Slowly varying media: Ray theory
Suppose the medium is not homogeneous (gravity waves impinging on a beach,
i.e. a varying depth). Then a pure plane wave whose properties are constant in space and
time is not a proper description of the wave field.
However, if the changes in the background occur on scales that are long and slow
compared to the wavelength and period of the wave, a plane wave solution may be
locally appropriate. (Fig. 2.1) This means: λ << Lm where Lm is the length scale over
which the medium changes. Consider the local plane wave
r
r
r
φ(x,t) = a(x,t)e iθ(x ,t)
a varies on the scale Lm while θ varies on the scale λ.
1 1 ∂a
1
∂θ
)
= 0( );
= 0(
λ a ∂x i
Lm
∂x i
⇒ ∇φ = ae iθ •∇θ+0 (
λ
)
Lm
r r
θ = k • x − wt
with
Define the local wavenumber and the local frequency as:
r
k = ∇θ |t
ω=−
∂θ
|x
∂t
From these definitions it follows that:
r
∇ × k = 0 the local wave number is irrotational.
Conservation of crests in a slowly varying medium.
Suppose we go from point A to point B over the curve C1.
1
B
C2
A
C1
Figure by MIT OpenCourseWare.
Figure 1
slowly varying wave fronts
The number of wave cress we pass along C1 is
n c1 =
1 Br r 1 r r
∫ k • ds =
∫ k • ds
2π A
2π c1
The number of wave crests we pass along C2 is
nc2 =
1 Br r 1 r r
∫ k • ds =
∫ k • ds
2π A
2π c 2
rr
w
Before for plane waves ω = Ω(k ) only, now ω = Ω(k, x,t).
r
As (ω, k ) are slowly varying functions of space/time, the dispersion relation is explicitly
dependent on space/time. Now we can introduce the group velocity in another way
∂ω r ∂Ω r r ∂Ω r ∂k i r
|x =
| +
|x ,t
|x =
∂t k,x ∂k i
∂t
∂t
∑i
=
∂k
∂Ω r
|k,xr +c gi i |xr
∂t
∂t
Where we use the summation convention over repeated indices,
and
c gi =
∂Ω
∂k i
by definition i = 1, 2, 3 = x,y,z
2
r
c g = ∇ kr Ω
group velocity
The difference is:
r r
r r 1
r r
r
1
1
n c1 − n c 2 = [( ∫ − ∫ )k • d s ] = [( ∫ + ∫ )k • d s ] =
∫ c total k • d s = ∫∫ A ∇xk • nˆ dA ≡ 0
π c1 c 2
π c1 − c 2
2π
nˆ = unit vector normal to C
r
We have used Stokes theorem relating the line integral of the tangential component of k
to the area integral of its curl over the area bounded by the closed contour C.
The
increase of phase is the same on C1 and C2. This means the number of crests along C1 is
the same as the number of crests along C2, that is the number of cress inside the area A is
conserved. Crests are neither created nor destroyed inside A. The crests have no ends, so
the number of crests within a wave group will be the same for all time. This is obviously
true only for slowly varying plane waves.
r
From the definition of k and w it follows:
r
∂k
+ ∇ω = 0
∂t
(1)
We have seen that the number of crests we cross from A to B is the same along any path
connecting A and B. Then:
n=
1 Br r
∫ k • ds
2π A
r
r 1
∂n 1 B ∂k r
1 B
• ds = −
=
∫
∫ ∇ω • d s = (ω(A) − ω(B)]
∂t 2π A ∂t
2π A
2π
This says that the rate of change of the number of wave crests between A and B is equal
to the frequency of crest inflow at A minus the frequency crest outflow at B.
3
Crests are neither created nor destroyed in the smoothly varying function φ. The
number in any local region increases or decreases solely due to the arrival of pre-existing
crests at A, not to the creation or destruction of existing crests.
We now introduce the dynamics by asserting that the wavenumber and frequency
must be related by a dispersion relation in the same way as for a plane wave.
Since by eq. (1)
∂k i
∂ω
=−
∂x i
∂t
we have
∂ω ∂Ω
∂ω
=
− c gi
∂t
∂x i
∂t
or
∂Ω r
∂ω r
+ c g • ∇ω =
| r
∂t k ,x
∂t
(1) equation for ω
Similarly from (1)
∂Ω r r ∂k j
∂k i r ∂Ω r
|x +
|k ,t +
|
=0
∂t
∂x i
∂k j x, t ∂x i
∂k i ∂Ω ∂k j
∂Ω
+
=−
∂x i
∂t ∂kj ∂x i
or
r
r
∂k r
+ c g • ∇k = −∇Ω |kr ,t
∂t
(2)
The “ray equation” gives the velocity at which the wave packet, or wave group, moves:
r
dx
cg =
dt
or c gx =
dx
dy
; c gy =
dt
dt
in two dimensions. Then the ray path in the (x,y) plane is
4
dy c gy
=
dx c gx
d ∂ r
= + cg • ∇
dt ∂t
r
r
dx
cg =
dt
(I)
∂Ω r r
∂ω
+ c g • ∇ω =
|
(II)
∂t t ,x
∂t
r
r
∂k r
+ c g • ∇k = −∇Ω |kr ,t
(III)
∂t
rr
r
Ω = Ω (k, x,t) has an explicit parametric dependence on ( x,t ), for instance when waves
enter in water of changing depth. The ray equations give the evolution of the local
r
wavenumber k and the local frequency ω as we move along the ray, i.e. we move with
r
the wave packet at the local group velocity c g . Is a plane wave a particular solution of
r
the ray theory formulation? Suppose the medium is homogeneous, no changes in ( x,t )
r
ω = Ω(k ) only
r r
Solution: plane wave φ = ae i(k•x −ωt)
r
where (k, ω) do not change but are constant in space
r
r
r
Initial condition φ( x) = ae ik• x
As
r
∂k
≡0
∂x 2
and
gives
r
k (t = 0)
∂Ω
≡0
∂x i
The ray equation (III) gives
r
r
r
∂k
= 0 : k never changes along the ray and remains equal to k (t=0).
∂t
r
ω = Ω(k ) gives ω at t = 0
.
5
As
∂ω
∂Ω
=0 ;
=0
∂x i
∂t
∂ω
=0
∂t
eq. (II) gives
ω = ω(t=0)
The frequency never changes along the ray.
Thus the plane wave solution in a
homogeneous medium is entirely consistent with the ray theory formulation.
6
MIT OpenCourseWare
http://ocw.mit.edu
12.802 Wave Motion in the Ocean and the Atmosphere
Spring 2008
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