Part 4 RAYLEIGH AND LAMB WAVES

Part 4 RAYLEIGH AND LAMB WAVES

Rayleigh Surfae Wave x x 1 x 3 urfae wave x 1 x 3

Partial Wave Deompoition Diplaement potential: u = ϕ + ψ Wave equation: 1 ϕ 1 ψ ϕ = = k ϕ an ψ = = k t t ψ Wave veloitie: = λ+ μ an μ = ρ ρ Wave propagate in the x 1 iretion: Simplifie wave equation: u 0 = = 0, ψ 1 = ψ 3 = 0, ψ = ψ x ϕ ϕ + + k ϕ = x1 x3 0 ψ ψ + + k ψ = x1 x3 0

Axial iplaement (tangential to the urfae) u 1 ϕ ψ = x1 x3 Tranvere iplaement (normal to the urfae): u 3 ϕ ψ = + x3 x1 Normal tre in the propagation iretion: ϕ ϕ ϕ ψ τ 11 = λ ( + ) + μ( ) x x 1 x3 x1 1 x3 Normal tre parallel to the urfae: ϕ ϕ ϕ ψ τ 33 = λ ( + ) + μ ( + ) x x 1 x3 x3 1 x3 Shear tre: ϕ ψ ψ τ 31 = τ 13 = μ ( + ) x 1 x3 x1 x3 We eek olution in the following form ϕ= ψ= F( x ( 1 1 ) 3) ei k x ωt Gx ( ( 1 1 ) 3) ei k x ωt k1 kr = ω R

From the wave equation, ϕ ϕ 0 F + + k ϕ = = ( kr k ) F x1 x3 x3 ψ ψ 0 G + + k ϕ = = ( kr k ) G x1 x3 x3 Solution: 3 R x k k F( x3) = Ae an 3 R x k k Gx ( 3) = Be The potential funtion an be rewritten a follow ϕ= Ae κ x3 i( kr x1 ωt) e ψ= Be κ x3 i( kr x1 ωt) e kr k κ = an κ = k k The urfae wave olution i a ombination of ouple partial (longituinal an hear) wave. The partial wave amplitue ratio (B/A) i etermine by the onition that the urfae i tration free. Both partial wave are evaneent, i. e., kr > k > k or R < < R

Bounary onition: Both normal an tangential tree vanih on the urfae ( x 3 = 0 ) ϕ ϕ ϕ ψ τ 33 = λ ( + ) + μ ( + ) x x 1 x3 x3 1 x3 = ( λ+ μ ) κ ϕ λkr ϕ μiκkrψ ϕ ψ ψ τ 31 = τ 13 = μ ( + ) x 1 x3 x1 x3 = μκ i krϕ μκ ( + kr ) ψ i( k x t) At the urfae, without the ommon term of e R 1 ω : 0 ( λ+ μ) κ λkr μiκkr A 0 = i kr ( kr ) B μ κ μ κ + kr kr k kr k kr ( λ+ μ ) κ λ = μ ( λ+ μ ) = μ μ = μ ( κ + ) Charateriti equation: R R i kr kr 0 ( κ + k ) iκ k A 0 = ( ) B κ κ + kr kr ( κ + ) 4 κ κ = 0

Rayleigh Veloity 1 ν ξ= ( = ) (1 ν) η= R Exat Rayleigh equation: Approximate expreion: η6 8η 4+ 8(3 ξ) η 16(1 ξ ) = 0 η 0.87+ 1.1ν 1+ν Rayleigh Veloity / Shear Veloity 0.96 0.95 0.94 0.93 0.9 0.91 0.9 0.89 0.88 exat approximation 0 0.1 0. 0.3 0.4 0.5 0.6 0.7 Shear Veloity / Longituinal Veloity

Partial Wave Compoition The partial wave amplitue ratio: B κ + kr iκk = = R A iκ kr κ + kr By ubtituting the Rayleigh wave number k R from the harateriti equation: B A κ = i = iζ κ ζ =ζ() v 1.5 1.6 Ae κ x 3 ϕ= an ψ= ζ i 3 Ae κ x i( k x t) (without the ommon e R 1 ω term)

Diplaement Axial iplaement (tangential to the urfae): φ ψ u1 = = ia k x Re κ ζκe κ x1 x3 ( 3 3) x Tranvere iplaement (normal to the urfae): φ ψ u3 = + = A κ x e κ ζkre κ x3 x1 ( 3 3) x u3 κ ζk = i R = ζ i u1 k x = 0 R ζκ 3 Normalize Partile Diplaement 1. 1 0.8 0.6 0.4. 0. 0-0. normal (u 3 ) ν = 0.3 tangential (u 1 ) 0 0.5 1 1.5 Depth / Rayleigh Wavelength

Stree 13 R ( κ x3 κ x3) τ = iaμ k κ e e τ 33 = Aμ k x R + κ e κ e κ ( )( 3 3) x τ13 ikr κ = = iζ τ 33 kr +κ 4μκκkR 11 A k R x3 x 3 κ + kr τ = ( λ+ μ) λκ exp[ κ ] exp[ κ ] 1. Normalize Stre 1 0.8 0.6 0.4 0. τ 11 τ 13 τ 33 ν = 0.3 0-0. 0 0. 0.4 0.6 0.8 1 1. 1.4 1.6 1.8 Depth / Rayleigh Wavelength

LAMB PLATE WAVES Lowet-orer Lamb wave in a plate a) ymmetri b) aymmetri x 3 x wave iretion x 1

Potential Metho ϕ= A exp[ ik ( x+ k x)] + Aexp[ ik ( x k x)] 1 1 1 3 3 1 1 3 3 k 3 = k k 1 = i κ or κ = k1 k exp[ ( )] exp[ ( )] ψ= B1 ik1x1 + k3x3 + B ik1x1 k3x3 k 3 = k k 1 = iκ or κ = k 1 k k1 = k1 = k, κ = k k, κ = k k oh( )exp( ) inh( )exp( ) ϕ= A κ x3 ikx1 + Aa κ x3 ikx1 inh( )exp( ) oh( )exp( ) ψ= B κ x3 ikx1 + Ba κ x3 ikx1 A = A + A1, Aa = A A1, B B B1 = an B = B + B1 a Bounary Conition τ ( x = ) = τ ( x = ) = τ ( x = ) = τ ( x = ) = 0 33 3 33 3 31 3 31 3 normal tre: hear tre: ϕ ϕ ϕ ψ τ 33 = λ + + μ + x x 1 x3 x3 1 x3 ( ) ( ) ϕ ψ ψ τ 31 = τ 13 = μ + x 1 x3 x1 x3 ( )

Example of Bounary Conition Normal tre on the upper urfae of the plate: τ 33( x3 = ) = 0 33 ( ϕ ϕ ) ( ψ ϕ τ / μ = + + ) x x 1 x 3 1 x3 x1 Partial erivative (without the ommon exp( ikx 1) : ϕ = k A κ x3 + Aa κ x3 x1 [ oh( ) inh( )] ϕ =κ A κ x 3 + A a κ x 3 x3 [ oh( ) inh( )] ψ = ik κ B κ x3 + Ba κ x3 x1 x3 [ oh( ) inh( )] Normal tre on the upper urfae of the plate: τ 33 / μ = ( k +κ)oh( κ ) A+ ( k +κ)inh( κ ) Aa + ikκoh( κ ) B+ ikκinh( κ ) Ba = 0

Full 4 4 Bounary Conition Matrix a A 0 A a 0 = B 0 Ba 0 a = +κ κ +κ κ κ κ κ κ +κ κ +κ κ κ κ κ κ κ κ κ κ +κ κ +κ κ ikκ i κ κ κ +κ κ +κ κ ( k )oh( ) ( k )inh( ) ik oh( ) ik inh( ) ( k )oh( ) ( k )inh( ) ik oh( ) ik inh( ) ik inh( ) ik oh( ) ( k )inh( ) ( k )oh( ) nh( ) ik oh( ) ( k )inh( ) ( k )oh( ) by umming an ubtrating the 1t an n row a well a the 3r an 4th row +κ κ κ κ 0 ( k +κ )inh( κ) 0 ikκinh( κ) κ κ +κ κ ikκ κ k +κ κ ( k )oh( ) 0 ik oh( ) 0 a = 0 ik oh( ) 0 ( k )oh( ) inh( ) 0 ( )inh( ) 0 Symmetri ubytem: ik k B ( k +κ )oh( κ ) ikκ oh( κ ) A 0 = inh( ) ( )inh( ) 0 κ κ +κ κ Aymmetri ubytem: a ik k B a ( k +κ )inh( κ ) ikκ inh( κ ) A 0 = oh( ) ( )oh( ) 0 κ κ +κ κ

Symmetri ubytem: Charateriti Equation Aymmetri ubytem: ( k +κ ) tanh( κ ) 4k κ κ tanh( κ ) = 0 ( k+κ) tanh( ) 4 κ k κ κtanh( κ ) = 0 Group veloity: g ω = k g = p + k k p g = 1 p p ω p ω

Potential Amplitue Ditribution #1 Potential amplitue ratio: A / B an A / B from the bounary onition a a Symmetri wave: ϕ= Aoh( κ x3)exp( ikx1) ikκ inh( κ ) ψ= A inh( κ x3 )exp( ikx1 ) ( k +κ )inh( κ ) Aymmetri wave: ϕ= A inh( κ x )exp( ikx ) a 3 1 ikκ oh( κ ) ψ= Aa oh( κ x3)exp( ikx1) ( k +κ )oh( κ )

Stre an Diplaement Ditribution # iplaement omponent: u 1 ϕ = x ψ x 1 3 u 3 ϕ = + x ψ x 3 1 #3 tre omponent: ϕ ϕ ϕ ψ τ = λ ( + ) + μ( ) x x 11 x 1 x3 x1 1 3 ϕ ϕ ϕ ψ τ = λ ( + ) + μ ( + ) x x 33 x 1 x3 x3 1 3 ϕ ψ ψ τ = τ = μ ( + ) 31 13 x 1 x3 x1 x3

Diperion Curve normalize veloity: η= Normalize Phae Veloity 5 4 3 1 0 S 0 A 0 normalize frequeny: Ω= k = π f A 1 S 1 S A 0 1 3 4 5 6 Normalize Frequeny ymmetri moe aymmetri moe Normalize Group Veloity 1 0 S 0 ymmetri moe aymmetri moe A 0 A 1 S 1 S A 0 1 3 4 5 6 Normalize Frequeny (teel, = 5,95 m/, = 3,141 m/)

Low-Frequeny Aymptote Lowet-Orer Symmetri Moe (0): Aymptoti Behavior E lim p = an 0 (1 g = p ω ρ ν ) ( k+ κ) tanh( ) 4 κ k κ κtanh( κ ) = 0 ( k+κ) 4 κ k κ κκ = 0 ( k+ κ) 4 k κ = 0 κ = k k an κ = k k ( k k) 4 ( k k k ) = 0 4k4 4kk 4 4 4 4 + k k + k k = 0 4kk 4 4 + k + k k = 0 k = k4 4( k k ) = 1

= 1 ν ν 1 ν μ (1 + ν) μ = 1 = = = ν 1 ν (1 ν) ρ (1 ν) ρ E = = (1 ν) ρ plate Lowet-Orer Aymmetri Moe (a0): E 3 ρ(1 ν ) lim p = 4 ω 0 an g = p ω ( k +κ ) tanh( κ ) 4k κ κ tanh( κ ) = 0 1t-orer approximation lim tanh( θ ) =θ+... θ 0 ( k+κ) 4 κ k κ κκ = 0 ( ) 0 k ( k + κ ) 4 κ = 0 k κ = or k = 0 (only at ω = 0 ) 3r-orer approximation θ3 lim tanh( θ ) = θ θ 0 3

3 3 3 3 κ ( ) ( ) 4 κ ( k +κ κ k κ κ κ ) = 0 3 3 κ ( ) (1 ) 4 ( κ )(1 k k k k k ) = 0 3 3 3 k4 ( ) ( ) 4 ( ) k k k k + k k k = 0 after further algebrai manipulation: 3 k4 (4 4 4 4)( ) 4 ( 4 4) k k k + k k k + k k k k + k = 0 3k4 4k6+ 4k4k 4 4 4 4 4 4 6 8 4 4 4 k k + k k k kk + kk + k k k + k k = 0 3k4 kk4 4 4 4 4 4 4 4 4 + k k k kk + kk k k + k k = 0 after negleting all but the highet power term in k (ine k i large) 3k4 4k4k 4 4 + k k = 0 4 4 = ω (1 ) 3 4 4 1 1 (1 ν = ω ) = ω 3 (1 ν) 3 1 ν

4 = E ω 3 ρ(1 ν ) plate ω = = f 3

Low-Frequeny Aymptote of the Funamental Lamb Moe teel plate ( = 5,900 m/, = 3,00 m/) 6 5 Phae Veloity [km/]. 4 3 1 0 funamental ymmetri moe (S0) thin-plate ilatational wave approximation funamental aymmetri moe (A0) thin-plate flexural wave approximation 0 1 3 4 5 Frequeny Thikne [MHz mm]

High-Frequeny Aymptote of the Funamental Moe Symmetri Moe: ( k +κ ) tanh( κ ) 4k κ κ tanh( κ ) = 0 Aymmetri Moe: ( k+κ) tanh( ) 4 κ k κ κtanh( κ ) = 0 k κ = k i pure real ine < κ = k k i pure real ine < lim tanh( θ ) = 1 θ ( k+κ) 4 k κ κ = 0 both funamental moe approah the Rayleigh veloity

High-Frequeny Aymptote of the Higher- Orer Moe Symmetri Moe: Aymmetri Moe: ( k +κ ) tanh( κ ) 4k κ κ tanh( κ ) = 0 ( k+κ) tanh( ) 4 κ k κ κtanh( κ ) = 0 k κ = k i pure real ine < κ = k k i pure imaginary ine > tanh( iθ ) = itan( θ ) κ = 0 all higher-orer moe approah the hear veloity

Cut-Off Frequenie k ω =, k ω =, k ω =, κ = k k, κ = k k, k 0 ik Symmetri Moe: κ = i pure imaginary, κ = i pure imaginary ik tanh( iθ ) = itan( θ ) ( k+ κ) tanh( ) 4 κ k κ κtanh( κ ) = 0 ( k k ) itan( k ) + 4k k k itan( k ) = 0 k 4 in( k)o( k ) = 0 o( k ) = 0 or in( k ) = 0 k π = (n 1) or k = nπ λ = (n 1) or = nλ f = (n 1) or 4 f = n where n = 1,,... u1( x3) = u1( x3) an u3( x3) = u3( x3)

Aymmetri Moe: ( k +κ ) tanh( κ ) 4k κ κ tanh( κ ) = 0 k 4 o( k)in( k ) = 0 o( k ) = 0 or in( k ) = 0 k π = (n 1) or k = nπ λ = (n 1) or = nλ f = (n 1) or 4 f = n u1( x3) = u1( x3) an u3( x3) = u3( x3) Cut-Off Frequenie ymmetri aymmetri hear f = n f ( 1) = n 4 ilatational ( 1) f = n f = n 4

general ymmetri Lamb moe pure tranvere longituinal reonane pure tranvere hear reonane general aymmetri Lamb moe pure tranvere longituinal reonane pure tranvere hear reonane