Angular Frequency Is Not a Convention

Response Across Multiple Domains

Response Amplitude Operators (RAOs) are foundational to offshore and subsea engineering. They can be expressed either as responses evolving in time, or as relationships defined over inverse time—frequency.

Both representations are valid. They are not interchangeable.

Inverse-time representations compress assumptions about phase evolution, energy accumulation, and causality into compact expressions. When those assumptions are either violated during conversion or left implicit, the resulting errors are deterministic, structural, and often invisible to standard validation checks.

This brief documents three closely related failure modes that arise when RAOs are moved across domains without enforcing their underlying operators.

Failure Mode 1: Angular Frequency Treated as Generic Inverse Time

In offshore practice, environmental inputs are typically described using ordinary frequency $f$ (Hz), while RAOs are defined in angular frequency $\omega$ (rad/s).

The conversion is exact:

The error begins when angular frequency is treated as interchangeable with a generic “per-second” quantity, or when $\omega$-based RAOs are applied as if they directly describe time-domain response.

Although radians are dimensionless, angular frequency is not merely a unit-scaled frequency. It is the generator of phase evolution. Confusing $\omega$ with $f$, or treating $\omega$ as a pointwise time-domain gain, alters how time enters the system.

The consequences are severe:

• phase no longer accumulates correctly
• energy does not build cycle by cycle
• resonance is suppressed
• dynamics collapse

In extreme cases, amplitudes trend toward zero across the spectrum. The system appears unrealistically stable—a “super-vessel” that remains motionless under energetic wave conditions.

This is not robustness. It is the removal of causality.

Failure Mode 2: Correct Conversion, Wrong Ordering

A more subtle failure occurs when the governing equations are formulated in the time domain, while RAOs are provided in angular frequency.

The necessary conversion to a time scale is:

Applying this transformation is necessary, but not sufficient. Angular frequency is ordered in ascending $\omega$. The corresponding time periods $T = 2\pi / \omega$ are therefore ordered in descending time.

If the converted values are used directly without reindexing the RAO table, the mapping is applied correctly but the lookup is not.

RAOs are typically tabulated and stored as functions of angular frequency, ordered by increasing $\omega$. After converting to period, the corresponding $T$ values decrease as $\omega$ increases. The independent variable remains monotonic, but it is monotonic in the opposite direction relative to the original table ordering.

If interpolation is then performed “as usual” (e.g., by searching for neighboring rows under the assumption of ascending $T$), the algorithm brackets the query with the wrong rows from the RAO table. The computation remains numerically smooth, but it is built from incorrect RAO samples.

This is not a unit error. It is a table-indexing error: the query is evaluated against the wrong neighbors because the RAO data were not re-sorted (or reindexed) for the domain actually used by the solver.

The result is a second, independent error:

• long-period behavior is blended with short-period response
• amplitude and phase relationships are scrambled
• dynamics are distorted without triggering numerical instability

The simulation runs. The outputs are smooth. The physics is wrong.

This error is particularly dangerous because the unit conversion itself is correct. Only the ordering assumption is violated.

Failure Mode 3: Treating RAOs as Pointwise Gains Instead of Operators

Or why frequency-domain results often look acceptable? In purely frequency-domain or generalized stability methods:

• scaling errors partially cancel
• conservative margins absorb discrepancies
• sequencing and memory are irrelevant

As a result, both failure modes can remain hidden. The absence of obvious failure is often mistaken for correctness.

Time-domain formulations do not offer this protection.

Operator-Level Explanation: Why These Failures Are Inevitable

At the operator level, the distinction between time, frequency, and angular frequency is unavoidable.

An RAO is a frequency-domain transfer function relating environmental excitation to system response:

Each term has a specific and non-interchangeable meaning:

• $X(\omega)$ is the frequency-domain representation of a single vessel response component, typically one of the six rigid-body degrees of freedom (surge, sway, heave, roll, pitch, or yaw). It represents how response energy is distributed across angular frequencies, not how the vessel moves in time.

• $\eta(\omega)$ is the frequency-domain representation of the environmental excitation experienced by the vessel, most commonly wave elevation at the reference point. It encodes the spectral content of the sea state, including amplitude and phase, but contains no information about vessel dynamics.

• $H(\omega)$ is the Response Amplitude Operator (RAO). It is a complex-valued transfer function that maps excitation to response under steady-state, linear assumptions. $H(\omega)$ encapsulates inertia, restoring forces, damping, and coupling effects, and is defined only in the frequency domain.

This relationship is algebraic in frequency space, but it is not causal in time.

Physical response requires one of two consistent constructions:

• spectral integration

  $$\sigma_x^2 = \displaystyle\int |H(\omega)|^2 S_\eta(\omega)\, d\omega$$

• time-domain convolution

  $$x(t) = \displaystyle\int h(\tau)\,\eta(t-\tau)\, d\tau$$

where $\sigma_x^2$ is the variance of the response $x(t)$ and $h(t)$ is the inverse Fourier transform of $H(\omega)$.

Variace of the response is not a pointwise quantity, it is a statistical property of the response over time. $S_\eta(\omega)$ is the power spectral density of the excitation $\eta(t)$, measured in m$^2$/(rad/s). The time domain convolution implies causality: the vessel responds at time $t$ as a weighted sum of past excitation.

Applying $H(\omega)$ pointwise in time, or interpolating it on a misordered domain, bypasses the operator entirely. The system is no longer evolving in time; it is being sampled symbolically.

When the measure is lost, energy cannot accumulate. When ordering is violated, causality cannot be enforced.

The mathematics does exactly what it is told.

Practical Implications

These failure modes affect:

• dynamic lateral stability analyses
• large-displacement design
• soil–structure interaction with memory
• AI-assisted simulation pipelines
• real-time operational decision support

They do not introduce noise. They introduce deterministic distortion.

Once embedded, the error propagates across model updates and downstream analyses while remaining numerically well-behaved.

Practical Guardrails

The following checks are essential:

• Every frequency axis SHALL explicitly state [Hz] or [rad/s]
• RAOs and spectra SHALL be expressed in the same domain prior to integration or convolution
• Any conversion from $\omega$ to $T$ SHALL be followed by explicit reordering of the domain
• Interpolation SHALL only be performed on strictly monotonic ascending independent variables
• Any unexplained amplitude collapse or phase loss SHALL trigger a domain-consistency audit

These checks are inexpensive. Their absence is not.

Key Takeaway

Angular frequency is not an alternative representation of time. It is a generator of time evolution.

Treating rad/s as a generic 1/s, or ignoring ordering after conversion, does not degrade results gradually. It removes dynamics altogether.

The system does not fail loudly. It behaves implausibly well.