Guo Formula of Linear Dispersion Relation Calculator

Guo Formula:

\[ kd = \frac{\omega^2 \cdot d}{g} \cdot \left(1 - \exp\left(-\left(\omega \cdot \sqrt{\frac{d}{g}}\right)^{\frac{5}{2}}\right)^{-\frac{2}{5}}\right) \]

Linear Dispersion Relation (kd):

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1. What is the Guo Formula of Linear Dispersion Relation?

The Guo Formula of Linear Dispersion Relation is an important expression in coastal and ocean engineering that shows how the dimensionless wave speed depends on the relative wavelength. It provides a mathematical relationship between wave angular frequency, coastal mean depth, and gravitational acceleration.

2. How Does the Calculator Work?

The calculator uses the Guo Formula:

\[ kd = \frac{\omega^2 \cdot d}{g} \cdot \left(1 - \exp\left(-\left(\omega \cdot \sqrt{\frac{d}{g}}\right)^{\frac{5}{2}}\right)^{-\frac{2}{5}}\right) \]

Where:

\( kd \) — Linear Dispersion Relation (dimensionless)
\( \omega \) — Wave Angular Frequency (rad/s)
\( d \) — Coastal Mean Depth (m)
\( g \) — Gravitational acceleration (9.80665 m/s²)

Explanation: The formula accounts for the relationship between wave characteristics and water depth, incorporating exponential and square root functions to model the dispersion behavior.

3. Importance of Linear Dispersion Relation Calculation

Details: Accurate calculation of linear dispersion relation is crucial for wave propagation studies, coastal engineering design, offshore structure analysis, and understanding wave behavior in different water depths.

4. Using the Calculator

Tips: Enter wave angular frequency in rad/s and coastal mean depth in meters. Both values must be positive numbers greater than zero for accurate calculation.

5. Frequently Asked Questions (FAQ)

Q1: What is the physical significance of linear dispersion relation?
A: Linear dispersion relation describes how wave speed depends on wavelength and water depth, which is fundamental for predicting wave behavior in coastal and ocean environments.

Q2: When is this formula most applicable?
A: This formula is particularly useful for intermediate water depth conditions where both deep and shallow water wave effects need to be considered.

Q3: What are typical values for wave angular frequency?
A: Wave angular frequency typically ranges from 0.1 to 2.0 rad/s for ocean waves, corresponding to periods of about 3-60 seconds.

Q4: How does water depth affect the dispersion relation?
A: In shallow water, waves become non-dispersive (speed depends only on depth), while in deep water, waves are dispersive (speed depends on wavelength).

Q5: What are the limitations of this formula?
A: The formula assumes linear wave theory and may have limitations for extreme wave conditions, very shallow water, or highly nonlinear wave phenomena.