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While ft zv due to a forward propagating wave

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Unformatted text preview: F(t-z/V) due to a forward propagating wave propagating in the positive z direction. Wave propagation from t=0 to t=t1 Putting : , (8) (9) then, The shear stress τ is also expressed by: (10) The coefficient G/V denotes impedance: : Impedance Eq.(10) is expressed by: ∂E(ς ) ∂F(η) τ( z, t ) = ρV { − } ∂ς ∂η (11) (12) [problem-1] Proof that Eq.(7) is the solution of Eq.(6). (proof) (a) (b) (c) (d) Substituting Eq.(b) and Eq.(d) into Eq.(6): (e) (3) Transmission and Reflection Consider the wave propagation in two semi-infinite media as shown in the figure. E11((t+z/V1) E t +z/V1 : Transmission Wave G1 ρ1 G2 ρ2 Medium 1 Medium 2 z E2(tt+z/V22) E ( +z/V ) : Incident Wave F (t - z/V 2 F22(t-z/V2)) : Reflection wave When the wave E2(t+z/V2) propagates upward and reaches at the interface (z=0) between two media, this wave is divided into the transmission wave E1(t+z/V1) and the reflection wave F2(t-z/V2) . The displacement and shear stress in medium 1: (13) ρ1,V1 Medium 1 (14) E1(t+z/V1) u1(z,t) τ1(z,t) where, (15) z Medium 2 ρ2,V2 Similarly, for medium 2: (16) (17) ρ1,V1 where, (18) Medium 1 (19) τ2(z,t) u2(z,t) z E2(t+z/V2) F2(t-z/V2) Medium 2 ρ2,V2 The boundary conditions at the interface(z=0): (20) (21) Medium 1 ρ1,V1 E1(t+z/V1) τ1(z=0,t) u1(z,t) τ2(z=0,t) u2(z,t) z E2(t+z/V2) Medium 2 F2(t-z/V2) ρ2,V2 Substitutin...
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