Wave Propagation Fundamentals -1
Propagation of Plane and Spherical Waves
Learning Objectives
Plane waves in fluids and solids
Elastic wave potentials
Spherical waves in fluids and solids
Plane Waves - Fluid
Waves in a fluid satisfy the wave equation
1 ∂2 p
∇ p− 2 2 =0
c ∂t
p = f% ( t − x / c ) is an arbitrary plane wave"pulse" solution
of the wave equation traveling in the + x direction.
The function p = F% ( f ) exp ⎡⎣ 2π if ( x / c − t ) ⎤⎦ is a harmonic
plane wave traveling in the + x direction.
2
They are simply related through the Fourier Transform:
amplitude
f% ( t − x / c ) =
+∞
∫
−∞
phase
F% ( f ) exp ⎡⎣ 2π if ( x / c − t ) ⎤⎦ df
Plane Waves-Fluid
we saw previously we can write a plane harmonic wave in
different forms:
F( f ) exp[2πif ( x / c − t )]
= F ( f )exp[ik( x − ct )]
= F (ω ) exp[iω ( x / c − t )]
For the last form we can also write
1 +∞
f (t − x / c) =
∫ F(ω ) exp[iω ( x / c − t )]dω
2π −∞
Plane Waves-Fluid
Summary: for a harmonic wave of exp( −iωt ) time
dependency, a plane wave traveling in the ± x direction
is given by (dropping the common time term)
F exp[±2πi f x / c ]
= F exp[±iω x / c ]
= F exp[±ik x ]
For a plane wave traveling in an arbitrary direction , n
n
n . x = constant
plane
F exp[ikn ⋅ x]
= F exp[ik ⋅ x ]
where k = k n
Plane Waves - Fluid and Solid Media
Waves in a fluid are governed by the scalar wave equation
n (direction of propagation)
1 ∂2 p
∇ p− 2 2 =0
c ∂t
p = f (t − x ⋅ n / c )
2
Waves in an isotropic elastic solid are governed by the
vector Navier's equations
∂u
μ∇ u + (λ + μ )∇(∇ ⋅ u) − ρ 2 = 0
∂t
2
2
u … displacement
λ, μ … Lame constants
ρ … density
Plane Waves-Solid
2
∂
u
2
μ∇ u + (λ + μ )∇(∇ ⋅ u) − ρ 2 = 0
∂t
Navier's equations also have plane wave solutions. There are
two types – P-waves and S-waves. These are "bulk" waves.
n
compressional
u = n f t − x ⋅ n / cp
u
(P) waves
(
polarization
n
shear
(S) waves
d
u
u = n × d g ( t − x ⋅ n / cs )
)
Plane Waves-Solid
Polarizations
Plane P-waves are longitudinally polarized
in the direction of propagation
SV
SH
Plane S-waves are polarized in the plane
perpendicular to the direction of propagation.
Vertically polarized S-waves are called
SV-waves while horizontally polarized
S-waves are called SH waves
Plane Waves-Solid
Bulk wave speeds
λ + 2μ
cp =
=
ρ
E (1 −ν )
E … Young's modulus
(1 + ν )(1 − 2ν ) ρ ν … Poisson's ratio
μ
G
E
=
=
cs =
ρ
ρ
2 (1 + ν ) ρ
Compressional wave speed in a bar
G = μ .. Shear Modulus
cbar =
Compressional wave speed in a plate c plate
E
ρ
E
=
(1 − v 2 ) ρ
Plane Waves-Solid
∂2 u
μ∇ u + (λ + μ )∇(∇ ⋅ u) − ρ 2 = 0
∂t
2
Although Navier's equations are not wave equations, solutions
to Navier's equations can be found in terms of potential functions
that do satisfy wave equations
u = ∇φ + ∇ × ψ
2
1
∂
φ
2
∇ φ− 2 2 =0
c p ∂t
1 ∂ψ
∇ ψ− 2 2 =0
cs ∂t
φ … scalar potential
ψ … vector potential
P-waves
2
2
S-waves
cp
Material
Compressional
(P-wave)
wave speed
(m/s x 103)
cs
Shear
(S-wave)
wave speed
(m/s x 103)
ρ
Density
ρ
(kgm/m3 x
103)
ρcp
Impedance
(P-wave)
( kgm/(m2-s)
x 106)
Air
0.33
--
0.0012
Aluminum
6.42
3.04
2.70
17.33
Brass
4.70
2.10
8.64
40.6
Copper
5.01
2.27
8.93
44.6
Glass
5.64
3.28
2.24
13.1
Lucite
2.70
1.10
1.15
3.1
Nickel
5.60
3.00
8.84
49.5
Steel, mild
5.90
3.20
7.90
46.0
Titanium
6.10
3.10
4.48
27.3
Tungsten
5.20
2.90
19.40
101.0
Water
1.48
--
1.00
Table D.1 Acoustical properties of some common materials.
0.0004
1.48
Plane Waves-Solid
Elastic Wave Potentials
u = ∇φ + ∇ × ψ
For two-dimensional waves
displacements
φ = φ (x, y,t )
ψ z = ψ (x, y,t ), ψ x = ψ y = 0
stresses
∂φ ∂ψ
ux =
+
∂x ∂y
⎡ 2 2
⎛ ∂ 2ψ ∂ 2φ ⎞ ⎤
− 2 ⎟⎥
τ xx = μ ⎢κ ∇ φ + 2 ⎜
⎝ ∂x∂y ∂y ⎠ ⎦
⎣
∂φ ∂ψ
uy =
−
∂y ∂x
uz = 0
⎡ 2 2
⎛ ∂ 2ψ ∂ 2φ ⎞ ⎤
+ 2 ⎟⎥
τ yy = μ ⎢κ ∇ φ − 2 ⎜
⎝ ∂x∂y ∂x ⎠ ⎦
⎣
⎡ ∂ 2φ ∂ 2ψ ∂ 2ψ ⎤
τ xy = μ ⎢ 2
+ 2 − 2⎥
∂x ⎦
⎣ ∂x∂y ∂y
cp
κ=
cs
τ zz = ν ⎡⎣τ xx + τ yy ⎤⎦ , τ xz = τ yz = 0
Plane Waves-Solid
Plane wave solutions
P-waves
u x , vx ,τ xx
k p = ω / cp
φ = Φ exp ( ik p x − iω t )
potential
u x = U x exp ( ik p x − iω t )
displacement
vx = Vx exp ( ik p x − iω t )
velocity
τ xx = Txx exp ( ik p x − iω t )
stress
U x = ik p Φ , Vx = −iωU x ,
Txx = − ρ c pVx
x
Plane Waves-Solid
S-waves
k s = ω / cs
s = ex × t
s
ψ = Ψt exp ( ik s x − iω t ) potential
u = U s s exp ( iks x − iω t ) displacement
v = Vs s exp ( ik s x − iω t )
velocity
τ xs = Txs exp ( ik s x − iω t ) stress
U s = ik s Ψ , Vs = −iωU s ,
Txs = − ρ csVs
us , vs ,τ xs
x
Spherical Waves-Fluid
Spherical waves arise physically from "point" sources. If a
point source emits waves uniformly in all directions, then we
expect the waves to depend only on the radial distance, r , from
the point source.
r
Spherical Waves-Fluid
Consider spherical harmonic waves of exp(-iωt) time
dependency. The equation of motion of the fluid and the
wave equation are given by
−∇p = −iωρ v
∇ p+
2
ω2
c
2
p=0
In terms of the distance, r, from the source, in spherical
coordinates these equations are
∂p
= iωρ vr
∂r
∂ 2 p 2 ∂p ω 2
+ 2 p=0
2 +
∂r
r ∂r c
Spherical Waves-Fluid
There are two solutions of the wave equation of the
form
B
A
p=
r
exp(ikr ) +
k=ω/c
r
exp(−ikr )
The first term represents a wave traveling in the outward
radial direction while the second term represents a wave
converging on the source. Since the source only generates
outward going waves, we must set B = 0. The pressure and
radial velocity in the out going wave are then
A
p = exp(ikr )
r
A⎡
1 ⎤ exp(ikr )
vr =
1−
⎢
ρc ⎣ ikr ⎥⎦
r
Spherical Waves-Fluid
In many cases we are only interested in the waves many
wavelengths from the source, in which case kr >> 1 and we
can write approximately
exp(ikr )
p=A
r
A exp(ikr)
vr =
ρc
r
r
Compare this to a plane wave traveling in the +z direction where
p = A exp(ikz )
A
vz =
exp(ikz)
ρc
z
Spherical Waves-Elastic Solid
Spherical waves from point sources in an elastic solid have a
more complex structure in general, but for kr >>1 they are
similar to that of a fluid:
u=A
A
er
exp ( ik p r )
r
exp ( iks r )
+B
r
S-wave
B ⊥ er
A
B
P-wave
r
er