4 π( λ+ μ) u so F 4 πμ

28 Dictivity Pattn of a Point Souc in an Infinit Solid Mdium longitudinal sha F

29 Point Souc in/on Infinit Solid Half-Spac, Lamb's Poblm Rcipocity lation : Longitudinal Wav Dictivity R ds u d P θ O Fnomal solid vacuum F adial P θ θd I d u n θ s O θd R dd ud(, θ) un(, θ) F F nomal adial u (, θ ) u ( ) [cos θ R ( θ)cos θ R ( θ)sin θ ] n i dd ds s c sin θ s s sin θ c d u () i u do ikd R dd (0) 1 u (,0) u () n 1 Dd( θ ) [cos θ Rdd( θ)cos θ Rds( θ)sin θ s] i

30 Sha Wav Dictivity Rcipocity lation : u s P θ O Fnomal solid vacuum θ P F azimuthal I s θs u n O θs θ d R ss R sd us(, θ) un(, θ) Fnomal Fazimuthal u (, θ ) u ()[sin θ + R ()sin θ θ + R ()cos θ θ ] n i ss sd d c sin θ d d sin θ cs ui() uso iks D ( θ ) sin θ + R ( θ)sin θ + R ( θ)cosθ s ss sd d Ds ( θ 0) 0 Ds ( θ π / ) 0

31 Dictivity Pattn of a Nomal Point Foc on a Solid Half-Spac a) longitudinal o o 0 o 15 o 30 o b) sha o o 0 o 15 o 30 o