Improved Numerical Modeling for Perturbations in Homogeneous and Stratified Flows - Advances in Control and Automation of Water Systems - page 150

( ω

* k --damping vibrations in length. In other words, ω --waves - waves
with stationary in time but varying in length amplitudes. Cases

Here,

k < 0 k > 0

and

*

> kk consistent with attenuation of amplitude of the disturbance regime in
the direction of phase fluctuations or phase velocity.
Let us obtain the characteristic equation, linking

0 < 0

*

k and ω . After substituting

(10) in the system of equations (9) we obtain:

ω

δ

0 *

+ U

0

= 0

k

*

ω
k

0 * 0

U + δ = 0

(13)

*

From the condition of the existence of a system of linear homogeneous algebraic equa-
tions (13) with respect to perturbations of a nontrivial solution implies the desired
characteristic, or dispersion, which has one solution:

v

Φ =

gh

(14)

0

Thus, we obtain a solution representing a sinusoidal in time and coordinate free un-
dammed oscillations. Such behaviors of the waves are due to the absence of any dis-
sipation in the fluid. The fluid is incompressible and ideal. There is no heat-mass
transfer. Equations (9) with respect to perturbations take the form of wave equations:

2 2

∂ξ ∂ξ

2 2

∂ u ∂ u

= gh and

= gh

(15)

0

2 0 2

∂ t

2

∂ x

2

∂ t ∂ x

Note that in gas dynamics equivalent to the speed of sound.

v

Φ =

gh

0

KeyWords

• euler equation

• gravity field

• Plumbing systems

• Quadratic equation

• telegraph equation

• Wavelength

references
1. Nagiyev, F. B. (1993). Dynamics, heat and mass transfer of vapor-gas bubbles in a two-compo-
nent liquid . Turkey-Azerbaijan petrol semin, Ankara, Turkey.
2. Nagiyev, F. B. (1995). The method of creation effective coolness liquids. Third Baku interna-
tional Congress. Baku, Azerbaijan Republic, September 19-22.