Wavelength For Distance From Bottom To Wave Trough Calculator

Formula Used:

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

Water Depth for Cnoidal Wave (d_c):

Complete Elliptic Integral of the First Kind (K_k):

Complete Elliptic Integral of the Second Kind (E_k):

Distance from the Bottom to the Wave Trough (y_t):

Height of the Wave (H_w):

Wavelength of Wave (λ):

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1. What is the Wavelength Calculation Formula?

The wavelength calculation formula determines the distance between consecutive corresponding points of the same phase on a cnoidal wave. This specialized formula accounts for water depth, elliptic integrals, and wave characteristics to provide accurate wavelength estimation.

2. How Does the Calculator Work?

The calculator uses the wavelength formula:

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

Where:

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

Explanation: The formula incorporates elliptic integrals and wave geometry to calculate the precise wavelength of cnoidal waves in varying water depths.

3. Importance of Wavelength Calculation

Details: Accurate wavelength calculation is crucial for coastal engineering, wave energy conversion, harbor design, and understanding wave propagation characteristics in shallow water environments.

4. Using the Calculator

Tips: Enter all values in appropriate units. Water depth, distance to trough, and wave height must be in meters. All elliptic integral values and dimensional parameters must be positive numbers for valid 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, characterized by their peaked crests and flat troughs, commonly found in shallow water.

Q2: How do elliptic integrals relate to wave calculations?
A: Elliptic integrals appear naturally in the solution of nonlinear wave equations and help describe the complete period and shape of cnoidal waves.

Q3: What is the typical range of wavelengths for ocean waves?
A: Ocean wavelengths can vary from centimeters (capillary waves) to hundreds of meters (swell waves), with most wind-generated waves having wavelengths between 10-150 meters.

Q4: When is this formula most applicable?
A: This formula is particularly useful for intermediate water depth conditions where both nonlinear and dispersive effects are significant.

Q5: What are the limitations of this calculation?
A: The formula assumes ideal cnoidal wave conditions and may not accurately represent breaking waves, very steep waves, or waves in extremely shallow water.