Shoaling Coefficient

Shoaling is bookkeeping. A wave train travelling towards shore carries energy at a rate $E c_g$ per metre of crest, and — ignoring for the moment friction, breaking and any change in the width of the wave ray — that flux has to be the same at every depth. $E$ goes as $H^{2}$, so if $c_g$ falls, $H$ must rise as $1/\sqrt{c_g}$. Hence $K_s = H/H_0 = \sqrt{c_{g0}/c_g}$. The wave gets taller because it gets slower, and for no other reason.

The behaviour is not monotonic, and this catches people out. On the way in from deep water, the group velocity first RISES: in deep water $c_g = c/2$, and as the wave enters intermediate depth the ratio $n = c_g/c$ climbs from ½ towards 1 faster than the phase speed falls. So $K_s$ dips below one, reaching a minimum of about 0.913 near $d/L_0 = 0.16$, before the wave slows in earnest and the height climbs. A 9% dip in height on the way in is real and it is measurable. Only after that does the familiar growth begin, and it accelerates sharply in the last stretch — in shallow water $K_s$ tends towards $(4d/L_0)^{-1/4}$, which grows without bound as depth goes to zero.

Of course it does not actually grow without bound, because the wave breaks. The depth-limited criterion is what stops it, and the interaction between the two is the whole structure of a surf zone: shoaling drives the height up until $H$ reaches $\gamma_b d$, at which point the wave breaks and the height thereafter simply tracks the falling depth.

Shoaling is only one of four transformations, and it is often not the largest. The full nearshore chain is $H = K_s K_r K_f K_d H_0$. $K_r$ is refraction, the crest bending to face the beach, which concentrates energy on headlands and spreads it in bays and can move the height by tens of percent in either direction. $K_f$ covers bottom friction and percolation, which matter over long shallow shelves and coral, and are negligible over a short steep approach. $K_d$ is diffraction, the spreading of energy into the lee of a breakwater or headland. Reporting a shoaled height without asking about refraction is the standard way to be confidently wrong on a coast with any relief to its contours.

The arithmetic trap here is the factor of one half. $c_{g0} = gT/4\pi$, which is HALF the deep-water phase celerity $gT/2\pi$. Using the phase speed by mistake makes $K_s$ come out $\sqrt{2} = 1.41$ times too large — a 41% error in wave height, which would be glaring in a survey and completely invisible in a spreadsheet. Compute the local $c_g$ properly from the dispersion relation as $c_g = nc$ with $n = \tfrac{1}{2}\left[1 + \dfrac{2kd}{\sinh 2kd}\right]$, and the whole calculation stays honest.

One closing piece of honesty. Shoaling is linear theory applied in the place linear theory is least valid. A shoaling wave has a peaked crest and a flat trough, its height is becoming a large fraction of the depth, and the small-amplitude assumption is visibly failing. Real waves grow faster just before breaking than Airy theory predicts, and cnoidal or stream-function theory is what a serious nearshore calculation uses. $K_s$ is the right first estimate and the wrong last word.