Minor Vertical Semi-Axis given Wavelength, Wave Height and Water Depth Calculator

Formula Used:

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

Wave Height (H):

Distance above the Bottom (DZ):

Wavelength (λ):

Water Depth (d):

Minor Vertical Semi-Axis (B):

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1. What is Minor Vertical Semi-Axis?

The Minor Vertical Semi-Axis in wave theory refers to the vertical component of the elliptical motion of water particles beneath surface waves. It represents half of the vertical diameter of the elliptical orbit that water particles follow as waves pass through.

2. How Does the Calculator Work?

The calculator uses the following formula:

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

Where:

\( B \) — Minor Vertical Semi-Axis (m)
\( H \) — Wave Height (m)
\( DZ \) — Distance above the Bottom (m)
\( \lambda \) — Wavelength (m)
\( d \) — Water Depth (m)
\( \pi \) — Mathematical constant pi (≈3.14159)
\( \sinh \) — Hyperbolic sine function

Explanation: This formula calculates the vertical semi-axis of water particle motion at a specific depth in wave conditions, considering wave characteristics and water depth.

3. Importance of Minor Vertical Semi-Axis Calculation

Details: Calculating the minor vertical semi-axis is crucial for understanding wave kinematics, designing coastal structures, predicting sediment transport, and analyzing wave forces on submerged objects.

4. Using the Calculator

Tips: Enter wave height in meters, distance above bottom in meters, wavelength in meters, and water depth in meters. All values must be positive (distance above bottom and water depth can be zero).

5. Frequently Asked Questions (FAQ)

Q1: What is the physical significance of minor vertical semi-axis?
A: It represents the maximum vertical excursion of water particles from their mean position during wave motion at a specific depth.

Q2: How does water depth affect the minor vertical semi-axis?
A: In deeper water, particle orbits are more circular, while in shallower water, they become more elliptical with reduced vertical motion.

Q3: What happens when the denominator approaches zero?
A: The formula becomes undefined when sinh(2πd/λ) = 0, which occurs in very shallow water conditions relative to wavelength.

Q4: How accurate is this calculation for real-world applications?
A: This is based on linear wave theory and provides good approximations for small-amplitude waves in uniform depth conditions.

Q5: Can this be used for breaking waves?
A: Linear wave theory assumptions break down near breaking conditions, so results may not be accurate for breaking waves.