Linear (and nonlinear) wave propagation in fluids
Christophe Bailly
Université de Lyon, Ecole Centrale de Lyon & LMFA - UMR CNRS 5509
http://acoustique.ec-lyon.fr
1
x General outline of the course q
Models for linear wave propagation in fluids
– Introduction, surface gravity waves, internal waves, acoustic waves,
waves in rotating flows, ...
– longitudinal and transverse waves, dispersion relation, phase velocity,
group velocity
Theories for linear wave propagation
– Fourier integral solution, asymptotic behaviour (stationary phase)
– Propagation of energy, ray theory (high-frequency approximation for inhomogeneous medium)
Introduction to nonlinear wave propagation
– Euler equations, N-waves, Weak shocks, Burgers
– Solitary waves (Korteweg - de Vries)
2
Wave propagation in fluids - S7 ECL 2A - déc. 2015 - cb1
x General outline of the course q
Schedule
monday 14/12/2015
friday 18/12/2015
monday 04/01/2016
friday 08/01/2016
monday 11/01/2016
friday 15/01/2016
monday 18/01/2016
thursday 21/01/2016
CM1 & TD1
TD2 (2h homework)
CM2 & TD3
TD4 (2h homework)
CM3 & TD5 (2h homework)
CM4
CM5 & TD6
Examen (8h00-10h15, room 105)
Laptop required for small classes !
3
Wave propagation in fluids - S7 ECL 2A - déc. 2015 - cb1
x Wave propagation in fluids q
Textbooks
Guyon, E., Hulin, J.P. & Petit, L., 2001, Hydrodynamique physique, EDP Sciences / Editions du CNRS, Paris Meudon.
Lighthill, J., 1978, Waves in fluids, Cambridge University Press, Cambridge.
Johnson, R. H., 1997, A modern introduction to the mathematical theory of water waves, Cambridge University Press,
Cambridge,
Morse, P.M. & Ingard, K.U., 1986, Theoretical acoustics, Princeton University Press, Princeton, New Jersey.
Ockendon, H. & Ockendon, J. R., 2000, Waves and compressible flow, Springer-Verlag, New York, New-York.
Pierce, A.D., 1994, Acoustics, Acoustical Society of America, third edition.
Rayleigh, J. W. S., 1877, The theory of sound, Dover Publications, New York, 2nd edition (1945), New-York.
Temkin, S., 2001, Elements of acoustics, Acoustical Society of America through the American Institute of Physics.
Thual, O., 2005, Des ondes et des fluides, Cépaduès-éditions, Toulouse.
Whitham, G.B., 1974, Linear and nonlinear waves, Wiley-Interscience, New-York.
4
Wave propagation in fluids - S7 ECL 2A - déc. 2015 - cb1
Waves in Fluids : models for linear wave propagation
5
Wave propagation in fluids - S7 ECL 2A - déc. 2015 - cb1
x Linear dispersive waves q
Introduction
Acoustic waves (in a homogeneous medium at rest),
hyperbolic wave equation
2 ′
∂ 2 p′
2 ∂ p
− c∞ 2 = 0
2
∂t
∂x1
General one dimensional solution p′(x1, t) = pl(x1 + c∞t) + pr (x1 − c∞t)
known as d’Alembert’s solution
vϕ
Dispersion relation
For a plane wave (Fourier component)
∼ ei(k1x1−ωt)
ω = ±c∞k1 non dispersive waves
pr ∼ eik1(x1−c∞t)
phase velocity vφ = ω/k1 = c∞
6
Wave propagation in fluids - S7 ECL 2A - déc. 2015 - cb1
x1
x Linear dispersive waves q
Introduction
1-D dispersive waves
η(x1, t) = Aei(k1x1−ωt) = Aeik1(x1−vφ t)
with ω = ω(k1)
The phase speed vφ = ω(k1)/k1 is not constant and depends on k1
ω = ω(k1) or D (k1, ω) = 0 dispersion relation, obtained by requiring the
plane waves to be solution of the linearized equations of motion.
The general solution is a superposition of modes ∝ ei(k1x1−ωt) through a Fourier
integral : wave packet characterized by a group velocity vg = ∂ω/∂k1
vg
x1
7
Wave propagation in fluids - S7 ECL 2A - déc. 2015 - cb1
x Surface gravity waves q
Body moving steadily in deep water
USS John F. Kennedy aircraft
carrier and accompanying destroyers
Ducks swimming across a lake
Kelvin’s angle of the wake 2α = 2asin(1/3) ≃ 39 deg !
8
Wave propagation in fluids - S7 ECL 2A - déc. 2015 - cb1
x Surface gravity waves q
Formulation
pa
h
ζ(x1, x2, t)
x3
U∞
x1
Flow velocity u = (u1, u2, u3), potential flow u = ∇φ
incompressibility ∇ · u = 0, Laplace’s equation ∇2φ = 0
Bernouilli’s equation for potential flows
∂φ 1
+ ρ∇φ · ∇φ + ρgx3 + p = cte
ρ
∂t
2
Boundary conditions
∂φ
= 0 on x3 = −h (impermeable wall)
u3 =
∂x3
and free boundary problem on x3 = ζ(x1, x2, t)
9
Wave propagation in fluids - S7 ECL 2A - déc. 2015 - cb1
(1)
x Surface gravity waves q
Surface tension
which introduces a pressure difference across a curved surface
Surface tension prevents the paper clip
(denser than water) from submerging.
10
Capillary waves
(ripples – short waves λ ≤ 2 cm)
produced by a dropplet of wine !
Courtesy of Olivier Marsden (2010)
Wave propagation in fluids - S7 ECL 2A - déc. 2015 - cb1
x Surface gravity waves q
Surface tension
Work δWt needed to increase the surface area of a mass of liquid by an amount
dS, δWt = γt dS (surface tension γt in J.m−2 = N.m−1)
Total energy variation δW
δW = −pw dVw − padVa + γt dS
3
2
dVw = d(4πr /3) = 4πr dr
2
dS = d(4πr ) = 8πrdr
dVa = −dVw
2γt
balance δW = 0 =⇒ pw − pa =
r
γt air-water ≃ 0.0728 N.m−1 ( 20oC)
r = 1 mm, ∆p/pa ≃ 0.14%
11
Wave propagation in fluids - S7 ECL 2A - déc. 2015 - cb1
dr
r
water drop
air, pa
x Surface gravity waves q
Young-Laplace equation (1805)
pa − p = γt Cf
where Cf = −∇ · n→a is the mean curvature in fluid mechanics
(the curvature is positive if the surface curves "towards" the normal, convex)
1 ∂(r 2 × 1) 2
=
∇·n= 2
r
∂r
r
1-D interface ζ(x1)
pa
ζ(x1)
p > pa
pa − p = γt Cf
Cf =
12
ζ x1 x 1
1+
for the water drop with n = er
2-D interface ζ(x1, x2)
(1 + ζx22 )ζx1x1 + (1 + ζx21 )ζx2x2 − 2ζx1 ζx2 ζx1x2
Cf =
(1 + ζx21 + ζx22 )3/2
≃ ζ x1 x1 + ζ x2 x2
by linearization
3/2
2
ζ x1
ζx1 ≡ ∂ζ/∂x1, ...
Wave propagation in fluids - S7 ECL 2A - déc. 2015 - cb1
x Surface gravity waves q
Formulation in incompressible flow : free boundary problem
(Kelvin, 1871)
Kinematic condition for the surface deformation ζ,
interface defined by f ≡ ζ(x1, x2, t) − x3 = 0
Fluid particles on the boundary always remain part on this free surface
(the free surface moves with the fluid), that is df/dt = 0
df
=0
dt
=⇒
∂ζ
∂ζ
∂ζ
+ u2
− u3 = 0
+ u1
∂t
∂x1
∂x2
∂ζ
∂ζ
∂ζ
+ u1
+ u2
u3 =
∂t
∂x1
∂x2
dζ(x1, x2, t)
= u3(x1, x2, ζ, t)
dt
13
Wave propagation in fluids - S7 ECL 2A - déc. 2015 - cb1
(2)
x Surface gravity waves q
Formulation in incompressible flow : free boundary problem
Kinematic condition for the surface deformation ζ, Eqs (1) - (2)

∂φ 1



 ρ ∂t + 2 ρ∇φ · ∇φ + ρgζ + pa − γt Cf = pa
on x3 = ζ(x1, x2, t)

∂ζ ∂φ ∂ζ
∂φ ∂ζ
∂φ


=
+
+

∂x3
∂t ∂x1 ∂x1 ∂x2 ∂x2
Linearization, velocity potential φ = U∞x1 + φ′,

2
2
′
′
∂
ζ
∂φ
∂
ζ
∂φ



+ 2 =0
 ρ ∂t + ρU∞ ∂x + ρgζ − γt
2
∂x1 ∂x2
1

∂φ′ ∂ζ
∂ζ



=
+ U∞
∂x3
∂t
∂x1
on x3 = 0
Wave equation obtained by applying D/Dt to eliminate ζ
2
2
′
γt ∂ ζ ∂ ζ
∂
∂
D Dφ
D
+ gζ −
≡
+
U
=
0
+
∞
Dt Dt
ρ ∂x12 ∂x22
Dt
∂t
∂x1
14
Wave propagation in fluids - S7 ECL 2A - déc. 2015 - cb1
x Surface gravity waves q
In summary : 2-D surface waves
∇2φ′ = 0
∂φ′
= 0 on x3 = −h (bottom)
∂x3
′
2
2
2 ′
′
∂
∂φ
γt ∂
D φ
∂φ
+
−
= 0 on x3 = 0 (surface)
+
g
2
2
2
Dt
∂x3
ρ ∂x1 ∂x2 ∂x3
« Phare des Baleines » (Lighthouse of
the Whales, by Vauban in 1682)
Photography taken by Michel Griffon
cross sea : two wave systems traveling at oblique angles
15
Wave propagation in fluids - S7 ECL 2A - déc. 2015 - cb1
(3)
(4)
(5)
x Surface gravity waves q
The dispersion relation for surface waves
Let us try to find a normal mode solution in Eq. (1), of the form
φ′ = ψ(x3)ei(k1x1+k2x2−ωt)
q
2
d
ψ
2
2
2
2
∇2φ′ = 0 =⇒
k
+
k
−
(k
+
k
)ψ
=
0
k
≡
1
2
1
2
dx32
– Waves on deep water (λ ≪ h, also kh → ∞)
With Eq. (2), ψ(x3) = A0ekx3 , dispersion relation with Eq. (3)
γt 3
2
−(k1U∞ − ω) + gk + k = 0
ρ
– Waves on water of a finite depth h
With Eq. (2), ψ(x3) = A0 cosh[k(x3 + h)], dispersion relation with Eq. (3)
γt
−(k1U∞ − ω)2 + gk + k 3 tanh(kh) = 0
ρ
16
Wave propagation in fluids - S7 ECL 2A - déc. 2015 - cb1
x Surface gravity waves q
The dispersion relation for surface waves (Kelvin, 1871)
with U∞ = 0, no running stream to simplify the discussion
2
ω =
γt 2
1+
k gk tanh(kh)
ρg
Capillary waves
r
γt
lc ≡
≃ 2.7 mm
ρg
dispersive waves
capillary length (air-water interface)
klc = 1 =⇒ λ = 2πlc ≃ 1.7 cm
Only important for short waves (‘ripples’)
2
2
λ ≥ lc , kh ≫ 1 ω ≃ 1 + (klc ) gk
2 1/2
Phase velocity vφ = (glc )/(klc ) 1 + (klc )
p
and minimum reached for klc = 1, vφ = 2glc ≃ 0.23 m.s−1
17
Wave propagation in fluids - S7 ECL 2A - déc. 2015 - cb1
x Surface gravity waves q
The dispersion relation for surface waves
2
2
ω = 1 + (klc ) gk tanh(kh)
2 gh
= 1 + (klc )
tanh(kh)
kh
deep water
(klc ≪ 1)
λ ≪ h, kh → ∞
shallow water
(klc ≪ 1)
ω2 = gk tanh(kh)
capillary waves
(klc ≥ 1)
18
vφ2
ω2 = gk
vφ2 = g/k
vg = vφ /2
vφ2 = g/k tanh(kh)
2
2
ω = 1 + (klc ) gk tanh(kh)
2
2
vφ = 1 + (klc ) g/k tanh(kh)
Wave propagation in fluids - S7 ECL 2A - déc. 2015 - cb1
x Surface gravity waves q
The dispersion relation for surface waves
10
deep water
p
vϕ = g/k
vϕ (m/s)
8
6
shallow
p water
vϕ = (g/k) tanh(kh)
4
2
capillary
waves
gravity
waves
0 −1
10
19
(h = 2.7 m)
10
klc = 1
0
1
10
10
k (1/m)
2
Wave propagation in fluids - S7 ECL 2A - déc. 2015 - cb1
10
3
10
4
x Surface gravity waves q
The dispersion relation for surface waves
For long waves (λ ≫ lc ), relation obtained by Lagrange
ω2 ≃ gk tanh(kh)
p
ω ≃ gk in deep water (kh → ∞)
p
vφ ≃ g/k, vg = vφ /2
Joseph Louis Lagrange
(1736–1813)
−1
T = 8 s, λ = 100 m, vφ ≃ 12.5 m.s
(propagation of crests)
0
−h
−4h
−2h
0
2h
Snapshot of the solution obtained for λ = 2h
20
Wave propagation in fluids - S7 ECL 2A - déc. 2015 - cb1
4h
x Surface gravity waves q
Wave refraction
Alignment of wave crests arriving near the shore
In deep water,
p
p
ω0 = gk vφ = g/k
Near the coast, h ց
ω02 = gk tanh(kh)
=⇒ λ, vφ ց
Reduction of both the wavelength and the wave speed near coasts
by shallow-water effects
for h = 10 m, λ ≃ 70.9 m, vφ = 8.9 m.s−1
for h = 1 m, λ ≃ 24.8 m, vφ = 3.1 m.s−1
21
Wave propagation in fluids - S7 ECL 2A - déc. 2015 - cb1
x Surface gravity waves q
Wave refraction and diffraction
Aerial photo of an area near
Kiberg on the coast of Finnmark in Norway (taken 12
June 1976 by Fjellanger Widerøe A.S.)
22
Wave propagation in fluids - S7 ECL 2A - déc. 2015 - cb1
x Surface gravity waves q
The dispersion relation for surface waves on a running stream
(k1U∞ − ω)2 = gk tanh(kh), stationnary waves ω → 0
2-D case (x1, x3), k3 = k = k1
tanh(kh)
= gh
kh
2
Only solutions for U∞
≤ gh, Froude number
U∞
Fr = p
< 1, subcritical flow
gh
3-D case, k3 = k =
q
0.8
tanh(ξ)/ξ
2
U∞
1.0
0.6
0.4
0.2
0.0
0
4
6
8
10
ξ
k12 + k22
2 2
U∞
k1 = gk tanh(kh)
2
h → 0, (U∞
− gh)k12 = ghk22, only solutions
for Fr > 1, supercritical flow (analogous to
supersonic flow)
23
2
Wave propagation in fluids - S7 ECL 2A - déc. 2015 - cb1
William
Froude
(1810 – 1879)
x Internal gravity waves q
Atmospheric internal gravity waves off Australia
(taken by Terra Satellite on Nov. 2003 - NASA)
24
Wave propagation in fluids - S7 ECL 2A - déc. 2015 - cb1
x Internal gravity waves q
Oscillations in the presence of gravity (atmosphere, ocean)
Stratified medium at rest, ρ0(x3), p0(x3) satisfying
the hydrostatic equation
dp0
= −ρ0g
dx3
Fluid particle moving from altitude x3 to x3 + δx3
x3 + δx3
x3
pp ?
ρp ?
The pressure of the fluid particle at x3 + δx3 is
pp = p0(x3 + δx3) ≃ p0(x3) − ρ0(x3)gδx3
pp = p0(x3)
ρp = ρ0(x3)
Assuming a reversible (adiabatic) process, the
density of the particle at x3 + δx3 is
ρp(x3 + δx3)
pp(x3 + δx3) 1/γ
ρ0gδx3
=
≃ 1−
ρp(x3)
pp(x3)
γp0
g
δx3
=⇒ ρp(x3 + dx3) ≃ ρ0(x3) − ρ0(x3) 2
c0 (x3)
25
Wave propagation in fluids - S7 ECL 2A - déc. 2015 - cb1
x Internal gravity waves q
Oscillations in the presence of gravity (atmosphere, ocean)
To observe wave propagation (oscillations : restoring force from the principle of
Archimedes), the density of the surrounding fluid at x3 + δx3 must be smaller than
the density of the fluid particle, ρ0(x3 + δx3) < ρp(x3 + δx3). And thus,
dρ0
g
δx3
(x3)δx3 < ρ0(x3) − ρ0(x3) 2
ρ0(x3) +
dx3
c0 (x3)
−
dρ0
g
− ρ0 2 ≥ 0
dx3
c0
The restoring gravitational force per unit volume may be written ρ0N 2δx3 where
N(x3) has the dimension of a frequency, known as the Väisälä-Brunt frequency
g dρ0 g2
N =−
− 2
ρ0 dx3 c0
2
Very low frequency – in the atmosphere, typically T = 2π/N ∼ 102 s
26
Wave propagation in fluids - S7 ECL 2A - déc. 2015 - cb1
x Internal gravity waves q
Oscillations in the presence of gravity (atmosphere, ocean)
Stratified fluid at rest ρ0(x3), incompressible perturbations
Linearized Euler equations
∇ · u′ = 0
∂ρ′
+ u′ · ∇ρ0 = 0
∂t
∂u′
ρ0
= −∇p′ + ρ′g
∂t
By cross-differentiation to eliminate ρ′, p′, u′1 and u′2, the following equation
can be derived for u′3
2
3 ′
N
∂2 2 ′
∂
u3
2 2 ′
0
∇ u3 = N0 ∇⊥u3 −
∂t 2
g ∂t 2∂x3
where ∇2⊥ ≡ ∂2x1x1 + ∂2x2x2
and N02 is the approximation of N 2 for incompressible perturbations,
N02(x3)
27
g dρ0 g2
− 2
=−
ρ0 dx3 c0
(c0 → ∞)
Wave propagation in fluids - S7 ECL 2A - déc. 2015 - cb1
x Internal gravity waves q
Oscillations in the presence of gravity (atmosphere, ocean)
Dispersion relation, by assuming that N0 ≃ cte (e.g. an isothermal atmosphere)
to find plane wave solutions w ′ = Aw ei(k·x−ωt)
2 2
ω k =
N02k⊥2
N02
+ iω k3
g
2
k⊥2 ≡ k12 + k22
In many interesting situations, kL ≫ 1, and
N02
g dρ0 g
∼
=−
ρ0 dx3
L
and the dispersion relation is then approximated by
k⊥
2 2
2 2
ω k ≃ N0 k⊥ or ω = N0
k
Waves are only possible in the case ω ≤ N0, and more surprisingly,
the wavelength is not determined by the dispersion relation
28
Wave propagation in fluids - S7 ECL 2A - déc. 2015 - cb1
x Internal gravity waves q
Oscillations in the presence of gravity (atmosphere, ocean)
Dispersion relation, k⊥ = k cos θ and k3 = k sin θ
ω = N0
k⊥
= N0| cos θ|
k
=⇒ vφ · vg = 0
x3
∇ · u′ = 0 =⇒ k · û = 0
transverse waves
u′(x1, x3)
vg
g
u′ = û(k)ei(k·x−ωt)
θ
θ
forcing
vφ
x⊥
phase velocity vφ = ω/k ≡ propagation of constant phase lines
in the k direction
29
Wave propagation in fluids - S7 ECL 2A - déc. 2015 - cb1
propagation
of phase fronts
ν = k/k
x Internal gravity waves q
Oscillations in the presence of gravity (atmosphere, ocean)
Mowbray & Rarity, J. Fluid Mech., 1967
Source : vertically oscillating cylinder (D = 2 cm) normal to the pictures
ω/N0 ≃ 0.615, 0.699, 0.900
=⇒
No gravity waves for ω/N0 ≃ 1.11
θ ≃ 52, 46, 26 deg
(θ = 56 deg)
30
Wave propagation in fluids - S7 ECL 2A - déc. 2015 - cb1
x Acoustic waves q
Sound propagation in a inhomogeneous atmosphere at rest
c0 = c0(x), ρ0 = ρ0(x), p0 = p0(x) and u0 ≡ 0 (no gravity effects)
Wave equation with a variable index of refraction
(Bergmann, J. Acoust. Soc. Am., 1946)
1 ∂ 2 p′
− ρ0(x)∇ ·
2
2
∂t
c0 (x)
p′
Introducing the new variable pr = √
ρ0
2 2
∇ p̂r + k∞n (x) + n1(x) p̂r = 0
2
k∞ =
ω
c∞
n=
c∞
c0(x)
1
∇p′
ρ0(x)
=0
pr = p̂r e−iωt
Helmholtz equation
with variable refraction index
n1 =
3
1 2
∇ ρ0 −
2ρ0
4
∇ρ0
ρ0
2
(See the exercises for further details)
31
Wave propagation in fluids - S7 ECL 2A - déc. 2015 - cb1
x Long-range propagation in Earth’s atmosphere q
Worldwide infrasound monitoring network developed to verify compliance with
the Comprehensive Nuclear-Test-Ban Treaty (CTBT)
Headquarter : Vienna, Austria
60 stations with 4 to 8 microbarometers over an area of 1 - 9 km2
45 actives, ✷ 5 under construction,
◦ 9 planned +1
(Hedlin & Walker, 2012)
32
Wave propagation in fluids - S7 ECL 2A - déc. 2015 - cb1
x Long-range propagation in Earth’s atmosphere q
Characteristics of infrasound
Infrasound : academic definition, 0.01 ≤ f < 20 Hz
Low-frequency waves can propagate over long distances (several hundreds of
km) in the Earth’s atmosphere. In practice, the relevant passband is about
0.02 ≤ f ≤ 4 Hz (and corresponds to a flat sensor response within 3 dB)
Infrasound sources : rocket launches or re-entry, re-entering space debris, supersonic aircraft, industrial or military explosions, bombings, meteors & bolides,
explosive volcanoes and meteorological events : localization and identification
Explosion in the atmosphere : approximately 50 % of the energy is released
into the atmosphere as a blast wave (Blanc & Ceranna, 2009)
Infrasound monitoring can also be used in the field of atmospheric tomography,
for instance to study gravity waves
Infrasound arrays : IMS (CTBT), USArray Transportable Array, ...
33
Wave propagation in fluids - S7 ECL 2A - déc. 2015 - cb1
x Long-range propagation in Earth’s atmosphere q
Propagation in the Earth’s atmosphere
Stratified atmosphere extending up to 180 km altitude
speed of sound c̄
180
150
x2 (km)
120
90
60
30
0
200
300
400
500
600
c̄ (m/s)
34
Wave propagation in fluids - S7 ECL 2A - déc. 2015 - cb1
x Long-range propagation in Earth’s atmosphere q
Propagation in the Earth’s atmosphere
Stratified atmosphere extending up to 180 km altitude
speed of sound c̄
180
150
x2 (km)
120
+
−
90
60
30
0
200
300
400
500
600
c̄ (m/s)
34
Wave propagation in fluids - S7 ECL 2A - déc. 2015 - cb1
x Long-range propagation in Earth’s atmosphere q
Propagation in the Earth’s atmosphere
Stratified atmosphere extending up to 180 km altitude
speed of sound c̄
effective speed of sound
c̄e = c̄ + ū1
180
150
x2 (km)
120
Waves naturally refracted towards
stratospheric and thermospheric wave
guides (x2 ≃ 44 km and x2 ≃ 105 km)
according to geometrical acoustics
through the Snell-Descartes law
90
60
30
0
200
300
400
500
600
c̄ (m/s)
34
Wave propagation in fluids - S7 ECL 2A - déc. 2015 - cb1
x Long-range propagation in Earth’s atmosphere q
Misty picture event : a reference case for atmospheric propagation
Gainville et al. (2010)
Alpine 248 km W
High explosive source (4.8 KT AFNO)
at White Sands Missile Range
May 14, 1987
(p − p̄) (Pa)
24
12
0
−12
−24
750
850
950
1050
1150
1300
1400
1600
1700
t (s)
White River 324 km W
(p − p̄) (Pa)
12
6
0
−6
−12
1000
1100
1200
t (s)
Roosevelt 431 km W
Meteorological data : T̄ (x2) and ū(x2)
Gainville & Marsden (2012)
❀ benchmark problems
35
(p − p̄) (Pa)
8
4
0
−4
−8
1300
Wave propagation in fluids - S7 ECL 2A - déc. 2015 - cb1
1400
1500
t (s)
x Long-range propagation in Earth’s atmosphere q
Nonlinear propagation with wind (NLW) : global view
nonlinear effects
p′/p̄ ∼ ρ̄−1/2 + absorption
80
180
c̄
c̄e
150
20
x2 (km)
120
90
60
60
30
0
200
300
400
500
600
c̄ (m/s)
source
fs ∼ 0.1 Hz
36
Sabatini et al. (2015)
0
0
100
200
300
400
x1 (km)
refraction induced by
c̄(x) and ū1(x)
Wave propagation in fluids - S7 ECL 2A - déc. 2015 - cb1
Various arrivals
at ground level