Wave Height For Simplified Vertical Fluid Particle Displacement Calculator

Wave Height Formula:

\[ H = \varepsilon' \times 2 \times \frac{\sinh(2\pi D / \lambda_{vp})}{\sinh(2\pi (DZ + d) / \lambda_{vp})} \times \cos(\theta) \]

Particle Displacement (ε'):

Water Depth (D):

Wavelength of Vertical Fluid Particle (λvp):

Distance above Bottom (DZ):

Phase Angle (θ):

rad

Wave Height (H):

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1. What is the Wave Height Formula?

The Wave Height formula calculates the vertical distance between the trough and crest of a wave based on particle displacement, water depth, wavelength, distance above bottom, and phase angle. It provides an accurate assessment of wave characteristics in fluid dynamics.

2. How Does the Calculator Work?

The calculator uses the Wave Height formula:

\[ H = \varepsilon' \times 2 \times \frac{\sinh(2\pi D / \lambda_{vp})}{\sinh(2\pi (DZ + d) / \lambda_{vp})} \times \cos(\theta) \]

Where:

\( H \) — Wave Height (m)
\( \varepsilon' \) — Particle Displacement (m)
\( D \) — Water Depth (m)
\( \lambda_{vp} \) — Wavelength of Vertical Fluid Particle (m)
\( DZ \) — Distance above Bottom (m)
\( \theta \) — Phase Angle (rad)
\( \pi \) — Archimedes' constant (≈3.14159)

Explanation: The formula accounts for the hyperbolic sine relationship between water depth, wavelength, and wave height, with cosine adjustment for phase angle.

3. Importance of Wave Height Calculation

Details: Accurate wave height estimation is crucial for coastal engineering, marine navigation, offshore operations, and understanding ocean dynamics and sediment transport.

4. Using the Calculator

Tips: Enter all values in appropriate units (meters for lengths, radians for angles). All values must be positive and valid for accurate calculation.

5. Frequently Asked Questions (FAQ)

Q1: What is particle displacement in wave mechanics?
A: Particle displacement refers to the movement of fluid particles from their equilibrium positions due to wave motion, typically following elliptical paths.

Q2: How does water depth affect wave height?
A: Water depth significantly influences wave characteristics through the sinh function, with waves experiencing shoaling (height increase) as they approach shallower water.

Q3: What is the significance of phase angle?
A: Phase angle determines the position of the wave at a specific time relative to a reference point, affecting the instantaneous wave height calculation.

Q4: Are there limitations to this equation?
A: This simplified formula works best for linear wave theory assumptions and may be less accurate for extreme wave conditions or complex bathymetry.

Q5: How is wavelength of vertical fluid particle determined?
A: Wavelength can be calculated from wave period and water depth using dispersion relationships, or measured directly from wave observations.