Image credit: Nautilus Advanced Intelligence, Inc.
Response Amplitude Operators
Response Amplitude Operators (RAOs) describe how vessels respond to incident waves. RAOs are typically treated as vessel fingerprints: six motions, six transfer functions, tightly coupled to a specific hull, draught, and heading.
That framing is convenient, and often misleading for cable installation.
For installation dynamics, the cable does not experience the vessel. It experiences a single driven boundary condition at the chute. Once that boundary is correctly characterized, vessel dynamics and cable dynamics can be cleanly separated without loss of physical fidelity.
The Default Assumption — and Why It Fails
Conventional thinking follows this chain:
- Waves excite the vessel
- Vessel motions propagate through the lay system
- Cable response is a downstream consequence
This leads to a high dimensional tightly coupled system sensitive to vessel-specific details that do not materially affect cable behavior.
The implicit assumption is that the cable must inherit the full vessel motion spectrum.
It does not.
What the cable actually “sees”
From the cable’s perspective, everything upstream collapses into a single mechanical object:
The chute behaves as a driven, damped harmonic oscillator.
- The cable is driven at the cable drop point
- The forcing has dominant frequency content
- The system exhibits inertia, stiffness, and damping
- Energy transmission depends on impedance, not hull geometry
This is not a numerical trick. It is classical mechanics.
Once the chute motion is defined, the downstream cable response follows deterministically.
Reframing RAOs: From Vessel Identity to Transfer Operator
RAOs are usually indexed by vessel, draught, and heading. That indexing reflects how RAOs are produced, not how they must be used.
For cable dynamics, only the following matter:
- Amplitudes of motion at the chute
- Phase relationships to wave excitation
- Frequency content of the response
If a modified or effective draught reproduces the same chute motion, then, from the cable’s point of view, the systems are dynamically equivalent. From the perspective of the cable, vessel identity matters only insofar as it determines the resulting chute motion; different vessels or operating draughts that produce dynamically equivalent boundary responses are therefore interchangeable for installation analysis.
In this framing, RAOs are not vessel identifiers.
They are transfer operators used to synthesize boundary excitation.
A Note on Damping and RAO Validity
Publicly available RAOs often neglect or oversimplify hydrodynamic damping. Without appropriate damping, resonance amplitudes and phase relationships become physically inconsistent, making such RAOs unsuitable for driving downstream cable dynamics.
For installation analysis, RAOs must represent not only excitation and restoring effects, but also the energy dissipation mechanisms that govern the stability of the driven response.
The Decoupling Argument
The decoupling follows directly:
- The cable responds to driving force
- The forcing enters only through the chute boundary
- The chute behaves as a driven, damped oscillator
- Any vessel configuration that reproduces the same chute response is dynamically equivalent for the cable
- Therefore, vessel and cable dynamics can be separated without violating physics
This is not an approximation layered on top of complexity.
It is a model reduction justified by first principles.
Why This Matters in Practice
This separation enables:
- Reduced model dimensionality without loss of relevance
- Cleaner sensitivity studies
- Better-conditioned validation and ML training datasets
- Explicit control over assumptions and uncertainties
Most importantly, it improves confidence:
- Confidence that the model explains why, not just what
- Confidence that results generalize beyond a single vessel
- Confidence that uncertainty is introduced deliberately, not implicitly
Closing Note
This insight did not come from adding sophistication.
It came from removing what did not matter.
When RAOs are treated as transfer operators rather than vessel signatures, the physics becomes simpler, the models become more robust, and the engineering decisions become easier to defend.