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Appendix. Formulas, Diagrams and Tables

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Appendix. Formulas, Diagrams and Tables Appendix. Formulas, Diagrams and Tables Formulas for Reflection and Transmission Coefficients The following formulas 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 a Solid (See Figs. 2.6, 2. 7 and Diagram 1). Symbols: the angles of the longi­ tudinal and transverse waves are a 1 and at, respectively. Thus for 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 _ (ctfcJ) 2 sin 2tXJ sin 2"'t- cos2 2tXt n- N (A.1) with the abbreviation for the denominator N = ( ~: rsin 2cx1 sin 2cxt + cos2 2cxt, for the reflected transverse wave: _ 2(ctfcJ) 2 sin 2tXJ cos 2tXt R (A.2) t1- N ' with incident transverse wave (oscillation plane parallel to plane of in­ cidence) (cf. Fig. 2. 7) for the reflected longitudinal wave: R __ sin 4tXt (A.3) It- N ' for the reflected transverse wave: R _ (ctfc!) 2 sin 2tXJ sin 2tXt - cos2 2tXt = R tt- N - n· (A.4) 606 Formulas Interface between two Liquids Symbols: the angles of the incident, reflected and transmitted longi­ tudinal waves are ae, ar and ad, respectively. The acoustic velocities and densities of materials 1 and 2 are Cv ~?! and c2, e2, respectively. The reflection coefficient of the acoustic pressure R = cos <Xe - (rhci/f!2C2) 1 - (c2/c1)2 sin2 <Xe V (A.5) cos <Xe + (rhc1/(!2c2) V1 - (c2/c1)2 sin2 <Xe ' the transmission coefficient of the acoustic pressure D = 2 cos <Xe (A.6) cos <Xe + ((hC1/(]2C2) V1 - (c2/c1)2 sin2 <Xe Interface between Liquids and Solids Symbols: angle of longitudinal wave in liquid a, angles of longitudinal and transverse waves in solid a1 and at, density and acoustic velocity in liquid !? and c, in solid !?f' c1 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 I(Ct)2 (!C cos iXI \ R = N \ - sin 2a1 sin 2at cos2 2at -- --/ (A.7) , C! + (?fC! COS X with the abbreviation Ct)2 (?C COSiX! N = (- sin 2a1 sin 2at cos2 2at C! + + -(!fCl --COS IX . (A.S) 2 (Ct \2 . Dtl =- N ~~ sm 2a1• (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 I( Ct )2 (!C cos iXI \ Ru = N \ - sin 2a1 sin 2at - cos2 2at + - -- , (A.lO) \ CJ Cf(!! COSiX 1 Formulas 607 for the reflected transverse wave: Ru = ~ ( :~ rsin 2ex1 cos 2cxt (A.ll) and for the transmitted longitudinal wave in the liquid: D - ~ f!C cos <XJ cos 2<Xt (A.12) ll - N (}fCJ cos <X In the case solid/liquid with an incident transverse wave: 1 I( Ct )2 . 2 f!C cos <XJ \ Rtt = N \ - sm 2ex 1 sm 2ext - cos 2ext -- --;' , (A.13) q ~~<X Rlt = -~sin 4cxt, (A.14) (A.15) 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 "'I cos2 2<Xt _i_ (A.16) II - N2 f!fCJ cos <X ' but if the transverse wave is used in the solid, E _ (~) 2 ~ cos <XJ sin 2<Xt sin 2<X1 _i_ (A.17) It - N2 CJ (}fCJ cos <X • Interface between two Solids The values in Diagrams 7 and 8 were calculated from a computer pro­ gram by Kuhn 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<Xrt 2e2 c~t e2 cuc~t cos2 2<X2t N = 2 cot ex +- +--cot ex + ------= (A.18) 1t 2 clt cos <Xu e1 cit Zt 2 e1 c1 t cos <Xu For the incident longitudinal wave in material 1: R = 1 - cos2 2<Xlt (A.HJ) n N cos "'U ' 608 Formulas (A.20) D = 2 1?2 C2J c2t ell cos2 2a t cos 2a t 2 1 (A.21) 11 e1 cit cos 2all sin a11N (A.22) For the incident transverse wave in material 1: R _ 4cu cos iXIt cos2 2<XIt (A.23) It - cit sin 2auN ' R = 4c11 cos "'It_ 1 (A.24) tt clt sin auN ' D = _ 4e2 c~t c21 en cos2 2a2t cos alt (A.25) It e1 ct t sin 2anN (A.26) Note: For the incident wave in material 2, the indices 1 and 2 should be interchanged. Velocity of Lamb Waves Referring to Diagram 9: Optimal sound propagation occurs if for the angle ex between the transverse wave and the perpendicular to the plate the following equation is fulfilled: F 1,2 = G (A.27) with the abbreviations: d a=n;.t; T 1 =tanh Vs 2 - 1a; Tq =tanh Vs 2 - q2 a; F - T1 for symmetric Lamb waves, 1 -Tq F - Tq for antimetric Lamb waves, 2- Tl 4s2 Vs2- 1 Vs2- q2 G = (282 _ 1)2 for both. Formulas 609 The correlated velocity of propagation (group velocity) then is [581]: Ct Up=-~· (A.28) 8 +ada The differential d8/da calculated from (A. 27) is: 8F1.2 d8 8a (A.29) da =~ oF1.2' ---- 08 88 and explicitly for the symmetric wave modes: oG 8(q2 -1)84 + 4(3- q2)82- 8q2 - = 8 --------,==,c-,= 88 (2s2 _ 1)3 V82 _ 1 V82 _ q2 oF1 = a: {Tq(Ty- 1) + T1(1- T~)}' os Tq V82_1 V8 2_q2 oF1 _ 1 {V-2-- 2 V-2--2 2 } &; - Tij_ 8 - 1 Tq (1- T1 ) + s - q T 1(Tq- 1) . For the antimetric forms the tanh function should be replaced by the coth function. 610 Diagrams Diagram 1. Reflection at free boundary of steel Rn, reflection coefficient for longitudinal wave versus incidence angle of longitudinal wave, <XJ (bottom scale), according to Eq. (A.l). Rtb reflection coefficient for transverse wave versus incidence angle of longitudinal wave, <Xt (top scale), according to Eq. (A.2). RJt, reflection coefficient for longitudinal wave versus incidence angle of transverse wave, <XJ, according to Eq. (A.3). Rtt. reflection coefficient for transverse wave versus incidence angle of transverse wave, <Xt, identical with R 11 , according to Eq. (A.4). a,- ,, !,' 6' 8' 10' 72' 74' 76' 78' 20' 22' 24' 26' 28' 30' 37' 32' 33' 312' f 500 10 0 ° ' % ~' % ' I 90 ~- ~ 450 I \1\,0'" I I v-- C-- / 1\ 1,00 80 [A, I \ I 1\ 350 70 1\ I ~ 300 I 1/\ I \ v / 250 t ~~ ~ / v 1,0 71\ I\ 200 !7 \ ~I 1/ \ 30 f--- v v 750 20 e-.--- I v r\ / 700 v- '\ i / 10 ' A "'-- 50 01£ 0 0 70' 20' 30' 1,0'ce,- 50' 60' 70' 80' 90' Diagrams 611 Diagram 2. Reflection and transmission of longitudinal waves at water- aluminium interface Longitudinal wave incident in water at angle ex furnishes longitudinal wave in alu­ minium with angle cx1 and transverse wave with ext· Calculated from Eqs. (A. 7) to (A.9) with the constants (!C ctfct = 0.488; ctfc = 4.26; - = 0.0888. (!fCl 100 R=lD_D_~ % Total A reflection 95 I I I 90 R) 85 I R~ \ 400 l----"""" 381,9%- / v \._. v 70 350 343,55/ ~ 65 60 I I 300 so• 5o• 90. a• w· 2o· 3o· ;,a·u,- 70'/' 250 1 ~ Du,y c::f v y v 200 ' - .- a· 2o· ~---1-a~~~---1----~~----~------~--------------~150 ~--~----~--~----~-------+------------~100 612 Diagrams Diagram 3. Reflection and transmission of longitudinal waves at aluminium- water interface Longitudinal wave incident in aluminium at angle cx1 furnishes reflected longitudinal wave in aluminium with angle cx1, reflected transverse wave in aluminium with angle <Xt and transmitted longitudinal wave in water with angle ex (Eqs. (A.lO) to (A.12), regarding constants, see Diagram 2). a:- 20 t,• 60 8. 1o o· 7o· w 72" 7J• 7158° 0 I I I I I I I % 90 I 1---. 80 I ~ 70 <II I ~ \._ 50 I I LRII \ __...- ...... I v "'>< v'X \ v ~ r--/ 30 / \ 20 v \ / ,...____ 10 / K- \ -1---- v r---...... _j 0 o· to• 20' 30' 1,0' so· 60° 70' eo· go· ----- a:,- o· r ,,. s· e· w· 12• w 15• te• 2o· 22• 21,• 26° 28° 29° 29)0 (X,t- Diagrams 613 Diagram 4. Reflection and transmission of transverse waves at aluminium­ water interface Transverse wave incident in aluminium at angle iXt furnishes reflected transverse wave with iXt. reflected longitudinal wave with cx1 and transmitted longitudinal wave in water with angle ex (Eqs. (A.13) to (A.15), regarding constants see Diagram 2). <X t- O' 10' 20' 30' 40' 50'60' 90' 100 I I I I I 500 % 90 \ I v 1\11 / 80 I\ v 400 ~ ~ 70 \ \ 60 300 \ v 200 I 30 J vRll K 1---. 20 / r 100 1/ ~ v J.....---.... 10 ~ V( \ 0 ~ ~ 0 0' 70' 20• Jo• 40° sa· 5o• 70° sa· go· at- o· 2• ;,• 5• 8' 10' 72' w 75' 78' 2a• 22' 24' 26' 28' 28,75' a- 614 Diagrams Diagram 5.
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