Wave Number given Wavelength, Wave Period and Water Depth Calculator

Wave Number Formula:

\[ k = \frac{\tanh^{-1}\left(\frac{L \cdot \omega}{g \cdot T}\right)}{d} \]

Wavelength (L):

Wave Angular Frequency (ω):

rad/s

Wave Period (T):

Water Depth (d):

Wave Number (k):

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1. What is Wave Number for Water Waves?

The Wave Number for Water Waves quantifies the number of waves per unit length along the surface of the water. It is a fundamental parameter in wave mechanics that describes the spatial frequency of a wave.

2. How Does the Calculator Work?

The calculator uses the wave number formula:

\[ k = \frac{\tanh^{-1}\left(\frac{L \cdot \omega}{g \cdot T}\right)}{d} \]

Where:

\( k \) — Wave number (rad/m)
\( L \) — Wavelength (m)
\( \omega \) — Wave angular frequency (rad/s)
\( g \) — Gravitational acceleration (9.80665 m/s²)
\( T \) — Wave period (s)
\( d \) — Water depth (m)

Explanation: The formula calculates the wave number by considering the relationship between wavelength, angular frequency, wave period, and water depth using inverse hyperbolic tangent function.

3. Importance of Wave Number Calculation

Details: Wave number is crucial for understanding wave propagation, energy distribution, and wave behavior in various water depths. It's essential for coastal engineering, oceanography, and offshore structure design.

4. Using the Calculator

Tips: Enter wavelength in meters, angular frequency in rad/s, wave period in seconds, and water depth in meters. All values must be positive. The ratio (L·ω)/(g·T) must be between -1 and 1 for valid calculation.

5. Frequently Asked Questions (FAQ)

Q1: What is the physical significance of wave number?
A: Wave number represents the spatial frequency of a wave, indicating how many wave cycles occur per unit distance. Higher wave numbers correspond to shorter wavelengths.

Q2: How does water depth affect wave number?
A: In deeper water, waves tend to have smaller wave numbers (longer wavelengths), while in shallower water, wave numbers increase (shorter wavelengths) due to wave shoaling effects.

Q3: What are typical wave number values for ocean waves?
A: Ocean wave numbers typically range from 0.01 to 1.0 rad/m, corresponding to wavelengths from about 6 meters to 600 meters.

Q4: Why use inverse hyperbolic tangent in this formula?
A: The inverse hyperbolic tangent function is used to solve the dispersion relationship for water waves, which relates wave number to wave frequency and water depth.

Q5: Can this calculator be used for all water wave types?
A: This calculator is primarily designed for linear gravity waves. For other wave types (capillary waves, tsunamis, etc.), different formulas may be required.