2.20 - Marine Hydrodynamics, Spring 2005
Lecture 21
2.20 - Marine Hydrodynamics
Lecture 21
6.4 Superposition of Linear Plane Progressive Waves
1. Oblique Plane Waves
v
k
kz
z
kx
v
k = (k x , k z )
Vp
θ
x
(Looking up the y-axis from
below the surface)
Consider wave propagation at an angle θ to the x-axis
k·x
η =A cos(kx cos θ + kz sin θ −ωt) = A cos (kx x + kz z − ωt)
gA cosh k (y + h)
φ=
sin (kx cos θ + kz sin θ − ωt)
ω
cosh kh
ω =gk tanh kh; kx = k cos θ, kz = k sin θ, k = kx + kz
1
2. Standing Waves
Same A, k, ω, no phase shift
+
η =A cos (kx − ωt) + A cos (−kx − ωt) = 2A cos kx cos ωt
2gA cosh k (y + h)
φ=−
cos kx sin ωt
ω
cosh kh
y
t = 0, T, 2T, …
2A
amplitude
90o at all times
x
node
t=
T 3T
, ,L
2 2
antinode
t=
T 3T 5T
, , L
4 4 4
∂η
∂φ
nπ
nλ
∼
= · · · sin kx = 0 at x = 0,
=
∂x
∂x
k
2
∂φ
Therefore,
= 0. To obtain a standing wave, it is necessary to have perfect
∂x
x
reflection at the wall at x = 0.
AR
Define the reflection coefficient as R ≡
(≤ 1).
AI
y
A I = AR
AR
R=
=1
AI
x
2
3. Oblique Standing Waves
ηI =A cos (kx cos θ + kz sin θ − ωt)
ηR =A cos (kx cos (π − θ) + kz sin (π − θ) − ωt)
z
ηR
θR
θ
θI
θ
x
θ R = π − θI
ηI
Note: same A, R = 1.
k x
k z−ωt
x
z
ηT = ηI + ηR = 2A cos (kx cos θ) cos (kz sin θ − ωt)
standing wave in x
propagating wave in z
and
λx =
2π
;
k cos θ
VPx = 0;
λz =
2π
;
k sin θ
VPz =
Check:
∂φ
∂η
∼
∼ · · · sin (kx cos θ) = 0 on x = 0
∂x
∂x
3
ω
k sin θ
4. Partial Reflection
ηR
ηI
+
ηI =AI cos (kx − ωt) = AI Re ei kx−ωt
ηR =AR cos (kx + ωt + δ) = AI Re R e−i kx
ωt
R: Complex reflection coefficient
AR
AI
ηT =ηI + ηR = AI Re ei kx−ωt 1 + Re−
|ηT | =AI 1 + |R| + 2 |R| cos (2kx + δ)
R = |R| e−iδ , |R| =
| ηT |
AI
λ
ikx
free surface
wave envelope
2
1+ | R |2
∇
x
node
antinode
At node,
|ηT | = |ηT |
= AI (1 − |R|) at cos (2kx + δ) = −1 or 2kx + δ = (2n + 1) π
At antinode,
|ηT | = |ηT |
= AI (1 + |R|) at cos (2kx + δ) = 1 or 2kx + δ = 2nπ
2kL = 2π so L =
|R | =
|ηT |
|ηT |
− |ηT |
+ |ηT |
4
λ
2
= |R (k)|
5. Wave Group
2 waves, same amplitude A and direction, but ω and k very close to each other.
VP1
η = Aei k1 x−ω1 t
η = Aei k2 x−ω2 t
VP2
ω
,
=ω
,
(k , ) and VP1 ≈ VP2
with δk = k − k and δω = ω − ω
ηT = η + η = Aei k1 x−ω1 t 1 + ei δkx−δωt
Vg
λg =
2A
VP1 ≈ VP2
T=
|ηT |
|ηT |
2π
δk
2π
ω
Tg =
2π
= λ1 ≈ λ 2
k1
= 2 |A| when δkx − δωt = 2nπ
= 0 when δkx − δωt = (2n + 1) π
5
2π
δω
⎫
⎬
⎭
xg = Vg t, δkVg t−(δω) t = 0 then Vg =
δω
δk
In the limit,
dω ,
δk, δω → 0, Vg =
dk k1 ≈k2 ≈k
and since
ω = gk tanh kh ⇒
ω 1 2kh
Vg =
1+
k2
sinh 2kh
Vp
⎫
⎪
⎪
(a) deep water kh >> 1
⎪
⎪
⎪
⎪
⎪
⎪
n = VVgp =
⎪
⎪
⎪
⎪
(b) shallow water kh << 1
⎬
⎪
n=
= 1 (no dispersion) ⎪
⎪
⎪
⎪
⎪
⎪
⎪
(c) intermediate depth
⎪
⎪
⎪
⎪
⎭
<n<1
Vg
Vp
n
Vg
VP
Vg ≤ Vp
Appear
6
Disappear
6.5 Wave Energy - Energy Associated with Wave Motion.
For a single plane progressive wave:
Energy per unit surface area of wave
• Potential energy PE
PE without wave = ρgydy = − ρgh
η
KEwave =
−h
η
dy ρ (u + v )
−h
Deep water = · · · =
ρgydy = ρg (η − h )
PE with wave
• Kinetic energy KE
−h
ρgA
to leading order
KE const in x,t
Finite depth = · · ·
PEwave = ρgη = ρgA cos (kx − ωt)
Average energy over one period or one wavelength
PEwave = ρgA
KEwave = ρgA at any h
• Total wave energy in deep water:
E = PE + KE = ρgA cos (kx − ωt) +
• Average wave energy E (over 1 period or 1 wavelength) for any water depth:
E = ρgA [
+
↑
PE
] = ρgA = Es ,
↑
KE
Es ≡ Specific Energy: total average wave energy per unit surface area.
• Linear waves: PE = KE = 12 Es
(equipartition).
Vp
x
• Nonlinear waves: KE > PE.
E
Es
1
Vp
PE = Es cos2 (kx − ωt)
PE = 12 E
KE = 12 E
½
x
Recall: cos x =
7
+ cos 2x
=
6.6 Energy Propagation - Group Velocity
S
Vp
x
E = E s per area
V
Consider a fixed control volume V to the right of ‘screen’ S. Conservation of energy:
dW
dE
=
=
Jdt
dt
rate of work done on S
rate of change of energy in V
energy flux left to right
where
η
J- =
pu dy with p = −ρ
−h
ω 1+
J- = ρgA
k E
Vp
n
Vg
e.g. A = 3m, T = 10 sec → J- = 400KW /m
8
dφ
+ gy
dt
kh
kh
and u =
∂φ
∂x
= E (nVp ) = EVg
6.7 Equation of Energy Conservation
∆x
2
1
1
x
E = E (x ),
2
=
(x )
h = h(x)
J- − J- ∆t = ∆E∆x
∂J-
∆x + · · ·
J- = J- +
∂x ∂E ∂J+
= 0, but J- = Vg E
∂t
∂x
∂E
∂ +
Vg E = 0
∂t
∂x
1.
∂E
= 0, Vg E = constant in x for any h(x).
∂t
2. Vg = constant (i.e., constant depth, δk << k)
∂
∂
+ Vg
∂t
∂x
E = 0, so E = E (x − Vg t) or A = A (x − Vg t)
i.e., wave packet moves at Vg .
9
6.8 Steady Ship Waves, Wave Resistance
D
U
Vp = U
2A
E = 21 ρgA2
E = 0 ahead of ship
(
= Vg E = ( 12 U ) 12 ρgA2
)
L
x=0
C.V.
• Ship wave resistance drag Dw
Rate of work done = rate of energy increase
d EL = EU
Dw U + J- =
dt
deep water
1
Dw = (EU − EU 2 ) = E = ρgA ⇒ Dw ∝ A
U
force / length
energy / area
• Amplitude of generated waves
The amplitude A depends on U and the ship geometry. Let ≡ effective length.
L
-
+l
To approximate the wave amplitude A superimpose a bow wave (ηb ) and a stern wave
(ηs ).
ηb = a cos (kx) and ηs = −a cos (k (x + ))
ηT = ηb + ηs
A = |ηT |
= 2a sin k
← envelope amplitude
g Dw = ρgA = ρga sin
k
⇒ Dw = ρga sin
U2
• Wavelength of generated waves To obtain the wave length, observe that the phase
speed of the waves must equal U . For deep water, we therefore have
g
U
ω
deep
−→
= U , or λ = 2π
Vp = U ⇒ = U water
k
k
g
10
• Summary Steady ship waves in deep water.
U = ship speed
g
U
g
Vp =
and λ = 2π
= U ; so k =
k
U
g
L =ship length, ∼ L
g 1
1
∼
∼
Dw =ρga sin
ρga sin
= ρga sin
U2 =
2FrL
2FrL
Fl =
max at:
Dw ρga 2
1
π
≈ 0.56 ⇒
U hull ≈ 0.56 gl ≅ 0.56 gL ⇒ U hull
∝ L
1
0
Fl =
1
π
Increasing U
Small speed U
• Short waves
• Significant wave cancellation
• Dw ~ small
11
U
, where l ≤ L
gl