Simocean | Simulating things in the ocean

Wave energy is the final frontier of renewable energy technology — the final boss. Solar and wind have achieved grid parity. Geothermal and hydropower hum along in their geographical niches. Tidal energy offers clockwork predictability. But wave energy — despite being one of the most abundant renewable resources on Earth — remains untapped.

“The millions of kilogram meters of energy which were hidden in the waves were used only for the stimulation of sweethearts! We obtained electricity from the amorous whisper of the waves!”— Yevgeny Zamyatin, We (1924)

A century after Zamyatin's vision, we're still chasing it. In this first post on the Simocean blog, I wanted to revisit first principles — the concepts that are key to unlocking wave energy. Energy, frequency and vibration.

Energy and Frequency

When a wave passes through the ocean, the water itself doesn't travel with it — individual fluid particles move in orbital paths, returning to roughly where they started. What does travel is energy — kinetic energy in the orbital motion of particles, and potential energy in their displacement against gravity. The two forms continuously convert into each other as the wave propagates.

Wave Height (H): 2.0m

Period (T): 10.0s

f = 0.100 Hzω = 0.63 rad/s

The visualization shows water particles moving in orbital paths as the wave passes from left to right. Notice that the water itself doesn't travel with the wave — each particle returns to roughly where it started. What travels is energy, handed from particle to particle through their coordinated motion.

Watch the energy transfer: at the crest, particles are raised against gravity (maximum potential energy). As they fall, that potential energy converts to kinetic energy in the orbital motion. At the trough, the cycle reverses. This continuous exchange between potential and kinetic energy is what propagates the wave forward.

The Mathematics of Wave Energy

Particle orbit & wave profile:

x(t) = r \cos(\omega t), \quad y(t) = r \sin(\omega t)

\eta(x,t) = A \sin(kx - \omega t)

These equations describe the same phenomenon from different perspectives. At the surface, the particle's vertical motion y(t) = r·sin(ωt) is exactly the wave profile η evaluated at that particle's position — and the orbit radius r equals the wave amplitude A. The wave “travels” because neighboring particles are phase-shifted by the kx term.

Potential Energy — from PE = mgh

Water displaced to height η against gravity:

PE = \int_0^{\eta} \rho g z \, dz = \frac{1}{2} \rho g \eta^2

Substituting η = A sin(kx − ωt):

PE = \frac{1}{2} \rho g A^2 \sin^2(kx - \omega t)

Averaging over one wavelength sweeps through a full cycle of sin², so ⟨sin²⟩ = ½ regardless of the phase (kx − ωt):

\langle PE \rangle = \frac{1}{2} \rho g A^2 \cdot \langle \sin^2 \rangle = \frac{1}{2} \rho g A^2 \cdot \frac{1}{2}

\bar{E}_p = \frac{1}{4} \rho g A^2

Kinetic Energy — from KE = ½mv²

Particle velocity at depth z (decays exponentially):

v = A \omega e^{kz}

Since KE = ½mv², for a fluid: KE = ½ρv² per unit volume. Integrating over the water column:

KE = \int_{-\infty}^{0} \frac{1}{2} \rho A^2 \omega^2 e^{2kz} \, dz

Constants come out — only e^(2kz) depends on z:

KE = \frac{1}{2} \rho A^2 \omega^2 \int_{-\infty}^{0} e^{2kz} \, dz

Using ∫e^(ax) dx = e^(ax)/a, then evaluating at limits (at z=0: e⁰=1; at z=−∞: e^(−∞)=0):

\left[ \frac{e^{2kz}}{2k} \right]_{-\infty}^{0} = \frac{1}{2k} - 0 = \frac{1}{2k}

Multiplying by the constants:

KE = \frac{1}{2} \rho A^2 \omega^2 \cdot \frac{1}{2k} = \frac{\rho A^2 \omega^2}{4k}

The deep-water dispersion relation ω² = gk describes how frequency relates to wavelength under gravity. Using it (so ω²/k = g):

\bar{E}_k = \frac{\rho A^2 \cdot g}{4} = \frac{1}{4} \rho g A^2

Total energy density:

E = \bar{E}_k + \bar{E}_p = \frac{1}{4} \rho g A^2 + \frac{1}{4} \rho g A^2 = \frac{1}{2} \rho g A^2

The kinetic and potential energies are exactly equal — a beautiful property of linear waves.

Wave power (energy flux):

Power is energy transported per second. Waves carry energy at the group velocity c_g — the speed at which a wave packet travels (half the phase velocity for deep water):

P = E \cdot c_g

For deep water, c_g = gT/(4π). Substituting our energy density E = ½ρgA²:

P = \frac{1}{2} \rho g A^2 \cdot \frac{gT}{4\pi}

\boxed{P = \frac{\rho g^2 A^2 T}{8\pi} = \frac{\rho g^2 H^2 T}{32\pi}} \approx H^2 T \;\text{kW/m}

Power per meter of wave crest for a regular deep-water wave (H = 2A, using ρ ≈ 1025 kg/m³, g ≈ 9.81 m/s²). Power scales with wave height squared and period — longer, taller waves carry dramatically more energy. For irregular seas, a similar formula with 64π uses significant wave height H_s and energy period T_e, giving P ≈ 0.5 H_s² T_e kW/m.

Symbol Reference

ρ — water density ≈ 1025 kg/m³

g — gravitational acceleration ≈ 9.81 m/s²

A — wave amplitude = H/2 (m)

H — wave height, peak to trough (m)

η — surface elevation (m)

ω — angular frequency (rad/s)

k — wavenumber (rad/m)

T — wave period (s)

z — depth (m, negative downward)

c_g — group velocity (m/s)

λ — wavelength (m)

E — energy density (J/m²)

P — wave power per meter crest (W/m)

This is why wave energy is so appealing: the ocean stores enormous amounts of energy in these orbital motions, continuously replenished by wind. The challenge is designing systems that can harness energy from complex wave combinations, resonate with typical operating conditions, survive harsh ocean environments, and withstand extreme storm forces. Designing such systems requires a deep understanding of energy, frequency, and vibration.

Wave Superposition

One of the most powerful principles in wave mechanics is superposition — the idea that when multiple waves occupy the same space, their amplitudes simply add together.

In the interactive demonstration below, three independent sine waves combine to form a complex waveform. Adjust the wave height and period of each component wave to see how they contribute to the resulting pattern:

Long-period swell — slow, powerful waves from distant storms.
Medium-period waves — generated by regional winds.
Short-period chop — quick ripples from local winds.

Wave 1: Long-period swell

Height: 5.0m

Period: 13.5s

Wave 2: Medium-period waves

Height: 3.0m

Period: 9.0s

Wave 3: Short-period chop

Height: 1.3m

Period: 5.5s

Hₛ

8.4m

Tₑ

12.0s

Power

427kW/m

by wave:

79%

19%

Particle Motion (Combined)

The Mathematics of Superposition

Principle of superposition:

When waves overlap, their surface elevations simply add together:

\eta(x,t) = \sum_{i} A_i \sin(k_i x - \omega_i t + \phi_i)

Each wave keeps its own amplitude A, wavenumber k, frequency ω, and phase φ — they pass through each other unchanged. The complex patterns we see are just many simple sine waves added together.

Significant wave height (H_s):

For irregular seas, H_s combines the energy from all component waves:

H_s = \sqrt{2(H_1^2 + H_2^2 + H_3^2 + \cdots)} = \sqrt{2 \sum_i H_i^2}

This follows from H_s = 4√m₀ where m₀ = Σ(H_i²/8) for sinusoidal components with height H_i.

Energy period (T_e):

The energy-weighted average period — waves with more energy contribute more:

T_e = \frac{H_1^2 T_1 + H_2^2 T_2 + H_3^2 T_3 + \cdots}{H_1^2 + H_2^2 + H_3^2 + \cdots} = \frac{\sum_i H_i^2 T_i}{\sum_i H_i^2}

Since energy E ∝ H², each wave's period is weighted by its energy fraction. Larger waves pull T_e toward their period.

Symbol Reference

η — surface elevation (m)

H_i — wave height of component i (m)

T_i — period of component i (s)

H_s — significant wave height (m)

T_e — energy period (s)

φ_i — phase of component i (rad)

If you've ever spent time at the beach, you might start to recognize the irregular pattern of the waves in the animation. The fact that these complex, seemingly chaotic patterns can be decomposed into simple sine waves was discovered by Joseph Fourier in 1822. Fourier analysis — the process of breaking down complex phenomena into simple sinusoidal functions — is now one of the most powerful tools in mathematics, science and engineering. The complex, irregular patterns we see in nature are often just many simple waves dancing together.

Vibration and Resonance

Every wave energy converter (WEC) is, at its core, a vibrating system. The orbital motion of water particles, the oscillation of a floating buoy, the pitch of a hinged flap — these are all forms of vibration; periodic motion repeating at a characteristic frequency.

The key to efficient energy capture is resonance — when a device's natural frequency matches the ocean's driving frequency. At resonance, energy transfers with maximum efficiency; even small waves can drive large device motions. Off-resonance, much of the wave's energy passes by uncaptured.

To illustrate these concepts, consider the simple oscillating system below. This system is absorbing energy from the vertical component of wave motion. The buoy is connected to the seabed by a spring (providing restoring force) and a damper (extracting energy). N.B. Some of the values may be unrealistic, but it's just to illustrate the concepts.

Wave

H = 2.0 m

T = 10.0 s

Device

● RES

m = 50t

k = 20 kN/m

c = 50 kNs/m

T₀9.9s

ζ0.79

Wave Power

39.2

kW/m

Power Absorbed

Capture

% efficiency

Try adjusting the sliders above to match the wave period with the device's natural period. Notice how the energy output spikes at resonance — the device amplifies the wave motion instead of fighting it.

The Mathematics of Vibration

Equation of motion:

The system above is a classic spring-mass-damper system:

m\ddot{x} + c\dot{x} + kx = F(t)

where m is mass, c is damping coefficient, k is stiffness, and F(t) is the wave forcing.

Natural period:

The system wants to oscillate at its natural frequency/period:

\omega_0 = \sqrt{\frac{k}{m}} \quad \Rightarrow \quad T_0 = 2\pi\sqrt{\frac{m}{k}}

Heavy devices (large m) oscillate slowly; stiff moorings (large k) oscillate quickly. The optimal stiffness for resonance is k = mω² = m(2π/T)².

Damping ratio:

The damping ratio ζ determines how oscillations decay:

\zeta = \frac{c}{2\sqrt{km}} = \frac{c}{c_{crit}}

•ζ < 1: Underdamped — oscillates before settling
•ζ = 1: Critically damped — fastest return without oscillation
•ζ > 1: Overdamped — sluggish return, no oscillation

Resonance and power extraction:

When driven at frequency ω, the response amplitude depends on the frequency ratio r = ω/ω₀:

\text{Amplification} = \frac{1}{\sqrt{(1-r^2)^2 + (2\zeta r)^2}}

At resonance (r = 1), amplitude peaks. The power extracted by the damper is:

P = c \cdot v^2 = c \cdot (\dot{x})^2

Maximum power extraction occurs when the damping is tuned to match the wave conditions — too little damping and the device moves freely but extracts nothing; too much and it barely moves at all.

Beyond the basics:

Real WECs must also account for radiation damping (the device radiates waves back), viscous losses, and the multi-frequency nature of real seas — topics for future posts.

Symbol Reference

m — device mass (kg)

c — damping coefficient (Ns/m)

k — stiffness (N/m)

ζ — damping ratio (dimensionless)

ω₀ — natural frequency (rad/s)

T₀ — natural period (s)

r — frequency ratio ω/ω₀

F(t) — wave forcing (N)

x — displacement (m)

ẋ, ẍ — velocity, acceleration

But here's the challenge: the ocean doesn't deliver a single clean frequency — real waves are complex combinations constantly shifting with weather and season. As a device oscillates, it radiates waves back, creating two-way coupling that must be understood. At resonance, motion can exceed physical limits — end stops, slack moorings, structural loads. And then there's the resource variability:

Calm Seas

P ≈ 1.5 kW/m

Storm Conditions

P ≈ 1,500 kW/m

A factor of 1,000× — a device must efficiently capture energy from gentle swells while surviving conditions that deliver a thousand times more power.

Cracking Wave Energy

Given the challenges outlined above — complex wave spectra, radiation damping, physical limits, and extreme load variability — I believe cracking wave energy will require advances across multiple disciplines simultaneously. This is not a problem that will be solved by optimizing a single component.

Wave energy demands state-of-the-art capabilities in:

System Dynamics
Hydrodynamics
Moorings
Structural Engineering
Resource Assessment
Control Systems
Power Conversion
...and more

The design space is huge. Different mechanical topologies, new control algorithms, different geometries, novel materials, alternative power take-offs — and each combination behaves differently in different sea states. You can't just test your way through it. Tank testing costs thousands per day. Sea trials cost millions. You can spend years and tens of millions validating a single concept, only to discover it doesn't work.

This is where simulation becomes essential. Not as a replacement for physical testing, but as a way to explore the design space before you commit to hardware. To try out ideas, break things virtually, iterate fast.

That's what we're building at Simocean with Project SEA-Stack — open-source simulation tools for wave energy converters, developed in collaboration with NREL and UW-Madison. The goal is to make it easier for researchers and developers to test a wider range of ideas faster — in both operational and extreme conditions — and to get as much data as possible before progressing to the ocean. We hope that this will help device developers ensure the best possible version of their concept progresses to the ocean, with the best possible chance of success.

That's why this topic felt like the right place to start this blog. To build better wave energy converters, we first need to simulate them. To do that we need to understand energy, frequency, and vibration.