Simocean | Simulating things in the ocean

Part 1 ended with a simple spring-mass-damper driven by a wave force. But we glossed over something important — when a buoy moves, it shoves water aside and radiates waves. Those waves carry energy away, and the forces they create depend not just on the body's current motion, but on its entire history of past motions — because the waves take time to travel away and keep pushing back on the hull.

Radiation Damping

Drop a sphere into still water and watch what happens. It bobs up and down and gradually settles, while concentric waves radiate outward — each oscillation smaller than the last.

Those ripples are radiated waves. Unlike viscous drag (which converts kinetic energy into heat), the energy here is carried away by the waves. And there's a useful rule of thumb: a good wave maker is a good wave absorber. To capture wave energy, a device has to be able to radiate it. In a real sea the motion isn't a single clean frequency — it's a mixture of many frequencies. Those components are all present at once, so there isn't a single $A(\omega)$ or $B(\omega)$ you can plug into the equation of motion. That's why we switch to a time-domain description that accounts for the whole spectrum.

This wave-making shows up as a force that opposes the motion — the radiation force. For a pure sinusoid you can describe it with frequency-domain coefficients. But for real, messy motion you need a time-domain description — because the waves created a moment ago still influence the body now.

Added Mass

When a body moves through fluid, it doesn't move alone — the surrounding water must accelerate as well, and some of it ends up moving with the body. How much depends on shape and oscillation frequency:

How fast it's bobbing

1.0 rad/s

At low frequency a wide region adjusts; at high frequency the free surface responds less and the effect tends to a limit. We'll put a proper curve on that in the next section.

Frequency-domain coefficients

The demo below shows both effects in one place — the near-field particle motion and the outgoing wave motion — with $A(\omega)$ and $B(\omega)$ underneath, driven by one $\omega$ slider.

Near body ↔ A(ω) (added mass)Outgoing waves ↔ B(ω) (radiation)

ω (rad/s): 1.0

A(ω)/A∞ = 1.46B(ω) = 0.359

$A(\omega)$ is the inertial part: it represents the “extra mass” of water that must be accelerated with the body. $B(\omega)$ is the wave-radiation part: it tells you how strongly the body radiates waves (and therefore power) at each frequency. Together they describe the radiation force in the frequency domain.

From Frequency to Time

For single-frequency motion, $A(\omega)$ and $B(\omega)$ are enough. But real seas are broadband — many frequencies at once — so we can't plug in a single value. We need a time-domain formulation that accounts for the radiation force from the whole spectrum.

The key quantity is the radiation impulse response: how much radiation force you feel now from a unit velocity applied $\tau$ seconds ago. That function of $\tau$ is $K(\tau)$ — the memory kernel. Converting $B(\omega)$ into $K(\tau)$ gives us a way to compute the radiation force for realistic broadband motion.

From B(ω) to K(τ) — the Cosine Transform

$B(\omega)$ lives in the frequency domain. The impulse response $K(\tau)$ is its time-domain counterpart — a single function of time lag that encodes the same information:

K(\tau) = \frac{2}{\pi} \int_0^{\infty} B(\omega)\,\cos(\omega\,\tau)\,d\omega

Each frequency contributes a cosine component weighted by $B$ at that frequency. Because $B(\omega)$ is peaked and finite, the superposition is a decaying oscillation — the high-frequency components cancel at large $\tau$ , and the whole function fades to zero.

$B(\omega)$ says: “How much energy is radiated at each frequency.”

$K(\tau)$ says: “How long the fluid remembers a disturbance.”

The explorer below shows how $B(\omega)$ assembles into $K(\tau)$ as you include more frequencies — that's the frequency-to-time conversion we need for simulation.

Include frequencies up to: 6.0 rad/s

Sweep to build K(τ) frequency by frequency

Hydrodynamic Memory and the Cummins Equation

With the memory kernel $K(\tau)$ in hand, we can write the full equation of motion in the time domain — the Cummins equation. Why does the radiation force depend on history? When the body moved a few seconds ago it launched waves, and those waves are still out there. As they propagate away they continue to push on the hull. That's why the radiation force at time $t$ depends on the recent history of the body's velocity, not just its value at that instant. To compute it, you look back in time: how fast was the body moving $\tau$ seconds ago?

Not all past motions matter equally. Recent velocities created waves still near the body, while waves generated long ago are far from the body, so their contribution to the pressure on the hull is small. Think of $K(\tau)$ as: “how much radiation force you feel now from a unit velocity $\tau$ seconds ago.” The function $K(\tau)$ captures this weighting: large for small $\tau$ , decaying toward zero as a damped oscillation (because the radiated waves themselves oscillate). It is the radiation impulse response of the fluid–body system. The radiation force is the sum of each past velocity $\dot{x}(t - \tau)$ weighted by $K(\tau)$ — a convolution.

The Cummins Equation

For a single frequency ω, we had:

\bigl(m + A(\omega)\bigr)\,\ddot{x} + B(\omega)\,\dot{x} + kx = F_{\text{exc}}(\omega)

The Cummins equation (1962) replaces the frequency-dependent coefficients with the convolution described above:

\bigl(m + A_{\infty}\bigr)\,\ddot{x}(t) + \int_0^t K(\tau)\,\dot{x}(t - \tau)\,d\tau + kx(t) = F_{\text{exc}}(t)

$A_{\infty}$ — the infinite-frequency added mass. A constant that replaces the frequency-dependent $A(\omega)$ .

$K(\tau)$ — the retardation kernel. Large for small $\tau$ (recent motions), decaying toward zero for large $\tau$ (distant past).

$K(\tau)$ is obtained from the cosine transform of $B(\omega)$ — see the previous section.

Tying it together

How the RIRF Unlocks Memory

Convolve $K(\tau)$ with the velocity history — multiply each past velocity by how much the fluid still “remembers” it, and sum. That single integral replaces infinitely many frequency-dependent coefficients and is the reason the Cummins equation works for arbitrary, broadband seas.

B(ω)→cosine transform→K(τ)→convolve with velocity→F_rad(t)

The demo below visualises this convolution. Move the time cursor to pick an instant. The top panel highlights the window of past velocity; the lower panels show it flipped in time ( $\dot{x}(t - \tau)$ ), multiplied point-by-point by $K(\tau)$ , and integrated — the shaded area is the radiation force at that instant:

Time cursort = 12.0s

Because the kernel decays over time, only the most recent seconds of velocity contribute significantly to the force. How quickly that memory fades depends on body geometry — a compact buoy forgets quickly, while a large flap can retain hydrodynamic memory for tens of seconds.

In practice the kernel is truncated, so only a finite window of past velocity is included in the convolution. Another common approach is to approximate the radiation dynamics with a state-space model, which reproduces the same physics without storing the full velocity history.

Radiation damping isn't a local drag force. It comes from the waves the body generates as it moves — waves that carry energy away from the body.

But that same wave-making ability is also what allows the body to interact with incoming waves. In linear wave theory, a body that can radiate waves efficiently can also absorb energy from them.

This is why radiation is central to WECs: the same mechanism that lets a body make waves is what lets it absorb them.