Wave Height Given Distance From Bottom To Wave Trough And Water Depth Calculator

Cnoidal Wave Formula:

\[ H_w = -d_c \times \left( \frac{y_t}{d_c} - 1 - \left( \frac{16 \times d_c^2}{3 \times \lambda^2} \right) \times K_k \times (K_k - E_k) \right) \]

Water Depth for Cnoidal Wave:

Distance from the Bottom to the Wave Trough:

Wavelength of Wave:

Complete Elliptic Integral of the First Kind:

Complete Elliptic Integral of the Second Kind:

Height of the Wave:

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1. What is the Cnoidal Wave Formula?

The Cnoidal Wave Formula calculates wave height based on water depth, distance from bottom to wave trough, wavelength, and elliptic integrals. It provides an accurate assessment of wave characteristics in shallow water conditions where cnoidal wave theory applies.

2. How Does the Calculator Work?

The calculator uses the Cnoidal Wave formula:

\[ H_w = -d_c \times \left( \frac{y_t}{d_c} - 1 - \left( \frac{16 \times d_c^2}{3 \times \lambda^2} \right) \times K_k \times (K_k - E_k) \right) \]

Where:

\( H_w \) — Height of the Wave (m)
\( d_c \) — Water Depth for Cnoidal Wave (m)
\( y_t \) — Distance from the Bottom to the Wave Trough (m)
\( \lambda \) — Wavelength of Wave (m)
\( K_k \) — Complete Elliptic Integral of the First Kind
\( E_k \) — Complete Elliptic Integral of the Second Kind

Explanation: The equation accounts for the complex relationship between wave characteristics and water depth in shallow water conditions, using elliptic integrals for precise calculation.

3. Importance of Wave Height Calculation

Details: Accurate wave height estimation is crucial for coastal engineering, offshore structure design, navigation safety, and understanding wave behavior in shallow water environments.

4. Using the Calculator

Tips: Enter water depth, distance to wave trough, wavelength, and both elliptic integral values. All values must be positive and valid for accurate results.

5. Frequently Asked Questions (FAQ)

Q1: What are cnoidal waves?
A: Cnoidal waves are exact periodic wave solutions of the Korteweg-de Vries equation that describe waves in shallow water with non-linear characteristics.

Q2: When should cnoidal wave theory be used?
A: Cnoidal wave theory is appropriate for intermediate water depths where both non-linearity and dispersion effects are important, typically for Ursell numbers greater than 26.

Q3: How are elliptic integrals obtained?

Q4: What are typical ranges for wave height calculations?
A: Wave heights can vary from centimeters to several meters depending on water depth, wavelength, and environmental conditions.

Q5: Are there limitations to this formula?
A: The formula is specifically designed for cnoidal waves in shallow water conditions and may not be accurate for deep water waves or breaking waves.