Wave height offshore as shown by the wave buoys does not equate directly into surf height at the break. What is important is the combination of the wave period and wave height. Period is directly related to the speed the wave is travelling. The longer the period the faster the wave and the more energy in contains.
The energy in the wave is comprised of kinetic and potential forms, [speed and height]
and this must remain constant, (Newtons first law, the conservation of energy).
In deep water [where h > L/2] the speed of waves is governed solely by their wavelength, long waves travel faster and have greater energy. In shallow water [h < L/20] the speed of waves is governed by the water depth rather than their wavelength. The shallower the water the slower they travel. The total energy in the waves has not changed so when the speed is decreased the height increases, the waves get bigger and steeper. However, some energy is lost through friction with the sea bed, causing the wave height to decease. This is particularly significant where there is a large continental shelf.
As a guide, the experience of surfers surfing Cornwall so far indicates that a wave period of about 9 seconds at the Lands End wave buoy is enough for the above factors to balance each other, so that the surf height is approximately the wave height at the buoy. Shorter period waves are smaller than indicated, and longer ones, (when they happen,) are bigger.
Thus long waves offshore, when they arrive on the beach turn into big surf. The wavelength is nearly impossible to measure at sea, but it is closely related to the wave period, which is easily measured by wave buoys. In fact L = 1.56 * T^2. (Longer waves have proportionately longer periods.)
Even more helpfully for us because C = L/T then C = 1.56 * T. So the speed of the waves can be calculated directly from the period measured by the buoys. Beware though that in the open ocean waves travel in groups, (which later become sets at the beach). The waves at the front of the group die away as they transfer their kinetic energy into potential energy by raising the surface of the water in front of them. They are replaced at the back of the group by new waves where the potential energy of the water turns back into kinetic energy. Because of this, the group travels at half the speed of the individual waves. So to get the speed of a swell indicated by the wave buoys or the forecast then use Cg = (1.56 * T) / 2 m/s. A typical 10 second period wave will travel in a group moving at 7.8 m/s
There is a limit on how steep a wave can be, [H/L ~ 1/7] when it gets beyond this it starts to break, thus reducing the height. This tends to produce spilling breakers.
Combined with this, as I said the speed of the wave depends on the water depth, so the crests of the waves travel faster than the troughs, [because the water is deeper] so the waves become asymmetric, with steeper fronts than backs. this increases the steepness of the face of the wave. Eventually the wave becomes unstable and the faster crest tries to overtake the trough and the wave breaks. This process tends towards producing plunging breakers, especially when it happens rapidly.
These two processes act together, [along with other smaller and even more complex ones]. The extent of each depends on things like the rate of shoaling, hence you see plunging breakers [+ tubes] on steep beaches where the depth changes rapidly.
h = water depth, (metres)
L = wavelength, (metres) (peak to peak)
H = wave height, (metres) (peak to trough)
C = wave speed, (metres/second)
Cg = Group speed, (metres/second)
T = wave period, (seconds).
1 m/s = 2 knots (nautical miles per hour) = 2 minutes of latitude =~ 2.3 statute miles per hour.