Continuing with the nonlinear Klein Gordon equation, using the average variational principle on equation, the wave action equation becomes by variation with respect to the wave phase \ theta:
12.
The primitive equations are identical to those used to develop the coastal Kelvin wave phase speed solution ( U-momentum, V-momentum, and continuity equations ) and the motion is unidirectional and parallel to the equator.
13.
The first equation states that the effective wavenumber \ boldsymbol { \ kappa } is irrotational, a direct consequence of the fact it is the derivative of the wave phase \ theta, a scalar field.
14.
Where a is the amplitude, H = 2 a is the wave height, \ omega = 2 \ pi / T is the angular frequency and \ phi ( x ) is the wave phase.
15.
Seismic wheel sensing works on the principle of seismic wave phase analysis, the sensors do not have to be installed on the rail track but can be installed within a convenient distance parallel to the track.
16.
In order to apply the Stokes expansion to progressive periodic waves, it is advantageous to describe them through Fourier series as a function of the wave phase " ? " ( " x ", " t " ):
17.
Now, " by definition ", the angular frequency \ omega and wavenumber vector \ boldsymbol { k } are expressed as the time derivative and gradient of the wave phase \ theta ( \ boldsymbol { x }, t ) as:
18.
Where a is the scattering length, k is the wave number, and \ delta ( k ) is the s-wave phase shift . The elastic cross section, \ sigma _ e, at low energies is determined solely by the scattering length,
19.
The two stream instability can be thought of as the inverse of Landau damping, where the existence of a greater number of particles that move slower than the wave phase velocity v _ { ph } as compared with those that move faster, leads to an energy transfer from the wave to the particles.
20.
Those particles having velocities slightly less than v _ { ph } will be accelerated by the electric field of the wave to move with the wave phase velocity, while those particles with velocities slightly greater than v _ { ph } will be decelerated losing energy to the wave : particles tend to synchronize with the wave.
