x σzz, σzy

σzx z = ±d

"u = ∇φ+∇× ψ

φ ψ

2dx

1

cL

∂2φ

∂t2= ∇

2φ =∂2φ

∂x2+

∂2φ

∂z2

1

cT

∂2ψ

∂t2= ∇

2ψ =∂2ψ

∂x2+

∂2ψ

∂z2

cL cT

φ = (A1 sin(pz) + A2 cos(pz))ei(kx−ωt)

ψ = (B1 sin(qz) + B2 cos(qz))ei(kx−ωt)

p2 =ω2

c2L− k2

q2 =ω2

c2T− k2

k =2π

λ=

ω

cP

ω cP

u =∂φ

∂x+

∂ψ

∂z

v = 0

w =∂φ

∂z−

∂ψ

∂x

σzx = µ

(

∂2φ

∂x∂z−

∂2ψ

∂x2+

∂2ψ

∂z2

)

σzz = λ

(

∂2φ

∂x2+

∂2φ

∂z2

)

+ 2µ

(

∂2φ

∂z2+

∂2ψ

∂x∂z

)

µ λ u v w

x y z

u = ikA2 cos(pz) + bqB1 cos(qz)

w = −pA2 sin(pz)− ikB1 sin(qz)

σzx = µ(−2ikpA2 sin(pz) + (k2− q2)B1 sin(qz))

σzz = −λ(k2 + p2)A2 cos(pz)− 2µ(p2A2 cos(pz) + ikqB1 cos(qz))

u = ikA1 sin(pz)− qB2 sin(qz)

w = pA1 cos(pz)− ikB2 cos(qz)

σzx = µ(2ikpA1 cos(pz) + (k2− q2)B2 cos(qz))

σzz = −λ(k2 + p2)A1 sin(pz)− 2µ(p2A1 sin(pz) + ikqB2 sin(qz))

tan(qh)

tan(ph)= −

4k2pq

(q2 − k2)2

tan(qh)

tan(ph)= −

(q2 − k2)2

4k2pq

sin θ =cwcp

cw

cp cw

cp

∆tij =

√

(xd − xi)2 − (yd − yi)2 −√

(xd − xj)2 − (yd − yj)2

vg

(xd, yd) (xi, yi) (xj, yj)

∆tij

vg

∂f

∂x=

f(x0 +δ

2)− f(x0 −

δ

2)

δ

∂P

∂t= −ρc2∇ · v

∂v

∂t= −

1

ρ∇P

P c v

ρ

∂P

∂t= −ρc2

(

∂vx∂x

+∂vy∂y

+∂vz∂z

)

∂vx∂t

= −1

ρ

∂P

∂x

∂vy∂t

= −1

ρ

∂P

∂y

∂vz∂t

= −1

ρ

∂P

∂z

P (x, y, z, t) = P (mδx, nδy, pδz, qδt) = P q[m,n, p]

vx(x, y, z, t) = vx([m+ 1/2]δx, nδy, pδz, [q + 1/2]δt) = vq+1/2x [m,n, p]

vy(x, y, z, t) = vx(mδx, [n+ 1/2]δy, pδz, [q + 1/2]δt) = vq+1/2y [m,n, p]

vx(x, y, z, t) = vx(mδx, nδy, [p+ 1/2]δz, [q + 1/2]δt) = vq+1/2z [m,n, p]

(m,n, p, q) (x, y, z, t)

δx = δy = δz = δ

P [m,n, p] = P q−1[m,n, p]− ρc2δt

δ(vq−1/2

z [m,n, p]− vq−1/2z [m− 1, n, p] +

vq−1/2y [m,n, p]− vq−1/2

y [m,n− 1, p] + vq−1/2z [m,n, p]− vq−1/2

z [m,n, p− 1]

vq+1/2x [m,n, p] = vq−1/2

x [m,n, p]−2δtδ

(P q[m+ 1, n, p]− P q[m,n, p])

ρ[m,n, p] + ρ[m+ 1, n, p]

vq+1/2y [m,n, p] = vq−1/2

y [m,n, p]−2δtδ

(P q[m,n+ 1, p]− P q[m,n, p])

ρ[m,n, p] + ρ[m,n+ 1, p]

vq+1/2z [m,n, p] = vq−1/2

z [m,n, p]−2δtδ

(P q[m,n, p+ 1]− P q[m,n, p+ 1])

ρ[m,n, p+ 1] + ρ[m,n, p+ 1]