Relative Motion and Period of Encounter

Vectors:

• Ship’s speed vector: $\vec{V}_s$
• Wave speed vector (phase velocity/Celerity): $\vec{V}_w$
• Angle between them: $\mu$

Key Concepts:

• Absolute period of waves: $T_w$ (true wave period)
• Period of encounter: $T_e$ (apparent period relative to ship)
• $T_e$ is the resultant of $T_w$ and the ship’s speed through waves

Direction Convention:

• Head seas (meeting waves): $\mu = 180^\circ$
• Following seas: $\mu = 0^\circ$

Relationships:

$$T_e \propto V_s \quad \text{(ship speed)}$$

$$T_e \propto \mu \quad \text{(encounter angle)}$$

Wave Group Velocity:

$$V_{group} = \frac{1}{2}V_w \quad \text{(waves travel at half phase speed)}$$

Note: A group of waves travels at $V_w/2$, while individual wave crests travel at $V_w$

Energy Distribution: Exactly half the energy of a wave is kinetic and half is potential.

Maximum Wave Height:

$$H_{w\text{-max}} \approx 1.6 \times \text{SWH} \quad \text{(over 10 minutes)}$$

(Note: The document also mentions $H_{w\text{-max}} = 2 \times \text{SWH}$ in another section—verify which coefficient your course uses)

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Sinusoidal and Trochoidal Wave Theory

Water Depth Classification

• Shallow water: $d < \frac{\lambda}{2}$
• Deep water: $d > \frac{\lambda}{2}$ (presume this if depth not given)

Wave Relationships

Deep Water Formulas:

$$V_w = \sqrt{\frac{g\lambda}{2\pi}}$$

$$T_w = \sqrt{\frac{2\pi\lambda}{g}}$$

$$\lambda = \frac{2\pi V_w^2}{g}$$

General Relationship:

$$\lambda = \frac{gT_w^2}{2\pi} \quad \text{(standard deep water dispersion)}$$

Empirical Relation:

$$T_w = 0.285 \times V_{wind} \quad \text{(where wind speed is in knots)}$$

Very Shallow Water:

$$V_w = \sqrt{g \times h_w}$$

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Trochoidal Wave Theory

Historical Background

• 1905: Froude theorized that irregular wave systems are composites of many regular wave systems
• 1952: Researchers such as Trucker applied Fourier analysis to regular wave theory based on trochoidal waveforms
• The Trochoidal wave theory is not a perfect replica of observed sea states but comes closest compared to other theories

Definition

A Trochoid is a curve traced by a fixed point inside a circle as the circle rolls along and under a straight line.

Geometric Properties

$$\lambda = L_w = 2\pi R \quad \text{(where \(R\) = radius of rolling circle)})$$

$$h_w = 2r \quad \text{(wave height, where \(r\) = radius of small circle)}$$

Physical Characteristics

• Waveform has sharper crests than troughs
• Theory assumes water has zero viscosity and is frictionless
• Only the waveform (phase) moves, not the water mass
• Water particles move in closed circular orbits (one revolution per wave period)
• At depths greater than $\frac{\lambda}{2}$, there is no water particle movement (wave base)

Amplitude Decay with Depth:

$$r = r_0 \cdot e^{-\left(\frac{2\pi z}{\lambda}\right)}$$

(where $z$ = depth below surface)

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Assumptions for Simple Wave Theory

1. Crests are straight, parallel, and equally spaced
2. Wave heights remain constant
3. Wave form advances perpendicular to crest lines with uniform velocity $V_w$
4. $V_w$ is “Celerity” (phase velocity), not individual particle velocity
5. Water has zero compressibility and zero viscosity
6. Trochoidal waves occur in deep waters
7. Deep sea waves: half kinetic, half potential energy
8. Individual waves rise behind the “dead water zone,” travel through the group at $2 \times V_w$ (relative to group), then subside in the dead water zone ahead
9. Waves travel in groups

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Significant Wave Height (SWH)

Definition: When wave heights $h_w$ are arranged in descending order, the mean of the highest one-third of the waves is called SWH.

Requirements:

• At least 20 observations needed for reliable assessment
• Waves not fully developed are excluded

Importance:

• Vital for master’s decisions in heavy weather (particularly quartering/following seas)
• MSC 1228 specifically mentions importance of SWH observation

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Summary of Key Formulas

Parameter	Formula	Notes
Deep water wave speed	$V_w = \sqrt{\frac{g\lambda}{2\pi}}$
Wave period	$T_w = \sqrt{\frac{2\pi\lambda}{g}}$
Wavelength	$\lambda = \frac{2\pi V_w^2}{g}$
Group velocity	$V_{grp} = \frac{V_w}{2}$
Shallow water speed	$V_w = \sqrt{gh}$	$h$ = water depth
Wave height	$H_{max} \approx 1.6\text{-}2.0 \times \text{SWH}$	See note above
Particle decay	$r = r_0 e^{-2\pi z/\lambda}$

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Heaving period

$$T_H = 2 \times \pi \times \sqrt(\frac{\Delta}{100\times TPC \times g})$$

Pitching period

$$T_P = 2 \times \frac{K}{\sqrt(GM_L)}$$

where K = 0.25 L

Rolling period

$$T_R = \frac{2 \times \pi \times K}{ \sqrt(g \times GM) }$$

where K = 0.35B