Wave Dynamics and Tidal Processes

Spec mapping (AQA 7037): Paper 1 (Physical), §3.1.3 Coastal systems and landscapes — marine processes and the energy inputs that drive the coastal system: wind-wave generation, wave refraction, the geomorphological work of constructive and destructive waves, and tides (including spring/neap variation, tidal range and storm surge). It draws on §3.1.1 systems theory (energy as a system input) and links to §3.1.6 Hazards through coastal flooding and surge management. Assessment objectives: AO1 (wave and tidal theory), AO2 (applying wave energy and refraction to real coasts) and AO3 (manipulating wave-power, surge and platform data).

The introductory course treats waves descriptively. This depth lesson treats them physically and quantitatively: wave power as an equation, the transformation of waves in shoaling water, the geometry of refraction, and the superposition of meteorological and astronomical water-level signals that produces extreme surges.

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Wind-Wave Generation and the Energy Budget

Wind transfers energy to the sea surface by tangential stress (friction dragging the surface) and by form drag (pressure differences across wave crests and troughs — the Miles–Phillips mechanism). The size of the resulting waves is set by three variables:

1. Wind speed — energy input rises with roughly the cube of wind speed, so small increases in gale strength produce disproportionately larger seas.
2. Wind duration — the wind must blow long enough for waves to develop toward equilibrium with the wind.
3. Fetch — the open-water distance over which wind acts. The Atlantic-facing southwest of England has a fetch of the order of 5,000 km; Holderness faces a North Sea fetch up to ~800 km from the north-east.

These three are not independent: a coast can have a long potential fetch yet rarely experience fully developed seas because the wind from that direction seldom blows long enough (duration-limited), while another coast with a short fetch may be regularly fetch-limited even in strong winds. The practical consequence is that you must reason about the dominant wave direction (the direction generating the largest waves) and the prevailing wind direction (the most frequent) separately — they need not coincide, and where they diverge, the geomorphically effective wave climate is set by the dominant direction even though it is not the commonest. On the south coast of England, for instance, the prevailing wind is south-westerly but the longest fetch — and hence the dominant, beach-shaping waves — can come from the south-west across the open Atlantic, which is why so much southern-English longshore drift runs west-to-east.

When duration and fetch are both large enough that waves stop growing, the sea is fully developed. Waves that travel beyond the generating area become swell — long-period, low-steepness, regular waves that arrive as constructive waves on distant coasts. A useful refinement is the distinction between a sea state (the chaotic, multi-directional, steep "wind sea" inside the generating area) and swell (the sorted, regular, long-period waves that have outrun the storm). Because longer waves travel faster (they are dispersive), the swell from a distant Atlantic depression arrives at a UK beach sorted by period — the longest, fastest waves first, the shorter ones following — which is why surfers can forecast a clean groundswell days ahead. The geomorphic upshot is that a single distant storm can deliver constructive, beach-building swell to a coast that is itself experiencing calm weather, decoupling the local wave climate from the local wind.

Wave terminology (precise definitions)

Term	Symbol	Definition
Wave height	$H$H	Vertical trough-to-crest distance
Wavelength	$L$L	Crest-to-crest horizontal distance
Wave period	$T$T	Time for successive crests to pass a fixed point
Wave celerity (speed)	$c$c	$c = L/T$c=L/T
Amplitude	$a$a	$H/2$H/2
Steepness	$H/L$H/L	Controls breaking; limit $\approx 1/7$≈1/7
Frequency	$f$f	Waves per minute (field measure)

In deep water (depth $d > L/2$d>L/2) celerity depends on wavelength: $c = \sqrt{gL/2\pi}$c=gL/2π​ — longer waves travel faster (they are dispersive). This is why swell sorts itself by period across an ocean basin.

Wave power

The energy transmitted per unit length of crest (the wave power, W m⁻¹) is approximately:

$P = \frac{\rho g^2 H^2 T}{32\pi}$P=32πρg2H2T​

or, in the form often quoted in textbooks using wavelength,

$P = \frac{\rho g H^2 L}{8T}$P=8TρgH2L​

where $\rho \approx 1025$ρ≈1025 kg m⁻³ (seawater density), $g = 9.81$g=9.81 m s⁻². The decisive feature for geomorphology is the $H^2$H2 term: power scales with the square of wave height. Doubling wave height quadruples wave power. A 4 m storm wave therefore carries sixteen times the power of a 1 m fair-weather wave — the single most important reason that a handful of storm days accomplishes more erosional work than months of calm.

This non-linearity drives the concept of geomorphological effectiveness: the events that do the work are not the most frequent (calm days) nor the rarest (the once-a-century superstorm) but those at the intersection of reasonable magnitude and reasonable frequency — the Wolman–Miller principle that moderately extreme, moderately frequent events accomplish most landscape change over time. On a high-energy UK coast this is the winter gale recurring several times each year, not the freak event. A corollary for fieldwork and modelling: a wave record averaged over a year (a mean $H_s$Hs​) systematically understates the work done, because the work is concentrated in the upper tail of the distribution that the mean smooths away. Whenever you are handed mean wave data in an AO3 resource, flag that the mean hides the storms that matter — a reliable route into the evaluative band.

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Orbital Motion and Wave Transformation in Shoaling Water

In deep water, water particles move in near-circular orbits whose diameter decays exponentially with depth, becoming negligible at the wave base (≈ $L/2$L/2). The wave form propagates, but the water itself barely advances — energy moves, mass does not.

flowchart LR
  A[Deep water<br/>d > L/2<br/>circular orbits] --> B[Shoaling<br/>d < L/2<br/>orbits flatten to ellipses]
  B --> C[Base of wave slowed by seabed friction]
  C --> D[L decreases, c decreases, H increases]
  D --> E[Steepness rises to ~1/7]
  E --> F[Wave breaks]

As a wave enters water shallower than half its wavelength (shoaling), the orbits are flattened into ellipses by the seabed; the base of the wave is retarded by friction while the crest continues, so wavelength and celerity fall while height rises. Steepness climbs until, at $H/L \approx 1/7$H/L≈1/7 (or when water depth $\approx 1.3H$≈1.3H), the crest can no longer be supported and the wave breaks.

Breaking-wave types and the surf-similarity parameter

The style of breaking is governed by beach gradient and deep-water steepness, summarised by the Iribarren (surf-similarity) number $\xi = \tan\beta / \sqrt{H_0/L_0}$ξ=tanβ/H0​/L0​​, where $\tan\beta$tanβ is beach slope and $H_0/L_0$H0​/L0​ is deep-water steepness:

Type	Description	Conditions	$\xi$ξ (approx.)	Dominant transport
Spilling	Crest foams gently down the front; energy released gradually over a wide surf zone	Gentle slope, steep (storm) waves	$\xi < 0.5$ξ<0.5	Suspended; net offshore (erosional)
Plunging	Crest curls and plunges into the trough; violent, localised energy release	Moderate–steep slope	$0.5 < \xi < 3.3$0.5<ξ<3.3	Strong, mixed; high erosive impact
Surging	Wave slides up the face without fully breaking; little air entrainment	Steep slope, low-steepness swell	$\xi > 3.3$ξ>3.3	Strong swash; net onshore (constructive)

Spilling breakers (storm conditions on gentle beaches) dissipate energy across a broad surf zone and tend to drive sediment offshore; surging breakers (swell on steep shingle) deliver strong swash and build beaches. Plunging breakers, the surfers' favourite, concentrate the most destructive instantaneous force on the bed.

Beyond the breakpoint lies the surf zone, where the broken wave continues shoreward as a turbulent bore, and then the swash zone, the wedge of beach alternately covered and exposed by swash and backwash. Two further controls on beach behaviour live here. The first is wave set-up: the momentum lost as waves break is balanced by a slight super-elevation of the mean water surface against the beach, raising the effective water level by tens of centimetres under storm conditions — a contribution to coastal flooding entirely separate from the surge. The second is the infiltration/exfiltration balance on the beach face: on a coarse, permeable shingle beach the swash drains into the berm before the backwash can mobilise sediment, favouring onshore transport and steep, reflective profiles; on a fine, less permeable sand beach the water returns as backwash, favouring offshore transport and gentle, dissipative profiles. This is the physical reason shingle beaches are typically steep and sand beaches gentle — grain size sets permeability, permeability sets the swash–backwash balance, and that balance sets the profile. The morphodynamic classification of beaches (reflective ↔ intermediate ↔ dissipative) formalises this coupling between waves, sediment and form.

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Wave Refraction and Diffraction

Refraction

Refraction is the bending of wave crests as they approach an irregular shoreline, caused by the part of the crest in shallower water slowing first.

flowchart TD
  W[Approaching wave crests, parallel in deep water] --> H[Crest over headland shoal slows first]
  W --> B[Crest over bay stays in deeper water, faster]
  H --> CONV[Crests bend to wrap the headland]
  B --> DIV[Crests stretch into the bay]
  CONV --> E1[Orthogonals CONVERGE on headland → energy concentrated → erosion]
  DIV --> E2[Orthogonals DIVERGE in bay → energy dispersed → deposition]

Drawing orthogonals (rays perpendicular to crests, representing equal energy in deep water): where orthogonals converge (over headlands, because the offshore bathymetry there shoals first) energy per unit length of coast rises and erosion is intense; where they diverge (into bays) energy is spread thin and deposition dominates. Refraction therefore tends, over time, to even out an irregular coast — eroding headlands and building bay-head beaches — a negative-feedback tendency toward a smoother, more energy-equilibrated plan form.

Exam Technique: A refraction sketch must show crests bending to parallel the shore, plus orthogonals converging on the headland and diverging in the bay, labelled "high energy / erosion" and "low energy / deposition". Markers reward the energy interpretation, not just the wavy lines.

Diffraction

Diffraction is the spreading of wave energy laterally into the sheltered zone behind an obstacle (a breakwater, an island, a headland tip). Energy bends into the geometric "shadow", which is why the lee of a breakwater is not perfectly calm and why sediment is supplied to the sheltered side of an island during tombolo formation. Refraction and diffraction together explain the wave climate in the lee of structures and islands — essential for predicting where engineered structures will trigger unexpected deposition or scour. A detached offshore breakwater, for example, casts a diffraction shadow in which longshore transport slackens and sediment accumulates, building a salient (a bulge in the beach) or, if the breakwater is close enough inshore, a full tombolo that ties the structure to the beach. Engineers exploit this deliberately to grow protective beaches without continuous hard defence — but the same diffraction can starve the next frontage downdrift, reproducing the groyne-field problem in a different geometry. The general lesson is that any structure that alters the wave field alters the sediment field, because the two are coupled through the breaker angle and the energy distribution along the shore.

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Constructive and Destructive Waves

Property	Constructive	Destructive
Origin	Distant swell, long fetch, calm	Local storm, short fetch
Height / wavelength	Low $H$H, long $L$L	High $H$H, short $L$L
Frequency	~6–8 min⁻¹	~10–14 min⁻¹
Breaker type	Surging / spilling	Plunging / spilling
Swash vs backwash	Strong swash, weak backwash	Weak swash, strong backwash
Net effect	Onshore — builds berms, widens beach	Offshore — combs sediment down, builds offshore bar
Beach profile produced	Wide, gentle, bermed	Steep upper beach, narrow, drawn-down

The mechanism is the swash–backwash asymmetry. Constructive waves have a long period, so swash drains/percolates before the next wave, leaving little backwash to remove what was carried up — net onshore movement. Destructive waves arrive too frequently for percolation; backwash combines with the next wave's drag to comb material offshore, building a breakpoint (longshore) bar while steepening the upper beach into a storm beach.

Seasonal profile. A British beach typically shows a summer profile (constructive swell dominant — wide, gently sloping, prominent berm) and a winter profile (destructive storms dominant — steep, narrow, sediment moved to an offshore bar). This is the coast's seasonal dynamic equilibrium in action.

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Wave-Driven Longshore Transport — the Angle of Approach

The volume of sediment driven along a coast depends not only on wave power but on the angle of wave approach to the shoreline. Where waves break obliquely, the longshore component of wave energy drives a current and a sediment flux parallel to the beach. In the engineering literature the longshore transport rate $Q_l$Ql​ is related to the longshore component of wave energy flux, which scales approximately with:

$Q_l \propto H_b^{5/2} \sin(2\alpha_b)$Ql​∝Hb5/2​sin(2αb​)

where $H_b$Hb​ is breaker height and $\alpha_b$αb​ is the breaker angle to the shoreline. Two features matter for the exam. First, the $\sin(2\alpha_b)$sin(2αb​) term is maximised at $\alpha_b = 45°$αb​=45° and falls to zero at both 0° (waves arriving parallel to the beach do no longshore work) and 90° (a geometric impossibility at breaking). Second, the high power of $H_b$Hb​ again confirms that storms dominate alongshore transport just as they dominate erosion. This is why a small change in the orientation of a beach relative to the dominant wave direction — for example, the smoothing produced by refraction, or the rotation of a beach by a new breakwater — can switch a frontage from net accretion to net erosion without any change in wave energy. The angle, not just the power, is the control.

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Tides

Tides are the long-period rise and fall of the sea driven by the gravitational attraction of the Moon and Sun modulated by Earth's rotation.

The two-bulge model

The Moon raises a tidal bulge on the sub-lunar side (direct attraction); a second bulge forms on the far side (where centrifugal effect in the Earth–Moon system exceeds the weaker lunar pull). As Earth rotates beneath the bulges, most coasts experience two highs and two lows per ~24 h 50 min — a semi-diurnal regime (the extra 50 minutes reflects the Moon's orbital motion). The Sun's tidal force is ~46% of the Moon's, so it modulates rather than dominates the pattern.

Spring and neap tides