1. What is Velocity of Propagation in Linear Dispersion Relation?

The Velocity of Propagation in Linear Dispersion Relation describes the speed at which waves travel through a fluid medium, considering the relationship between wave frequency and wave number. It's particularly important in coastal engineering and oceanography for understanding wave behavior.

2. How Does the Calculator Work?

The calculator uses the formula:

Where:

• \( C_v \) — Velocity of Propagation (m/s)
• \( [g] \) — Gravitational acceleration (9.80665 m/s²)
• \( d \) — Coastal Mean Depth (m)
• \( k \) — Wave Number for Water Wave (rad/m)
• \( \tanh \) — Hyperbolic tangent function
• \( \sqrt{} \) — Square root function

Explanation: The formula calculates wave propagation velocity by considering gravitational effects, water depth, and wave characteristics through the hyperbolic tangent function.

3. Importance of Velocity of Propagation Calculation

Details: Accurate calculation of wave propagation velocity is crucial for coastal engineering, tsunami prediction, offshore structure design, and understanding wave energy distribution in marine environments.

4. Using the Calculator

Tips: Enter coastal mean depth in meters and wave number in radians per meter. Both values must be positive numbers greater than zero for accurate calculation.

5. Frequently Asked Questions (FAQ)

Q1: What is the physical significance of the hyperbolic tangent function in this formula?
A: The tanh function accounts for the transition between deep water waves (where tanh(kd) ≈ 1) and shallow water waves (where tanh(kd) ≈ kd), providing accurate results across all depth regimes.

Q2: How does water depth affect wave propagation velocity?
A: In deep water, wave speed depends primarily on wavelength. In shallow water, wave speed becomes dependent on water depth, decreasing as depth decreases.

Q3: What are typical values for wave propagation velocity in ocean waves?
A: Ocean wave velocities typically range from 5-25 m/s for wind waves, while tsunami waves can travel at speeds up to 200-800 km/h in deep ocean.

Q4: Are there limitations to this linear dispersion relation?
A: The linear theory assumes small amplitude waves and may not accurately represent extreme wave conditions or nonlinear wave interactions.

Q5: How is this calculation used in practical applications?
A: This calculation is essential for coastal protection design, ship navigation, offshore platform design, and predicting wave arrival times for tsunami warning systems.