Group Velocity Of Wave Given Wavelength And Wave Period Calculator

Group Velocity for Shallow Water Formula:

\[ V_{gshallow} = 0.5 \times \frac{\lambda}{P} \times \left(1 + \frac{4\pi d/\lambda}{\sinh(4\pi d/\lambda)}\right) \]

Group Velocity for Shallow Water:

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1. What is Group Velocity for Shallow Water?

Group Velocity for Shallow Water is the speed at which a group of waves travels through shallow water, crucial for understanding how energy and momentum are transferred within wave groups. It differs from phase velocity and is particularly important in coastal engineering and oceanography.

2. How Does the Calculator Work?

The calculator uses the Group Velocity for Shallow Water formula:

\[ V_{gshallow} = 0.5 \times \frac{\lambda}{P} \times \left(1 + \frac{4\pi d/\lambda}{\sinh(4\pi d/\lambda)}\right) \]

Where:

\( V_{gshallow} \) — Group velocity for shallow water (m/s)
\( \lambda \) — Wavelength (m)
\( P \) — Wave period (s)
\( d \) — Water depth (m)
\( \pi \) — Archimedes' constant (≈3.14159)
\( \sinh \) — Hyperbolic sine function

Explanation: The formula accounts for the dispersion relationship in shallow water waves, where the group velocity depends on wavelength, wave period, and water depth.

3. Importance of Group Velocity Calculation

Details: Accurate group velocity calculation is essential for predicting wave energy propagation, designing coastal structures, understanding sediment transport, and forecasting wave behavior in shallow water environments.

4. Using the Calculator

Tips: Enter wavelength in meters, wave period in seconds, and water depth in meters. All values must be positive (wavelength > 0, period > 0, depth ≥ 0).

5. Frequently Asked Questions (FAQ)

Q1: What is the difference between phase velocity and group velocity?
A: Phase velocity is the speed of individual wave crests, while group velocity is the speed at which wave energy propagates. In dispersive media, they differ.

Q2: How does water depth affect group velocity?
A: In shallow water, group velocity decreases with decreasing depth. The formula accounts for this through the hyperbolic sine function term.

Q3: What is considered "shallow" water for this calculation?
A: Water is considered shallow when the depth is less than half the wavelength (d < λ/2). The formula applies to transitional depths between deep and shallow water.

Q4: Why is the hyperbolic sine function used?
A: The sinh function accurately models the dispersion relationship for water waves across all depth regimes, providing a smooth transition between deep and shallow water approximations.

Q5: What are typical values for group velocity in coastal waters?
A: Group velocities in coastal waters typically range from 1-10 m/s, depending on wave characteristics and water depth. Longer waves generally have higher group velocities.