Wave Speed Given Group Velocity Calculator

Wave Speed Formula:

\[ v = \frac{V_g}{0.5 \times \left(1 + \frac{k \times d}{\sinh(k \times d) \times \cosh(k \times d)}\right)} \]

Wave Speed (v):

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1. What is Wave Speed Given Group Velocity?

Wave Speed Given Group Velocity calculates the phase velocity of water waves based on the group velocity, wave number, and water depth. This relationship is important in coastal engineering and oceanography for understanding wave propagation characteristics.

2. How Does the Calculator Work?

The calculator uses the formula:

\[ v = \frac{V_g}{0.5 \times \left(1 + \frac{k \times d}{\sinh(k \times d) \times \cosh(k \times d)}\right)} \]

Where:

\( v \) — Wave Speed (m/s)
\( V_g \) — Group Velocity of Waves (m/s)
\( k \) — Wave Number for Water Wave
\( d \) — Coastal Mean Depth (m)
\( \sinh \) — Hyperbolic sine function
\( \cosh \) — Hyperbolic cosine function

Explanation: The formula relates the phase velocity (wave speed) to the group velocity through the wave number and water depth, using hyperbolic trigonometric functions to account for depth effects.

3. Importance of Wave Speed Calculation

Details: Accurate wave speed calculation is crucial for coastal engineering projects, tsunami warning systems, offshore structure design, and understanding wave energy propagation in various water depths.

4. Using the Calculator

Tips: Enter group velocity in m/s, wave number, and coastal mean depth in meters. All values must be positive numbers. The calculator will compute the corresponding wave speed.

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 or the wave envelope propagates.

Q2: When is this formula most applicable?
A: This formula is particularly useful for water waves in intermediate depths where both depth and dispersion effects are significant.

Q3: What are typical values for wave number in coastal waters?
A: Wave numbers typically range from 0.01 to 1.0 rad/m for ocean waves, depending on wavelength and water depth.

Q4: How does water depth affect wave speed?
A: In deep water, wave speed depends primarily on wavelength. In shallow water, wave speed depends primarily on water depth. Intermediate depths show transitional behavior.

Q5: Are there limitations to this formula?
A: This formula assumes linear wave theory and may not accurately represent extreme wave conditions or strongly nonlinear waves.