lect3 - 3.1.Reection of sound by an interface 1...

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3.1.Reflection of sound by an interface 1 1.138J/2.062J/18.376J, WAVE PROPAGATION Fall, 2006 MIT Notes by C. C. Mei CHAPTER THREE TWO DIMENSIONAL WAVES 1 Reflection and tranmission of sound at an inter- face Reference : L. M. Brekhovskikh and O. A. Godin: Acoustics of Layered Media I. Springer. § .2.2. The governing equation for sound in a honmogeneous fluid is given by (7.31) and (7.32) in Chapter One. In term of the the veloctiy potential de±ned by u = φ (1.1) it is 1 c 2 2 φ ∂t 2 = 2 φ (1.2) where c denotes the sound speed. Recall that the fluid pressure p = ρ∂φ/∂t (1.3) also satis±es the same equation. 1.1 Plane wave in InFnite space Let us ±rst consider a plane sinusoidal wave in three dimensional space φ ( x ,t )= φ o e i ( k · x ωt ) = φ o e i ( k n · x ) (1.4) Here the phase function is θ ( x k · x (1.5)
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3.1.Reflection of sound by an interface 2 The equation of constant phase θ ( x ,t )= θ o describes a moving surface. The wave number vector k = k n is deFned to be k = k n = θ (1.6) hence is orthogonal to the surface of constant phase, and represens the direction of wave propagation. The frequency is deFned to be ω = ∂θ ∂t (1.7) Is (1.4) a solution? Let us check (1.2). φ = µ ∂x , ∂y , ∂z φ = i k φ 2 φ = ∇·∇ φ = i k · i k φ = k 2 φ 2 φ 2 = ω 2 φ Hence (1.2) is satisFed if ω = kc. (1.8) Sound in an inFnite space is non-dispersive. 1.2 Two-dimensional reflection from a plane interface REfering to Fgure ?? , let us cnsider two semi-inFnite fluids separated by the plane interface along z = 0. The lower fluid is distinguished from the upper fluid by the subscript ”1”. The densities and sound speeds in the upper and lower fluids are ρ, c and ρ 1 ,c 1 respectively. Let a plane incident wave arive from z> 0 at the incident angle of θ with respect to the z axis, the sound pressure and the velocity potential are p i = P 0 exp[ ik ( x sin θ z cos θ )] (1.9) The velocity potential is φ i = iP 0 ωρ exp[ ik ( x sin θ z cos θ ] (1.10) The indient wave number vector is k i =( k i x ,k i z k (sin θ, cos θ ) (1.11)
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3.1.Reflection of sound by an interface 3 Figure 1: Plane wave incident towards the interface of two fluids. The motion is con±ned in the x, z plane. On the same (incidence) side of the interface we have the reflected wave p r = R exp[ ik ( x sin θ + z cos θ )] (1.12) where R denotes the reflection coefficient. The wavenumber vector is k r =( k r x ,k r z )= k (sin θ, cos θ ) (1.13) The total pressure and potential are p = P 0 { exp[ ik ( x sin θ z cos θ )] + R exp[ ik ( x sin θ + z cos θ )] } (1.14) φ = iP 0 ρω { exp[ ik ( x sin θ z cos θ )] + R exp[ ik ( x sin θ + z cos θ )] } (1.15) In the lower medium z< 0 the transmitted wave has the pressure p 1 = TP 0 exp[ ik 1 ( x sin θ 1 z cos θ 1 )] (1.16) where T is the transmission coefficient, and the potential φ 1 = iP 0 ρ 1 ω T exp[ ik 1 ( x sin θ 1 z cos θ 1 )] (1.17) Along the interface z = 0 we require the continutiy of pressure and normal velocity, i.e., p = p 1 ,z = 0 (1.18) k i k r k 1 θ θ 1 Figure by MIT OCW.
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3.1.Reflection of sound by an interface 4 and w = w 1 ,z =0 , (1.19) Applying (1.18), we get P 0 © e ikx sin θ + Re ikx sin θ ª = TP 0 e ik 1 x
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lect3 - 3.1.Reection of sound by an interface 1...

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