Velocity Of Propagation In Linear Dispersion Relation Given Wavelength Calculator

Velocity of Propagation Formula:

\[ C_v = \sqrt{\frac{g \cdot d \cdot \tanh\left(\frac{2\pi d}{\lambda''}\right)}{\frac{2\pi d}{\lambda''}}} \]

Velocity of Propagation (Cv):

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1. What is Velocity of Propagation?

The Velocity of Propagation is the speed at which a wave travels through a medium, indicating the rate of energy transfer. In coastal engineering, it represents how quickly wave energy propagates through water of varying depths.

2. How Does the Calculator Work?

The calculator uses the linear dispersion relation formula:

\[ C_v = \sqrt{\frac{g \cdot d \cdot \tanh\left(\frac{2\pi d}{\lambda''}\right)}{\frac{2\pi d}{\lambda''}}} \]

Where:

\( C_v \) — Velocity of propagation (m/s)
\( g \) — Gravitational acceleration (9.80665 m/s²)
\( d \) — Coastal mean depth (m)
\( \lambda'' \) — Deep water wavelength of coast (m)
\( \pi \) — Mathematical constant pi (3.14159)
\( \tanh \) — Hyperbolic tangent function

Explanation: The formula accounts for wave dispersion in varying water depths, where waves slow down as they approach shallower water.

3. Importance of Velocity Calculation

Details: Accurate velocity calculation is crucial for coastal engineering, wave energy prediction, tsunami modeling, and understanding wave behavior in transitional water depths between deep and shallow water conditions.

4. Using the Calculator

Tips: Enter coastal mean depth and deep water wavelength in meters. Both values must be positive numbers greater than zero for accurate calculation.

5. Frequently Asked Questions (FAQ)

Q1: What is the difference between phase velocity and group velocity?
A: Phase velocity is the speed at which individual wave crests travel, while group velocity is the speed at which wave energy propagates. This calculator computes phase velocity.

Q2: How does water depth affect wave velocity?
A: In deep water, waves travel faster. As water becomes shallower, wave velocity decreases due to bottom friction and wave dispersion effects.

Q3: What is transitional water depth?
A: Transitional water depth refers to depths where water is neither deep enough for deep water approximations nor shallow enough for shallow water approximations.

Q4: When is the tanh function significant in this calculation?
A: The tanh function becomes significant in transitional water depths where neither deep nor shallow water approximations are fully accurate.

Q5: What are typical velocity ranges for ocean waves?
A: Ocean wave velocities typically range from 5-25 m/s, depending on wavelength and water depth, with longer waves traveling faster than shorter waves.