DanielDuaneThesis - William & Maryphysics.wm.edu/Seniorthesis/SeniorTheses2015/Duane_Daniel.pdf ·...
Transcript of DanielDuaneThesis - William & Maryphysics.wm.edu/Seniorthesis/SeniorTheses2015/Duane_Daniel.pdf ·...
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x σzz, σzy
σzx z = ±d
"u = ∇φ+∇× ψ
φ ψ
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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)
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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
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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))
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tan(qh)
tan(ph)= −
4k2pq
(q2 − k2)2
tan(qh)
tan(ph)= −
(q2 − k2)2
4k2pq
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sin θ =cwcp
cw
cp cw
cp
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∆tij =
√
(xd − xi)2 − (yd − yi)2 −√
(xd − xj)2 − (yd − yj)2
vg
(xd, yd) (xi, yi) (xj, yj)
∆tij
vg
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∂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
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∂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]
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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]
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