approximately by 10% at the distance equal to the length of the tidal internal wave
(130–150 km) [11].
We can also estimate the influence of different factors, including the horizontal
inhomogeneity of density, on the IGW decay. In the framework of the theory
discussed above, we consider the evolution of IGW frequency ω corresponding to
the semidiurnal period T = 12 h, which also admits slow variations in the stratification along the wave propagation path. The real geometry of the experiment
allows us to assume that the problem under study is two-dimensional, which means
that the stratification depends only on two variables: depth z and distance x along
the wave propagation path.
Now we consider the case of constant depth H and stratification N linearly
depending only on x: NðxÞ = N 1 + ðN 2 − N 1 Þ x ̸ L, where L is the distance between
the two observation points, x = x 1 = 0 is the initial point, x = x 1 = L is the end point,
and N 1, 2 = Nðx 1, 2 Þ. We consider only the first mode η 1 ðz, xÞ of the amplitude of the
vertical displacement of particles and omit its index. We seek the amplitude ηðz, xÞ
in the form ηðz, xÞ = AðxÞ f ðz, xÞ, where f ðz, xÞ is the normalized eigenfunction of
the standard boundary-value problem for the equation of internal waves with the
normalization
R H
0 ðN
2
ðxÞ − ω
2
Þ f
2
ðz, xÞ dz = 1,
which
has
the
form
f ðz, xÞ =
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi
2
HðN 2 ðxÞ − ω 2 Þ
q
sinðπ z ̸ HÞ. Amplitude A(x) depending only on x is determined from the conservation law:
A
2 ðx 1 Þ
k 2 ðx 1 Þ daðx 1 Þ =
A
2 ðx 2 Þ
k 2 ðx 2 Þ daðx 2 Þ, where kðxÞ is the
absolute value of the horizontal wave vector, and daðxÞ is the width of an elementary wave tube. Since the problem is two-dimensional, the width of the ray tube
does not vary along the ray and the conservation law is simpler: AðxÞ ̸ kðxÞ = const.
Since we consider small values of ω, the velocity of wave propagation is close to
the maximum group velocity cðxÞ = NðxÞ H ̸ π; hence, the wave number is equal to
kðxÞ = ωπ ̸ NðxÞ H and the corresponding wave length is equal to
λðxÞ = 2NðxÞ H ̸ ω. Then, under the assumption that the observation points are at the
same depth, it follows from the conservation law ðA 1, 2 = Aðx 1, 2 ÞÞ that A 1 N 1 = A 2 N 2
or A 2 = A 1 λ 1 ̸ λ 2 . Then the total amplitude attains the following values
W 1, 2 = A 1, 2
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
2
H ðN 2
1, 2
− ω 2 Þ
q
, which implies W 2 = W 1
N 1
N 2
ffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi
ðN 2
1
− ω 2 Þ
ðN 2
2
− ω 2 Þ
r
or W 2 = W 1 λ
2
1 ̸ λ
2
2 ,
because ω ≪ N, i.e., the amplitude of the internal gravity wave is inversely proportional to the squared wave length. The wave travel time τ along the horizontal
ray is determined from the equation of characteristics
dx
dt = cðxÞ, where
cðxÞ = ðN 1 + axÞ H ̸ π and a = ðN 2 − N 1 Þ ̸ L. Integrating this equation, we obtain the
wave travel time τ =
π
a H ln
N 2
N 1
=
T L
ðλ 2 − λ 1 Þ ln
λ 2
λ 1
. The available data of full-scale
tests give the following values of the basic parameters of the problem: λ 1 = 167 km,
λ 2 = 156 km, L = 2000 km. The wave attenuation coefficient without the wave
length variations taken into account, which describes the amplitude decrease versus
wave length denoted by β, gives the value of β: β = 0.2
167 ̸ 2000 = 0.874 with regard
to relation W2 ̸ W2 = 0.2 ≡ β
t ̸ T = β
L ̸ λ derived from the observation results. The
attenuation with regard to the wave length variations along the ray, W 2 ̸ W 1 = β
τ ̸ T ,
116
V. V. Bulatov and Y. V. Vladimirov
(130–150 km) [11].
We can also estimate the influence of different factors, including the horizontal
inhomogeneity of density, on the IGW decay. In the framework of the theory
discussed above, we consider the evolution of IGW frequency ω corresponding to
the semidiurnal period T = 12 h, which also admits slow variations in the stratification along the wave propagation path. The real geometry of the experiment
allows us to assume that the problem under study is two-dimensional, which means
that the stratification depends only on two variables: depth z and distance x along
the wave propagation path.
Now we consider the case of constant depth H and stratification N linearly
depending only on x: NðxÞ = N 1 + ðN 2 − N 1 Þ x ̸ L, where L is the distance between
the two observation points, x = x 1 = 0 is the initial point, x = x 1 = L is the end point,
and N 1, 2 = Nðx 1, 2 Þ. We consider only the first mode η 1 ðz, xÞ of the amplitude of the
vertical displacement of particles and omit its index. We seek the amplitude ηðz, xÞ
in the form ηðz, xÞ = AðxÞ f ðz, xÞ, where f ðz, xÞ is the normalized eigenfunction of
the standard boundary-value problem for the equation of internal waves with the
normalization
R H
0 ðN
2
ðxÞ − ω
2
Þ f
2
ðz, xÞ dz = 1,
which
has
the
form
f ðz, xÞ =
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi
2
HðN 2 ðxÞ − ω 2 Þ
q
sinðπ z ̸ HÞ. Amplitude A(x) depending only on x is determined from the conservation law:
A
2 ðx 1 Þ
k 2 ðx 1 Þ daðx 1 Þ =
A
2 ðx 2 Þ
k 2 ðx 2 Þ daðx 2 Þ, where kðxÞ is the
absolute value of the horizontal wave vector, and daðxÞ is the width of an elementary wave tube. Since the problem is two-dimensional, the width of the ray tube
does not vary along the ray and the conservation law is simpler: AðxÞ ̸ kðxÞ = const.
Since we consider small values of ω, the velocity of wave propagation is close to
the maximum group velocity cðxÞ = NðxÞ H ̸ π; hence, the wave number is equal to
kðxÞ = ωπ ̸ NðxÞ H and the corresponding wave length is equal to
λðxÞ = 2NðxÞ H ̸ ω. Then, under the assumption that the observation points are at the
same depth, it follows from the conservation law ðA 1, 2 = Aðx 1, 2 ÞÞ that A 1 N 1 = A 2 N 2
or A 2 = A 1 λ 1 ̸ λ 2 . Then the total amplitude attains the following values
W 1, 2 = A 1, 2
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
2
H ðN 2
1, 2
− ω 2 Þ
q
, which implies W 2 = W 1
N 1
N 2
ffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi
ðN 2
1
− ω 2 Þ
ðN 2
2
− ω 2 Þ
r
or W 2 = W 1 λ
2
1 ̸ λ
2
2 ,
because ω ≪ N, i.e., the amplitude of the internal gravity wave is inversely proportional to the squared wave length. The wave travel time τ along the horizontal
ray is determined from the equation of characteristics
dx
dt = cðxÞ, where
cðxÞ = ðN 1 + axÞ H ̸ π and a = ðN 2 − N 1 Þ ̸ L. Integrating this equation, we obtain the
wave travel time τ =
π
a H ln
N 2
N 1
=
T L
ðλ 2 − λ 1 Þ ln
λ 2
λ 1
. The available data of full-scale
tests give the following values of the basic parameters of the problem: λ 1 = 167 km,
λ 2 = 156 km, L = 2000 km. The wave attenuation coefficient without the wave
length variations taken into account, which describes the amplitude decrease versus
wave length denoted by β, gives the value of β: β = 0.2
167 ̸ 2000 = 0.874 with regard
to relation W2 ̸ W2 = 0.2 ≡ β
t ̸ T = β
L ̸ λ derived from the observation results. The
attenuation with regard to the wave length variations along the ray, W 2 ̸ W 1 = β
τ ̸ T ,
116
V. V. Bulatov and Y. V. Vladimirov