Appendix. Formulas, Diagrams and Tables Formulas for Refiection and Transmission Coefficients The following fermulas give reflection and transmission coefficients for the acoustic pressure as a function of the incidence angle, calculated for plane waves on plane boundaries and disregarding absorption. The notation of the formulas is based on that of Schoch [34] where, how­ ever, they refer not to the acoustic pressure but to the particle displace­ ment. Figures 2.6 to 2.13 and the Diagrams only give numerical values, i.e. without regard to phase. Free Boundary of aSolid (See Figs. 2.6, 2.7 and Diagram 1). Symbols: the angles of the longi­ tudinal and transverse waves are iXl and iXt , respectively. Thusfor example Rlt signifies the reflection coefficient of the acoustic pressure for a reflected longitudinal wave referred to an incident transverse wave. For the reflected longitudinal wave we have: R _ (Ct!Cl)2 sin 2G\1 sin 2G\t - cos2 2G\t 11- N (A.1) with the abbreviation for the denominator N = ( ::r sin 2iXi sin 2iXt + cos2 2iXt, for the reflected transverse wave: (A.2) with incident transverse wave (oscillation plane parallel to plane of in­ cidence) (cf. Fig. 2.7) for the reflected longitudinal wave: R __ sin 4G\t It - N' (A.3) for the reflected transverse wave: (A.4) 606 Formulas Interface between two Liquids Symbols: the angles of the incident, reflected and transmitted longi­ tudinal waves are cxe' cxr and CXd' respectively. The acoustic v-elocities and densities of materials 1 and 2 are Cl' (11 and C2, (12' respectively. The reflection coefficient of the acoustic pressure R = COS (Xe - ((!IC1/(J2C2) 1 - (C2/C1)2 sin2 (Xe , V (A.5) cos (Xe -I- ((!IC1/(J2C2) VI - (C2/C1)2 sin2 (Xe the transmission coefficient of the acoustic pressure D= 2 cos (Xe (A.6) COS (Xe -I- ((J1C1/'!2C2) VI - (C2/C1)2 sin2 .xe Interface between Liquids and Solids Symbols: angle of longitudinal wave in liquid cx, angles of longitudinal and transverse waves in solid cxI and cxt ' density and acoustic velocity in liquid e and c, in solid ef' ~ and ct ' respectively. In the case liquid/solid (i.e. incident longitudinal wave in liquid) (see Fig. 2.8 and Diagram 2) we have: 1 l(ct)2 . (lC COS (XI \ R = N \ - sm 2cxI sin 2cxt cos2 2cxt -- (A.7) CI + (lfel --Icos x with the abbreviation ct)2 .. (JC cos (XI N = (- sm 2cxI sm 2cxt + cos2 2cxt + - --. CI (JfCI cos (X 2 Dn = N cos 2CXt, (A.8) 2 (ct\2 . Dtl = - N -;;; sm 2cxI' (A.9) In the case solid/liquid (see Figs. 2.9, 2.10 and Diagrams 3 and 4) with incident longitudinal wave, we have for the reflected longitudinal wave: 1 l(ct)2 .. (lC cos (XI \ Rn = N \ - sm 2cxI sm 2cxt - cos2 2cxt + - --I ' (A.10) \ CI Cf (11 COS (X Formttlas 607 for the reflected transverse wave: Rtl = ; (~; rsin 2cxI cos 2cxt (A.ll) and for the transmitted longitudinal wave in the liquid: D _ ~ (!C cos <XI COS 2<Xt (A.12) 11 - N (!fCI cos <X In the case solid/liquid with an incident transverse wave: 1 /( ct)2 •• 2 (!C COS <XI \ Rtt = N \ - sm 2CX I sm 2cxt - cos 2cx -- (A.13) cI t (!fCI --Icos <X ' Rlt = - ~ sin 4cxt , (A.14) D 2 (!C cos <XI sin 2<Xt (A.15) lt = N (!fCI ~<X-- The echo transmittances (see Fig. 2.13 and Diagrams 5 to 8) for both cases, solid/liquid and liquid/solid are identical. If in both materials the longitudinal wave is used, we have E _ ~ ~ cos <XI COs2 2<Xt (A.16) 11 - N2 (!fCI COS <X ' but if the transverse wave is used in the solid, _ ~ (~)2 ~ cos <XI sin 2<Xt sin 2<Xj E (A.17) lt - N2 CI (!fCj cos <X • Interface between two Solids The values in Diagrams 7 and 8 were calculated from a computer pro­ gram by Kühn and Lutsch [414, 5]. For the case of coupling through a thin liquid layer, and using the above symbols, the following reflection and transmission formulas apply. Abbreviations: denominator N (index 1 for material 1, index 2 for material 2) Cu cos2 2<Xlt 2(!2 C~t (!2 Cu C~t cos2 2<X2t N = 2 cot cx + - + --cot cx + --- (A 18) lt 2 clt cos <Xli (!1 cf t 2t 2 (!1 cf t cos <Xli • For the incident longitudinal wave in material 1: R 1 C082 2<Xlt (A.1U) 11 = - N cos <Xli ' 608 Formulas (A.20) D _ 2 (/2 ~l ~t Cu cos2 ~t cos 2(Xlt (A.21) 11 - €!l ctt COS 2(Xu sin (XUN ' D __ 2€!2~tCll cos 2(Xlt (A.22) tl - €!ICft sin 2(XUN For the ineident transverse wave in material 1: R _ 41:u cos (Xlt C082 2(Xtt (A.23) It - Clt sin 2(XuN ' R _ 4cu cos (Xtt _ 1 (A.24) tt - Clt Bin (XUN ' D _ _ 4€!2 I:~t c21 Cu cos2 2(X2t COS .(XI t (A.25) It - €!l ch sin 2(XllN ' D _ 4€!2 C~t Cu cos (Xlt (A.26) tt - €!l ctt sin (XUN . Note: For the ineident wave in material 2, the indices 1 and 2 should be interchanged. Velocity 01 Lamb Waves Referring to Diagram 9: Optimal sound propagation oeeurs if for the angle (X between tlw transverse wave and the perpendicular to the plate the following equation is fulfilled: F I,2 = G (A.27) with the abbreviations: Tl = tanh Vs2 - la; T q - tanhVs2 - !l...2 a ·, F - Tl for symmetrie Lamb waves, 1 -Tq q F 2 - TTl for antimetric Lamb waves, 4s2 VSI - 1 VSI - ql 2 I I G = ( s - 1) for both. Formulas 609 The eorrelated veloeity of propagation (group veloeity) then is [581]: ~= da· (A.28) 8 + a da The differential ds/da ealeulated from (A.27) is: oF1.2 d8 oa (A.29) da - ----oG oF1.2' 08 08 and expIieitly for the symmetrie wave modes: oG 8 (q2 - 1) 8' + 4 (3 - q2) 82 - 8 q2 -- S -"''----'---======-== 08 - (2s2-1)SYs2-1 Y82_q2 ' oFl as {Tq(Ti - 1) T l (l - T~)} es = Ta YS2 - 1 + Y82 _ q2 ' l 0:a = ;ö. {Vs2 - 1 Tq (1 - Ti) + VS2 - q2 T1(T! - 1)}. For the antimetrie forms the tanh funetion should be replaeed by the eoth funetion. 610 Diagrams Diagram 1. Reflection at free boundary 01 steel Rn, reflection coefficient for longitudinal wave versus incidence angle of longitudinal wave, <XI (bottom scale), according to Eq. (A.l). Rt!, reflection coefficient for transverse wave versus incidence angle of longitudinal wave, <Xt (top scale), according to Eq. (A.2). RIt , reflection coefficient for longitudinal wave versus incidence angle of transverse wave, <Xl> according to Eq. (A.3). Rtt, reflection coefficient for transverse wave versus incidence angle of transverse wave, <Xt, identical with Rn, according to Eq. (AA). at- 2' 4' 6' 8' 10' 12']1.' 16' 18' 20' 22' 24' 26' 28' JO' JI' J2' JJ' J3,2' 10 0 0 500 %, ~ % 450 1---- ~ 90 \ I II = RI/ I V K I I / 1\ 400 8O~- Ir, I \ / \' J50 70 1\ / 1/\fN JOD \ v / 250 t ~ / v 40 / 1\ 1\ 200 11 Ät / \ JO I '\ 750 v v 20 / v 1\ / IOD '\ v ~V 70 vA 50 olb 0 o 10' 20' JO' Cl,-50' 60' 70' 80' 90' Diagrams 611 Diagram 2. Reflection and transmission 01 longitudinal waves at water- aluminium interface Longitudinal wave incident in water at angle IX furnishes longitudinal wave in alu­ minium with angle IXj and transverse wave with IXt. Calculated from Eqs. (A.7) to (A.9) with the constants ec cdc = 4.26; - = 0.0888. efCj 700 R=700% % Tatal reflection 95 / I / 90 I I I R) \ 85 V/ Ry \ 1,00 r 80 J87,9%~ V <>: 75 I ./ ---- ,,/ I '- I 350 70 I I I I I JI,(,557 65 i 60 1/ 300 0° 70° 20° 30° 1,0° 50° 60° 70° 80° 90 ° lXI_ I I I i I i I I 250 ! I c::,- I I c::,::: I. [01/1 / 200 :t:1 ~ Y 0° 20° r---~----r---~-----r------~----------~700 IC.:----'------:':----'-----:'::---------:L---_---,L-_____---..J 0 --- 0° 5° 70° 25° 28,75° 90° 612 Diagrams Diagram 3. Reflectionand transmission of longitudinal waves at aluminium- water interface Longitudinal wave incident in aluminium at angle:X1 furnishes reflected longitudinal wave in aluminium with angle (Xh reflected transverse wave in aluminium with angle (Xt and transmitted longitudinal wave in water with angle (X (Eqs. (A.I0) to (A.12), regarding constants, see Diagram 2). eG- 10 O' 2' 6' 8' 10' 11" 12' 13' 1358' 0 I I I I I I I I %, 1 I I I 9 0 I I I I ! 8 O~ I ~ I il 70 K I ~ I 60 f\. / 1/ L RII \ - ........ / "'x V Y / \ V ~ ~ 3D / 1\ 20 / \ / 1/ 10 ~- -r--- \ V -- I--- o ...-/ ~ O' 10' 20' 30' 1,0' 50' 60' 70' 80' 90' eGl_ I I I I I I I I O' 2' 4' 6' 8' 10' 12' W 16' 18' 20' 22' 24' 26' 28' 29' 29,2' ex/- Diagrams 613 Diagram 4. Reflection and transmission 01 trans verse waves at aluminium­ water interfa,ce Transverse wave incident in aluminium at angle iXt furnishes reflected transverse wave with iXt, reflected longitudinal wave with iXI and transmitted longitudinal wave in water with angle iX (Eqs.