Oceanography

The concept of group velocity C_g is fundamental for understanding the propagation of linear and nonlinear waves. First, it is the velocity at which a group of waves travels across the ocean. More importantly, it is also the propagation velocity of wave energy whitham gives a clear derivation of the concept and the fundamental equation .

The definition of group velocity in two dimension is

C_g =  θω

          __

          θk

Using the approximations for the dispersion relation:

                 g           c

   Cg    =    _     =    _       

                2ω          2

         

Deep water group velocity

 C_g =     gd     = c

                  

Shallow-water group velocity

For ocean-surface waves, the direction of propagation is perpendicular to the wave crests in the positive x direction. In the more general case of other types of waves, such as Kelvin and Rossby  waves the group velocity is not necessarily in the direction perpendicular to wave crests.

Notice that a group of deep-water waves moves at half the phase speed of the waves making up the group. How can this happen? If we could watch closely a group of waves crossing the sea, we would see waves crests appear at the back of the wave train, move through the train, and disappear at the leading edge of the group.

Each wave crest moves at twice the speed of the group. Do real ocean waves move in groups governed by the dispersion relation? Yes. Walter Munk and colleagues (1963) in a series of experiments in the 1960s showed that ocean waves propagating over great distances are dispersive, and that the dispersion could be used to track storms. They recorded waves for many days using an array of three pressure gauges just offshore of San Clemente Island, 60 miles due west of San Digeo, California. Wave spectra were calculated for each day’s data. From the spectra,the amplitudes and frequencies of the low-frequency waves and the propagation direction of the waves were calculated. Finally, they plotted contours of wave energy on a frequency-time diagram.

Contours of wave energy on a frequency-time plot calculated from spectra of waves measured by pressure gauges offshore of southern California. The ridges of high wave energy show the arrival of dispersed wave trains from distant storms. The slope of the ridge is inversely proportional to distance to the storm. Δ is distance in degrees, θ is direction of arrival of waves at California. From Munk et al. (1963).

To understand the figure, consider a distant storm that produces waves of many frequencies. The lowest-frequency waves (smallest W) travel the fastest, and they arrive before other, higher –frequency waves. The further away the storm the longer the delay between arrivals of waves of different frequencies. The ridges of high wave energy seen in the figure degrees Δ along a great circle; and the phase information from the array gives the angle to the storm. The two angles give the storm’s location relative to San Clemente. Thus waves arriving from 15 to 18 September produce a ridge indicating the storm was 115⁰ away at an angle of 205⁰ which is south of new Zealand near Antarctica.

The locations of the storms producing the waves recorded from June through October 1959 were compared with the location of storms plotted on weather maps and in most cases the two agreed well.