The invisible force that waves carry: Understanding Longuet-Higgins Radiation Stress Tensor — Part 1

Why did Michael Longuet-Higgins call it radiation stress?
At first glance, the term might sound like something out of nuclear physics — but don’t worry, nothing’s glowing here. In this context, “radiation” simply refers to how waves carry or radiate momentum forward as they travel across the sea, much like light radiates energy through space. Surface waves aren’t just wiggling water up and down — they’re transporting actual momentum. And when those waves grow, shoal, or break near the coast, that momentum doesn’t vanish — it has to go somewhere. It gets transferred into the ocean itself, pushing the water around. That internal push — a force per unit area within the fluid — is what we call stress. Hence the term: radiation stress.

If you’ve ever stood waist-deep in the sea and felt a swell gently nudge you shoreward — even before it breaks — then you’ve already experienced radiation stress in action. It’s subtle, but it’s there. That invisible shove is the ocean’s response to momentum being transferred from the wave field into the water column. It’s not just surface motion — it’s bulk movement.

And this effect adds up. As wave groups approach the coast and begin to shoal or break, the momentum they’ve been carrying needs to be redistributed. It flows into the mean current, raising sea level through a process we call wave setup. This isn’t a sudden splash — it’s a slow but steady rise in the average water level, driven by internal stresses induced by the waves. What’s fascinating is that the system self-regulates: the incoming wave momentum must be balanced by an increase in barotropic pressure (i.e., pressure due to the weight of the water column), and this dynamic balance is what triggers the setup. This same redistribution of momentum is also responsible for driving longshore currents and those slow, rhythmic oscillations in the surf zone we know as surf beat.

Why is it a tensor?

To properly describe how momentum flows within the ocean, we need more than just a single number. Momentum can move in different directions, and it can push differently depending on which surface you're looking at. A tensor is just a mathematical object that keeps track of how something — like momentum — is transmitted in space along different axes.

For instance, in the case of waves, the radiation stress tensor includes components like:

• Sxx: the flux of x-momentum across a vertical surface perpendicular to x
• Sxy: the flux of y-momentum across a surface perpendicular to x

This structure is very similar to the stress tensor used in solid mechanics or the Reynolds stress tensor used in turbulent flows.

Waves vs. turbulence: two kinds of stress

Longuet-Higgins himself pointed this out: while Reynolds stress arises from chaotic, random velocity fluctuations (think turbulence), radiation stress arises from regular, coherent wave oscillations. Both describe the transfer of momentum through oscillatory motion — random in turbulence, rhythmic in waves.

Importantly, even though the water velocity oscillates back and forth with time, its square \( u^2(t) \) is always positive. When you average this squared velocity over a full wave period, it contributes a net flux of momentum. That’s why this effect must be accounted for, and why we describe it using a tensor.

Mathematical expressions

In waves, the radiation stress tensor can be expressed as:

\[ S_{ij} = \overline{ \rho u_i u_j + p\, \delta_{ij} } - \overline{p} \, \delta_{ij} \]

Where:

• \( \rho \) is the fluid density
• \( u_i \) are the wave-induced velocity components
• \( p \) is the pressure
• \( \delta_{ij} \) is the Kronecker delta

And for turbulence, the Reynolds stress is:

\[ \tau_{ij} = -\rho\, \overline{u'_i u'_j} \]

Here, \( u'_i \) represents the turbulent fluctuations from the mean flow.

What do tensors tell us?

Here’s the cool part: both of these tensors — whether from waves or turbulence — can be diagonalised. That means we can compute their eigenvalues and eigenvectors:

• The eigenvalues tell us how strong the internal forcing is in a given direction
• The eigenvectors tell us which directions the fluid is pushing itself towards

So even if the movement is complex — chaotic in turbulence or rolling in waves — the fluid still ends up redistributing momentum in a few clear directions. That’s the kind of insight we gain when we zoom out and look at the ocean through tensors — not just tracking the motion, but understanding where and how it pushes.

Radiation stress explains how wave energy becomes a wave-induced force acting on the mean flow — driving currents, reshaping coastlines, and giving your knees a gentle push next time you’re standing at the edge of the sea. Of course, we’ll take it step by step — because this is one of Michael Longuet-Higgins’ most celebrated contributions, and it deserves to be explained with apples and oranges.

The definition of radiation stress

The x-x component of radiation stress is given by:

\( S_{xx} = \left\langle \int_{-h}^{\eta} \left(p + \rho u^2\right) \, dz \right\rangle - \int_{-h}^{0} p_0 \, dz \)

Where:

• \( \rho \): water density
• \( p \): total pressure under the wave
• \( u \): horizontal velocity of wave motion
• \( p_0 \): hydrostatic reference pressure = \( \rho g (h - z) \)
• \( \eta \): wave elevation
• \( \langle \cdot \rangle \): time average over a wave cycle

This expression compares the total wave-induced momentum flux to what we’d have under still water. Now we decompose it into three parts.

1. Kinetic term (horizontal velocity squared)

This comes from squaring the horizontal wave velocity:

\( u(x,z,t) = a \omega \frac{\cosh k(z+h)}{\sinh kh} \cos(kx - \omega t) \)

Squaring and averaging in time gives:

\( \left\langle u^2 \right\rangle = \frac{1}{2} a^2 \omega^2 \left( \frac{\cosh k(z+h)}{\sinh kh} \right)^2 \)

Then we integrate vertically:

\( \left\langle \int_{-h}^0 \rho u^2 \, dz \right\rangle = \frac{1}{2} \rho a^2 \omega^2 \int_{-h}^0 \left( \frac{\cosh k(z+h)}{\sinh kh} \right)^2 dz \)

Physical meaning: Even though water particles move back and forth, their squared velocity is always positive. That means they carry momentum in a net direction — in this case, the direction of wave travel. As water depth decreases, the whole column participates in motion, making this term stronger. That’s why waves feel like they push harder as they approach shore.

2. Dynamic pressure correction

This comes from the non-hydrostatic part of the pressure field — the fact that pressure under a crest is higher than under a trough due to the curvature of the free surface. We integrate the deviation from hydrostatic pressure:

\( \left\langle \int_{-h}^{\eta} (p - p_0) dz \right\rangle = \frac{1}{2} \rho g a^2 \)

Physical meaning: Crests exert more pressure than troughs lift. In deep water, this term balances neatly. But in shallow water, wave crests sharpen and troughs flatten — the wave profile becomes asymmetric. That makes this pressure imbalance more intense and the dynamic pressure term more relevant.

3. Surface pressure term

This comes from the pressure applied by the mass of water raised above mean sea level (between 0 and \( \eta \)):

\( \left\langle \int_0^{\eta} p_0 dz \right\rangle = -\frac{1}{4} \rho g a^2 \)

Physical meaning: Water weighs more when it’s piled up. The elevated crest acts like a load on the surface, exerting a downward stress. This is a negative contribution.

Putting it all together

Now let’s add the three contributions:

\[ S_{xx} = \text{kinetic} + \text{dynamic pressure} + \text{surface pressure} \]

\[ S_{xx} = \rho g a^2 \cdot \left( \frac{kh}{\sinh 2kh} \right) + \frac{1}{2} \rho g a^2 - \frac{1}{4} \rho g a^2 \]

Combining terms:

\[ S_{xx} = \frac{1}{2} \rho g a^2 \left( \frac{2kh}{\sinh 2kh} + \frac{1}{2} \right) \]

Define wave energy:

\( E = \frac{1}{2} \rho g a^2 \)

Final form:

\[ S_{xx} = E \left( \frac{2kh}{\sinh 2kh} + \frac{1}{2} \right) \]

Physical meaning: This expression tells us how much extra momentum the wave field is carrying compared to still water. But what happens as the water depth changes?

Let’s look at limits of the term \( \frac{2kh}{\sinh 2kh} \):

• Deep water: when \( kh \gg 1 \), then \( \sinh 2kh \approx e^{2kh}/2 \), so the term becomes very small. That means the radiation stress approaches: \[ S_{xx} \approx E \cdot \frac{1}{2} \] So waves in deep water carry momentum — but not that much.
• Intermediate depth: here, the full expression matters. The term \( \frac{2kh}{\sinh 2kh} \) grows as depth decreases. So the total stress increases steadily.
• Shallow water: when \( kh \ll 1 \), then \( \sinh 2kh \approx 2kh \), so: \[ \frac{2kh}{\sinh 2kh} \approx 1 \] and we get: \[ S_{xx} \approx E \cdot \left(1 + \frac{1}{2} \right) = \frac{3}{2} E \] So in shallow water, the wave momentum flux is three times greater than in deep water.

Intuitive takeaway: As waves move from deep to shallow water, their ability to push momentum increases dramatically. That growing imbalance — the spatial change in radiation stress — is what drives wave setup and surf beat. So yes, when you're knee-deep at the beach and feel a pulse of water shoving toward shore, that's not just the crest passing. It's the momentum flux itself becoming concentrated and pushing the ocean around you.

How does this enter coastal models?

In real-world applications — especially when building long-term wave–current interaction datasets for coastal defence design or storm surge barriers — we don’t simulate every wave bump and splash directly. Why? Because it would take forever. Literally. We'd be old and grey before the models finished running. Instead, models like ADCIRC or ROMS use a neat trick: they average the waves out over time.

These models operate on time scales of hours or days, so they treat the detailed wave motion as background noise. But the effect of the waves — the momentum they carry and release — still needs to be accounted for. This is where radiation stress comes in.

The momentum equations get an extra term that captures the net force exerted by waves on the mean flow:

\[ \frac{\partial \mathbf{U}}{\partial t} + \cdots = -\frac{1}{\rho} \nabla \cdot \mathbf{S} \]

Here, \( \mathbf{U} \) is the depth-averaged velocity, and \( \mathbf{S} \) is the radiation stress tensor. The divergence of that tensor — how it varies in space — acts like a body force in the fluid. It's what drives wave setup, surf beat, and wave-driven currents.

So, although the waves aren’t explicitly resolved, their footprint is still felt. The stress tensor \( \mathbf{S} \) is typically provided by a spectral wave model like SWAN, which calculates wave energy, direction, and frequency at each point in space and time. From that, it builds the full tensor, which is then passed to the hydrodynamic model as a source term.

This coupling between averaged wave models and hydrodynamic solvers is what allows engineers and modellers to simulate complex phenomena — like flooding, overtopping, or harbour agitation — without needing to resolve every single ripple. Clever, right?

\[ \frac{\partial \mathbf{U}}{\partial t} + \cdots = - \frac{1}{\rho} \nabla \cdot \mathbf{S} \]

The tensor \( \mathbf{S} \) comes from external wave models like SWAN, which simulate wave energy spectra and provide values of \( S_{xx} \), \( S_{xy} \), etc. The gradients of these quantities are what drive setup, surf beat, and wave-induced currents. So even though the waves aren’t explicitly resolved in the model, their effect is fully felt through radiation stress.

Conclusion

Radiation stress might sound abstract — but it explains the very real push you feel as swells roll past. It underlies wave setup, surf beat, and wave-driven currents. Today we saw where it comes from, how it’s derived, and how each term plays a role. In the next post, we’ll explore how this concept is used in practice: how SWAN and ADCIRC share data, and how coastal engineers use radiation stress to simulate flooding and design defences.

Reference

Longuet-Higgins, M. S., & Stewart, R. W. (1964). Radiation stress in water waves; a physical discussion, with applications. Deep Sea Research and Oceanographic Abstracts, 11(4), 529–562. DOI Link