When electromagnetic waves pass through the ionosphere, the dispersion of waves takes place, and we must consider two kinds of the propagation velocities, that is, the phase and group velocities. The phase of the carrier of the signal wave transmitted from the satellite propagates with the phase velocity. However, the code modulated by the carrier travels with a velocity called the group velocity. Since the phase velocity becomes faster in the ionosphere than in vacuum, the phase range measured by using the phase of the carrier is estimated to be shorter. On the other hand, the pseudo range measured by using the code signal is estimated to be longer, since the group velocity becomes slower. When we start a study on GNSS, we encounter this phenomenon and here is an easy explanation beginning from an example in water waves (or gravity waves travelling on water surface).

Let us consider a water region ?₀ of constant depth h with a free surface S_F0 and bottom S_B. The water region extends infinitely in the horizontal direction. We take x and y axes on the free surface, and z axis vertically upward. The time is referred to by t. The velocity potential f satisfies a boundary value problem:

where g is gravitational acceleration.

This boundary value problem has a solution expressing a progressive wave:

where A is the amplitude of the wave,and the wave number k and the circular frequency w satisfies a dispersive equation:

The progressive wave travels to θ direction with the phase velocity w/k.The surface elevation ζ is given as

The phase velocity is defined as a velocity at which the equiphase line:

travels. In case of θ = 0, π/2, it may be easily understood. The wave travels keeping its form.

If we write the wavelength and phase velocity as λ and c, we have






According to equation (3), k decreases and λ and c increases, when w decreases.Hence, a water wave consisting of multiple frequency components disperses in a way that the component waves with
longer wave length go ahead of those with shorter wave length. The dispersion is determined by equation (3). So, this equation is called dispersive equation.

In case of a deep water wave (h/λ >>1), we have

And in case of a very shallow water wave (or tidal wave; h/λ ≈ 0),the following equations are obtained:

With respect to a tidal wave, the phase velocity does not depend on the wave length and the wave does not disperse. So, in case of a very long wave such as generated by a tsunami, for example, the wave due to Chili Tsunami in 1960 traveled to the Japanese coast from the wave source off the shores Chile over Pacific Ocean without changing its shape. In this case, the wave trough arrived at the Japanese coast first. It is said that the wave crest is replaced by the wave trough because of the dispersion effect due to Colioli force, when the tsunami wave passed the neighborhoods of Hawaii. The velocity of the tsunami is calculated by . So, if we assume the average depth of the ocean as 4000m, the velocity is 200m/s or 700m/h.

In case of a tidal wave, the wave travels without changing its form, since the phase velocity c that transmits the phase of the wave is equal to the group velocity cg that transmit the energy of the wave. However, the phase and group velocities are different in case of a dispersive wave. When you observe a swell surging against a beach, the peak disappears at the wave front. You may observe this phenomenon more clearly in a wave tank. If you run the wave generator for a short period of time, you can make a wave packet. If you watch a wave peak, the peak moves faster than the wave packet and disappears at the front of the wave packet. At the front, waves disappear one after another. If you observe the tail of the wave packet, waves emerge one after another. However, the wave packet advances with a constant speed or a group velocity. The velocity of the packet or the group velocity is slower than the velocity of the peak or the phase velocity in case of gravitational water waves.

The wave energy transmits into the direction of the wave propagation. A part of the wave energy becomes the kinematic energy and the rest is potential energy. In case of a deep water wave, both energies per wave length are equal. Hence, the group velocity should be half of the phase velocity for the balance of the energy.

Let's consider this phenomenon mathematically. We consider a wave composed of two component waves. One is a wave with the circular frequency w and the wave number k, and the other is a wave with the circular frequency w' and the wave number k'. The amplitudes of both waves are assumed equal, that is, A. So, the surface elevation ? of the synthesized wave is given as


If the difference between (w, k) and (w', k') are small, the above equation can be approximated as

Hence, a low speed wave with the velocity (w - w')/(k - k') is modulated by a high speed wave with the velocity w/k. Namely, the peaks and troughs of the wave advances fast, and the whole wave advances slow. The faster velocity w/k is the phase velocity c, and the slower velocity is the group velocity c_g. We now have

Substituting the dispersive equation (3), we obtain






In case of a deep water wave, we have from equations (11) and (7b)

For a tidal wave, the group velocity is given from equations (11) and (8b) as

The velocity potential and the surface elevation of a deep water wave advancing into x -axis (or ? = 0) direction can be written as

Substituting the above expressions into the kinetic energy T and the potential energyU per unit width and wave length:

Hence, we know that the kinematic energy is equal to potential energy in case of a deep water wave.