Major Horizontal Semi-Axis Given Wavelength, Wave Height And Water Depth Calculator

Formula Used:

\[ A = \frac{H}{2} \times \frac{\cosh\left(\frac{2\pi(DZ + d)}{\lambda}\right)}{\sinh\left(\frac{2\pi d}{\lambda}\right)} \]

Wave Height (H):

Distance above the Bottom (DZ):

Wavelength (λ):

Water Depth (d):

Horizontal Semi-axis of Water Particle (A):

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1. What is the Horizontal Semi-axis of Water Particle?

The Horizontal Semi-axis of Water Particle is a key parameter in wave mechanics that describes the horizontal movement amplitude of water particles in wave motion. It represents half of the maximum horizontal distance that water particles travel during wave oscillation.

2. How Does the Calculator Work?

The calculator uses the following formula:

\[ A = \frac{H}{2} \times \frac{\cosh\left(\frac{2\pi(DZ + d)}{\lambda}\right)}{\sinh\left(\frac{2\pi d}{\lambda}\right)} \]

Where:

\( A \) — Horizontal Semi-axis of Water Particle (m)
\( H \) — Wave Height (m)
\( DZ \) — Distance above the Bottom (m)
\( \lambda \) — Wavelength (m)
\( d \) — Water Depth (m)
\( \pi \) — Mathematical constant pi (≈3.14159)

Explanation: The formula accounts for the hyperbolic variation of wave particle motion with depth, where cosh and sinh functions describe the vertical distribution of wave-induced motions.

3. Importance of Horizontal Semi-axis Calculation

Details: Calculating the horizontal semi-axis is crucial for understanding wave kinematics, designing coastal structures, predicting sediment transport, and analyzing wave forces on offshore installations.

4. Using the Calculator

Tips: Enter all values in meters. Wave height, wavelength, and water depth must be positive values. Distance above bottom must be non-negative. Ensure water depth is sufficient to avoid division by zero errors.

5. Frequently Asked Questions (FAQ)

Q1: What happens when water depth approaches zero?
A: As water depth approaches zero, the denominator sinh(2πd/λ) approaches zero, making the result undefined. This reflects the physical reality that waves break in shallow water.

Q2: How does distance above bottom affect the result?
A: The horizontal semi-axis decreases with increasing distance above the bottom due to the cosh function in the numerator, reflecting reduced wave motion near the seabed.

Q3: What are typical values for horizontal semi-axis?
A: Values vary widely depending on wave conditions, but typically range from centimeters for small waves to several meters for large storm waves.

Q4: Can this formula be used for deep water waves?
A: Yes, the formula is valid for all water depths. In deep water (d/λ > 0.5), the formula simplifies as sinh(2πd/λ) approaches cosh(2πd/λ).

Q5: What are the limitations of this formula?
A: The formula assumes linear wave theory, which may not accurately represent extreme wave conditions, breaking waves, or highly nonlinear wave phenomena.