1.138J/2.062J, WAVE PROPAGATION CHAPTER THREE TWO DIMENSIONAL WAVES 1 Reflection and tranmission of sound at an inter- face Reference: Brekhovskikh and Godin §.2.2. The governing equation for sound in a homogeneous fluid is given by (7.31) and (7.32) in Chapter One. In term of the velocity potential defined by u = Vo it is 102 2012 =V² (1.1) where cedenotes the sound speed. Recall that the fluid pressure pe po Otealso satisfies the same equation. We first generalize the plane sinusoidal wave in three dimensional space (x,t) = kx-wt) where n is the unit vector in the direction of k. Here the phase function is 0(x,t) = kx-wte (1.2) (1.3) The equation of constant phase 6(x,t) 0, describes a moving surface. The wave mumber vector kkn is defined to be kkn Ve (1.4) hence is orthogonal to the surface of constant phase, and represents the direction of wave propagation. The frequency is defined to be 00 Ote (1.5) 2 Is (1.2) a solution? Let us check (1.7). V= = де Hence (1.1) is satisfied if wkee (1.6) Consider two semi-infinite fluids separated by the plane interface along ze 0. The sound speeds in the upper and lower fluids are crand c, respectively. Let a plane incident wave Argive from at the incident angle of 6 with respect to the zeaxis, p.explik(sin cos)] (1.7) implying that k(kk)k(sin, cos) (1.8) The motion is confined in the r, zeplane. On the same (incidence) side of the interface we have the reflected wave p. Rexplik(xsin 0+ cos 0)] (1.9) where Redenotes the reflection coefficient. The wavenumber vector is k(kk)k(sin, cos (0) (1.10) In the lower medium the transmitted wave has the pressure pTexplik (rsin fcos0)] (1.11) where Te is the transmission coefficient. Along the interface ze 0 we require the continuity of pressure and normal velocity. i.e., p=0,0 (1.12) and [w] = 10 ze 0,0 (1.13) Note that the governing equations for φ and H_z are governed by waves uncoupled from those for H_x and H_y, hence the in-plane displacement components (u_x, u_y) are independent of the out-of-plane component u_z. The in-plane displacements (u_x, u_y) are associated with dilatation and in-plane shear, represented respectively by φ and H_z, which will be referred to as the P wave and the SV wave. The out-of-plane displacement u_z is associated with H_x and H_y and will be referred to as the SH wave. Different physical situations arise for different boundary conditions. We shall consider first the half plane problem bounded by the plane y = 0. 5 Reflection of elastic waves from a plane boundary Several types of boundary conditions can be prescribed on the plane boundary: (i) dynamic: the stress components only (the traction condition); (ii) kinematic: the displacement components only; or (iii) a combination of stress components and displacement components. Most difficult are (iv) the mixed conditions in which stresses are given over part of the boundary and displacements over the other. We consider the simplest case where the plane y = 0 is completely free of external stresses, τ_yy = τ_xy = 0, τ_yx = 0 (5.23) and (5.24) It is clear that (5.23) affects the P and SV waves only, while (5.24) affects the SH wave only. Therefore we have two uncoupled problems each of which can be treated separately. 5.1 P and SV waves Consider the case where only P and SV waves are present, then H_x = H_z = 0 Let all waves have wavenumber vectors inclined in the positive z direction: φ = f(y)e^(ikz-iωt) H_z = h_z(y)e^(ikz-iωt)

Two Dimensional Waves

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