Part 3 REFLECTION AND TRANSMISSION

Part 3 REFLECTION AND TRANSMISSION

Normal Inidene Inident Wave Refletion ρ, 1 1 ρ, Transmission u e i ( k1 x t ) i A ω ( 1 ) i τ 1 1 ei k i uz i iωaz i x ω t u e i ( k 1 x t ) r A ω ( 1 ) r τ r u r Z1 iωar Z1 ei k x ω t t e i ( k x t ) ( ) t τ ei k t uz t iωaz t x ω t u A ω ( physial sign onvention) Boundary Conditions: for any value of t at x ui + ur ut τ i +τ r τ t Ai Ar At A Z + A Z A Z i 1 r 1 t

Refletion/Transmission Coeffiients Rd Ar Ai Td At Ai Z 1 + R d Td 1 Rd Td Z 1 displaement: stress: Rd Td Rs Ts Ar Z Z 1 Ai Z + Z1 At Ai Z1 Z1 + Z τr Z Z 1 τi Z + Z1 τ t τi Z Z1 + Z steel-water interfae ( ρ 46.5 16kg/m ss s, 6 ρ 1.5 1 kg/m s ) w w a) p i steel water p t p r b) water steel p i p t p r

Power Coeffiients: Pr + Pt Pi (Instantaneous) Intensity: Ir + It Ii τ Zv I τ v Zv ωzu Z Z1 Z1 Z1 + Z Z1 Z + Z1 Z1 + Z Speial ases: solid/vauum ( Z ) Rd solid/rigid ( Z R 1, T, R 1, T s ) d s s Rd R 1, T, T s d s Shear wave at normal inidene: displaement: Ar Zs Z R s1 d Ai Zs + Zs1 Td t Z A s1 Ai Zs1 + Zs stress: Rs τr Zs Z s1 τi Zs + Zs1 Ts τt Z s τi Zs1 + Zs

Impedane-Translation Theorem Z 1 inident wave refleted wave Z input A + d Z o,k o A_ x Z load Z transmitted wave τ ( x) A exp( ik x) + A exp( ik x) + o o τ/ x 1 v ( x) [ A+ exp( ikox) A exp( ikox)] iωρ Z The input impedane of the layer: o o Zinput τ () A A Z + + o v() A+ A Zload ik d τ ( d ) o Z A + e + A e v( d) A e o A e ik d o ik d ikod + o A Z o o load e ik d Z ik d o e + + A Z ikod ikod load e Zo e

Translation Formula: os( ) sin( ) os( ) sin( ) Zload kd o izo kd Z o input Zo Z o kd o iz load kd o Refletion Coeffiient: R Zinput Z1 Zinput + Z1 Z Z load Immersed/Embedded Layer: Z Z 1 R itan( k od)( Zo Z 1 ) itan( k )( od Zo + Z1 ) ZoZ1 T (1 R ) R ξsin( kd o ) ξ sin ( kd o ) + 1 T 1 ξ sin ( kd o ) + 1 impedane ontrast: ξ ½ Zo/ Z1 Z1/ Zo

Refletion/Transmission at a Layered Interfae Inident Wave Refletion ρ, 1 1 ρ o, o ρ, 1 1 Transmission T 1 ξ sin ( kd o ) + 1 Transmission Coeffiient 1.8.6.4. in water Plexiglas Steel.5.5.75 1 1.5 Thikness / Wavelength

Refletivity of Thin Craks in Solids R ξsin( kd o ) ξ sin ( kd o ) + 1 lim d R ξ k d o 1 Refletion Coeffiient.8.6.4. air gap in steel water-filled rak in steel -1-8 -6-4 - log {Frequeny x Thikness [MHz mm]}

Impedane Mathing Z os( kd) iz sin( kd) os( ) sin( ) Z load o o o input Zo Z o kd o iz load kd o d (n+ 1) λ / 4 o ko d π (n+ 1) Zinput Z o Zload Perfet mathing by quarter-wavelength layer: enter frequeny fo d Zo Z1Z λo o 4 4 fo Bandwidth: R( f) Zinput ( f) Z1 Zinput ( f) + Z1 Zinput ( f) Zo π f π f Zload os( d) izo sin( d) o o π f π f Zo os( d) izload sin( d) o o

Zo Z1Z and Zload Z Zinput ( f) Z1 Z π f π f Zos( d) i Z1Zsin( d) o o π f π f Z1Zos( d) izsin( d) o o R R R( fo) + ( f fo) f f f o Z ( f ) Z input o 1 R( f o) r Z/ Z1 π f π f ros( d) i rsin( d) input ( ) o Z f Z o 1 π f π f os( d) i rsin( d) o o sin( kd o ) 1, and os( kd o ) Δ π fo fo f Z ( f) Z input 1 rδ i r Δ i r

R( f) rδ i r Z1 Z1 Δ i r Δ( r 1) rδ i r Δ ( r + 1) i r Z1 + Z1 Δ i r R( f) r r 1 r 1π i r r 4 + 1 i Δ fo fo f Tenergy ( r 1) ( r 1) π f 1 1 o f Δ 4r 4r fo Br 1 f f1 4 r 1.8 r Q fo π( r 1) r 1 f 1 and f are the half-power (-3 db) points

Quarter-Wavelength Mathing Layer quarter-wavelength mathing layer between quartz and water 1 Energy Transmission.8.6.4. exat unmathed approximate.5.5 Thikness / Wavelength quarter-wavelength mathing layer between steel and water 1 Energy Transmission.8.6.4. exat unmathed approximate.5.5 Thikness / Wavelength

Continuous Transition ultrasoni transduer inident wave eho from the bottom j N lear water ρ, 1 1 mud ρ oj, oj j 1 solid rok ρ, For the jth layer: Zoj ρ oj oj, koj π f, (j 1,,... N) oj Reursive relationship: d N Zload1 Z ρ Zinp j Zoj Zload j os( kojd) izoj sin( kojd) Zoj os( kojd) izload j sin( kojd) Refletion oeffiient: Z load j+ 1 Zinp j R Zinp N Z1, where Z1 ρ 11 Zinp N + Z1

Imperfet Interfae, Finite Interfaial Stiffness u e i ( k1 x t ) i A ω ( 1 ) i τ 1 1 ei k i uz i iωaz i x ω t u e i ( k 1 x t ) r A ω ( 1 ) r τ r u r Z1 iωar Z1 ei k x ω t u e i ( k x t ) t A ω ( ) t τ ei k t uz t iωaz t x ω t x ρ 1, 1 Inident Wave Refletion K ρ, Transmission Boundary Conditions: ui + ur +Δ u ut τ i +τ r τ t τ +τ Δ u K τ K i r t K denotes the normal K n or transverse K t interfaial stiffness Slip boundary onditions: Kn/ Kt Low-density interphase layer: K / K 3 6 Kissing bond: K / K 3 Partial bond: K / K.5 1 n n n t t t

Refletion and Transmission Coeffiients Continuity of displaement: A A A i r t τt K Ai Ar At 1 iωz K Continuity of stress: A Z + A Z A Z i 1 r 1 t Stress refletion and transmission oeffiients: Imperfet interfae: R τr At Z Z1 + iωz1z/ K τi Ai Z + Z1 iωz1z/ K T τt At Z Z τi Ai Z1 Z1 + Z iωz1z/ K Ideal interfae (K ): R T τ r τi τ t τi Z Z Z + Z 1 1 Z Z + Z 1 lim R ω R and lim T T ω lim R 1 and lim T ω ω

Frequeny Dependene Moduli of the refletion and transmission oeffiients of an imperfet steelaluminum bond of K 114 N/ m3 for longitudinal wave at normal inidene Refletion and Transmission Coeffiients 1.9.8.7.6.5.4.3..1 Refletion Transmission 4 6 8 1 Frequeny [MHz] For similar materials ( Z 1 Z Z): R T iωz/ K iωω / 1 iωz/ K 1 iω/ Ω 1 1 1 iωz/ K 1 iω/ Ω Ω K/ Z is the harateristi transition frequeny

Oblique Inidene, Snell s Law 1 θ 1 λ 1 Λ λ θ λ Λ sin θ λ sin θ 1 1 f sinθ f sinθ 1 1 sinθ sinθ 1 1

Refletion and Transmission y y θ s1 R s I s θ si θ s1 R s I d θ di θd1 R d θd1 R d solid 1 z solid 1 z solid solid θ s θ s θ d T s T d θ d T s T d Snell's Law: sin θ sin θ sinθ sin θ sin θ sinθ Constitutive relationships: di si d1 s1 d s d1 s1 d1 s1 d s u u z y τ yy λ + ( λ + μ) z y uy u τ ( z zy μ + ) z y μ 1 ρ1s 1, λ 1 + μ 1 ρ1d 1, μ ρs, and λ + μ ρ d

Boundary Conditions both normal and transverse veloity and stress omponents must be ontinuous at the interfae () (1) uy u y () (1) uz u z () (1) τ yy τ yy () (1) τzy τzy longitudinal inidene: ( d1) ( d) ( s1) ( s) ( i) uy + uy uy + u y u y ( d1) ( d) ( s1) ( s) ( i) uz + uz uz + u z u z ( d1) ( d) ( s1) ( s) ( i) τ yy +τyy τ yy +τ yy τ yy ( d1) ( d) ( s1) ( s) ( i) τ zy +τzy τ zy +τzy τzy shear inidene Id 1, Is Is 1, Id a11 a1 a13 a14 Rd b1 1 a1 a a3 a 4 T d b or a31 a3 a33 a34 Rs b3 3 a41 a4 a43 a44 Ts b4 4 longitudinal [b] or shear wave inidene []

The matrix elements aij, bi, and i an be easily alulated from simple geometrial onsiderations: a osθd1 osθd sinθs1 sinθs sin θd1 sin θd osθs1 osθ s Zd1osθs1 Zdosθs Zs1sin θs1 Zssin θs s1 Z 1 sin s s θd1 Zs sin θd Zs1osθs1 Zsosθ s d1 d (the ommon -iω fator was omitted in the last two rows) Cramer's rule: os θdi sinθsi sinθ di osθ si b Zd1osθsi and Zs1 sin θsi Z s1 s1 sin θ di Zs1 osθ si d1 a(1) a() a(3) a(4) d, d, s, s a a a a det[ ] det[ ] det[ ] det[ ] R T R T det[ ] det[ ] det[ ] det[ ]

Speial Cases a) fluid-vauum b) fluid-fluid ( d > d1 ) I d R dd I d R dd θ i θ r θ i θ r fluid vauum fluid 1 fluid θ d T dd ) solid-vauum d) solid-vauum (longitudinal inidene) (shear inidene) R ds I s R ss I d θ i θ s θr R dd θ i θ r θ d R sd solid solid vauum vauum

e) fluid-solid I d R dd θ i θ r fluid solid θ d T dd θ s T ds f) solid-fluid g) solid-fluid (longitudinal inidene) (shear inidene) θs1 R ds I s θ i θ r θ s1 R ss I d solid θ i θ r θd1 R dd solid θd1 R sd fluid fluid θ d T dd θ d T sd

h) solid-solid i) solid-solid (longitudinal inidene) (shear inidene) θ s1 R ds I s θ i θ r θ s1 R ss I d θ i θ r θd1 R dd θd1 R sd solid 1 solid 1 solid solid θ s θ s θ d T ds T dd θ d T ss T sd Fluid-vauum: R 1, θ dd r θ i Fluid-fluid: sin sin θ, d θi θr θ i d d1 sin θd d d1 sin θ i then d < d1 θd < θ i then d > d1 θd > θ i There exists one ritial angle ( sin θd 1, θ d 9 ) sin θ r1 d1 d

Solid-Vauum Interfae, Mode Conversion P-wave inident (no ritial angle): sin θ sin ( ), s θi θr θd θ i s d S-wave inident: sin θ sin ( ), d θi θr θs θ i There exists one ritial angle (sin θd 1 or θd 9 ) d s sin θ r1 s d The boundary onditions require that both normal and transverse stress disappear at the surfae. Zd osθs Zssin θs Zd osθs Rdd s Z sin os s s θd Zs θs R ds Zs sin θd d d Rdd os θ s s sin θ sin s θd d os θ s s + sin θ sin s θd d depends on the Poisson ratio of the solid R dd ( ) R (9 ) 1 dd

Longitudinal and Shear Wave Refletion Coeffiients ν.3 (solid) and ν.35 (dashed) Refletion Coeffiient 1. 1.8.6.4. longitudinal-tolongitudinal longitudinal-toshear 1 3 4 5 6 7 8 9 Angle of Inidene [deg] 1. 1 shear-to-shear Refletion Coeffiient.8.6.4. shear-to-longitudinal 5 1 15 5 3 35 Angle of Inidene [deg]

Polar diagrams longitudinal inidene o 3 45 o 15 o o 15 o 3 o longitudinal shear 45 o 75 o 9 o 6 o 6 o 75 o 9 o shear inidene o 3 45 o 15 o o 15 o 3 o longitudinal shear 45 o 75 o 9 o 6 o 6 o 75 o 9 o

Fluid-Solid Interfae ( d1) ( d) ( s) ( i) uy + uy + u y u y ( d1) ( d) ( s) ( i) τ yy +τ yy +τ yy τ yy ( d) ( s) +τ zy +τ zy a11 a1 a14 Rdd b1 a31 a3 a 34 T dd b 3 a4 a44 Tds osθ osθ sin θ R osθ i d s dd i Zd1 Zdos s Zssin s T dd Z θ θ d1 s T sin os ds Zs θd Zs θ s d Rdd det[ a (1)] det[ a] osθi osθd sin θs Zd1 Zdosθs Zssin θs Z s s sinθd d Zsosθs osθi osθd sin θs Zd1 Zdosθs Zssin θs Z s s sinθd d Zsosθs

Rdd osθi osθd sin θs ρf d osθs ssin θs ssinθd d osθs osθi osθd sin θs ρf d osθs ssin θs ssinθd d osθs ρ ρ1/ ρ f d1, d d, s s, i di d1 θ θ θ, θ d θ d, and θ s θ s R dd i d s s d s f d s d s d s i d s s d s f d s d s d s os θ ( os θ + sin θ sin θ ) ρ ( os θ osθ + sin θ sin θ ) os θ ( os θ + sin θ sin θ ) + ρ ( os θ osθ + sin θ sin θ )

Displaement, Stress, Intensity, and Power Coeffiients ( stress) ( displaement) Zβj αβ αβ Zα1 Γ Γ ( stress) Z Γ αβ Γ αβ Z βj α1 Γ stands for either R (j 1) or T (j ) α and β are either d or s ( intensity) ( displaement) ( stress) Z βj Γ αβ Γαβ Γ αβ Γ αβ Z α 1 os Z os ( power) ( intensity) θβj βj θβj Γ αβ Γ αβ Γ osθ αβ α1 Zα1 osθα1 R ( power) ( power) ( power) ( power) αd + R αs + T αd + T αs 1 Law of reiproity: ( power) ( power) αβ βα Γ Γ

Energy Refletion and Transmission Coeffiients aluminum in water Energy Refletion and Transmission Coeffiients 1.8.6.4. refletion longitudinal transmission 5 1 15 5 3 Angle of Inidene [deg] shear transmission Energy Refletion and Transmission Coeffiients 1.8.6.4. refletion longitudinal transmission steel in water 5 1 15 5 3 Angle of Inidene [deg] shear transmission

Energy Refletion and Transmission Coeffiients 1 Plexiglas/aluminum interfae Energy Refletion Coeffiients.9.8.7.6.5.4.3..1 longitudinal refletion shear refletion 1 3 4 5 6 7 8 9 Angle of Inidene [deg].6 Energy Transmission Coeffiients.5.4.3..1 longitudinal transmission shear transmission 1 3 4 5 6 7 8 9 Angle of Inidene [deg]

Slip Boundary Conditions ( d1) ( d) ( s1) ( s) uy + uy uy + u () i y u normal displaement y ( d1) ( d) ( s1) ( s) tangential displaement uz + uz uz + u () i z u z normal tration ( d1) ( d) ( s1) ( s) () τ yy +τyy τ yy +τ i yy τ yy tangential tration ( d1) ( d) ( s1) ( s) ( i) τ zy +τzy τ zy +τzy τzy a11 a1 a13 a14 Rd b1 1 a1 a a3 a 4 T d b or a31 a3 a33 a34 Rs b3 3 a41 a4 a43 a44 Ts b4 4 Slip boundary onditions: ontinuity of the normal displaement and tration vanishing tangential tration on both sides a11 a1 a13 a14 Rd b1 1 a31 a3 a33 a 34 T d b or a41 a43 Rs b4 4 a4 a44 Ts

Angle-Beam Transduers wedge θ i transduer ouplant θ s speimen s i sin θ sin θ s i Plexiglas/Aluminum, longitudinal-to-shear transmission Energy Transmission.7.6 "slip" boundary.5.4.3 "rigid" boundary..1 3 4 5 6 7 8 9 Angle of Refration [deg]

SH Wave Refletion and Transmission at a Solid-Solid Interfae I θ i y θ s1 θ i R solid 1 solid z θ s T () i ( r) () t ux + ux ux and () i ( r) () t τ xy +τ xy τ xy ( r) ( t) ( i) x x x ( r) ( t) ( i) xy xy xy u + u u τ + τ τ or a11 a1 R 1 a13 a 14 T All displaements are in the x diretion only (without the ommon i t e ω term): () i i( os i ks1 y sin i ks1z) x u e θ + θ () i i( os i ks1 y sin i ks1z) x u e θ + θ () t i( os t ks y sin t ks z) x u Te θ + θ θ t θ s, sin θ t s/ s1sin θi

Stress omponents: xy xy s x/ τ με ρ u y () i ( os 1 sin 1 ) 1 os i i k s y i k s xy i Zs i e θ + θ z τ ω θ ( r ) (os 1 sin 1 ) 1 os i i k s y i k s xy i Zs i Re θ + θ z τ ω θ () t ( os sin ) os i t k s y t k s xy i Zs t Te θ + θ z τ ω θ Z s sρ is the speifi aousti impedane of the medium 1 1 R 1 Z osθ Z osθ T Z osθ s1 i s t s1 i (the seond row was divided by -iω) (Displaement) refletion and transmission oeffiients: R T 1 1 Zs1osθi Zsosθt Zs1osθi Zsosθ t 1 1 Zs1osθ i + Zsosθt Zs1osθi Zsosθt 1 1 Zs1osθi Zs1osθi Zs1 osθ i 1 1 Zs1osθ i + Zsosθt Zs1osθi Zsosθt Normal omponent of the aousti impedane Z' s Zsosθ R Z ' ' s1 Zs Z ' ' s1 + Zs and T Z ' s1 Z ' ' s1 + Zs

Solid-vauum interfae (free surfae): Rayleigh Wave Zd osθs Zssin θs Rd Z s s sin θd Zsosθs R s d Nontrivial solution exists if: os θ s s + sinθ sin s θ d d sin θs sin θd 1 s d R Relative veloities: s 1 ν ξ ( ) d (1 ν) Exat Rayleigh equation: η R s η6 8η 4+ 8(3 ξ) η 16(1 ξ ) Approximate expression: η.87+ 1.1ν 1+ν