Improved Numerical Modeling for Perturbations in Homogeneous and Stratified Flows - Advances in Control and Automation of Water Systems - page 147

Thus, we obtain the following system of equations describing the fluid flow in the
channel:

¶x ¶ u ¶ u ¶x

+ h = 0, + = > -¥< <¥ g 0, t 0, x

(8)

0

¶ t ¶ x ¶ ¶ t x

8.3 results And discussion
8.3.1 velocity Phase of the Harmonic Wave
The phase velocity

h expressed in terms of frequency Φ

v

and wavelength f (or the

angular frequency) λ and wave number

= k . The concept of
phase velocity can be used if the harmonic wave propagates without changing shape.
This condition is always performed in linear environments. When the phase velocity
depends on the frequency, it is equivalent to talk about the velocity dispersion. In the
absence of any dispersion the waves assumed with a rate equal to the phase velocity.
Experimentally, the phase velocity at a given frequency can be obtained by determin-
ing the wavelength of the interference experiments. The ratio of phase velocities in
the two media can be found on the refraction of a plane wave at the plane boundary of
these environments. This is because the refractive index is the ratio of phase velocities.
It is known that the wave number k satisfies the wave equation are not any values ω
but only if their relationship. To establish this connection is sufficient to substitute the
solution of the form

ω 2 = π

formula

2 π/ λ

( )

ω− in the wave equation [3]. The complex form is
the most convenient and compact. We can show that any other representation of har-
monic solutions, including in the form of a standing wave leads to the same connection
between ω and k . Substituting the wave solution into the equation for a string, we
can see that the equation becomes an identity for

exp i t kx

ω= . Exactly the same rela-
tion follows from the equations for waves in the gas, the equations for elastic waves in
solids and the equation for electromagnetic waves in vacuum.

2 22

kv Φ

8.4 conclusion
The presence of energy dissipation, [4] leads to the appearance of the first deriva-
tives (forces of friction) in the wave equation. The relationship between frequency and
wave number becomes the domain of complex numbers. For example, the telegraph
equation (for electric waves in a conductive line) yields

ω = + ⋅ω . The
relation connecting between a frequency and wave number (wave vector), in which the
wave equation has a wave solution is called a dispersion relation, the dispersion equa-
tion or dispersion. This type of dispersion relation determines the nature of the wave.
Since the wave equations are partial differential equations of second order in time and
coordinates, the dispersion is usually a quadratic equation in the frequency or wave
number. The simplest dispersion equations presented above for the canonical wave
equation are also two very simple solutions

2

kv

22

i R L /

Φ

ω=+ and kv Φ

ω=− .We know that
these two solutions represent two waves traveling in opposite directions. By its physi-
cal meaning the frequency is a positive value so that the two solutions must define
two values of the wave number, which differ in sign. The Act permits the dispersion,
generally speaking, the existence of waves with all wave numbers that is of any length,

kv Φ