Wave Period Calculator
Compute wave period using validated scientific equations. See step-by-step derivations, unit analysis, and reference values.
Formula
T = sqrt(2*pi*L/g) [deep water] | T = L/sqrt(g*d) [shallow water]
Where T is wave period (seconds), L is wavelength (meters), g is gravitational acceleration (9.81 m/s2), and d is water depth (meters). The deep water formula applies when d/L > 0.5 and the shallow water formula when d/L < 0.05.
Worked Examples
Example 1: Deep Water Period from Wavelength
Problem: A deep water wave has a wavelength of 100 meters. Calculate the wave period, celerity, and frequency.
Solution: Period T = sqrt(2*pi*L/g) = sqrt(2*3.1416*100/9.81) = sqrt(64.08) = 8.005 s\nCelerity C = L/T = 100/8.005 = 12.49 m/s\nAlternatively C = sqrt(gL/2*pi) = sqrt(9.81*100/6.2832) = 12.49 m/s\nFrequency f = 1/T = 1/8.005 = 0.1249 Hz\nAngular frequency = 2*pi*f = 0.7849 rad/s
Result: Period: 8.005 s | Celerity: 12.49 m/s | Frequency: 0.1249 Hz
Example 2: Wind Wave Period Estimation
Problem: A sustained wind of 15 m/s blows over a fetch of 200 km. Estimate the dominant wave period and height that would be generated.
Solution: Using empirical fetch relations:\nFetch = 200,000 m, Wind speed = 15 m/s\nWave height estimate: H = 0.0016 * sqrt(F * U) = 0.0016 * sqrt(200000 * 15) = 2.77 m\nWave period estimate: T = 0.2857 * F^(1/3) / g^(1/3) = 0.2857 * 58.48 / 2.147 = 7.78 s\nThese are approximate values using simplified empirical formulas
Result: Estimated Period: ~7.78 s | Estimated Height: ~2.77 m | Developing Sea State
Frequently Asked Questions
What is wave period and how is it measured?
Wave period is the time interval between two consecutive wave crests passing a fixed point, measured in seconds. It is one of the most fundamental wave parameters along with wave height and wavelength. Wave period can be measured directly using wave buoys, pressure sensors on the seabed, or visual observation from a fixed structure. Modern wave buoy networks operated by NOAA and other agencies continuously record wave period data at hundreds of locations worldwide. In practice, oceanographers use several statistical measures of wave period including the peak period (associated with the spectral peak), the mean zero-crossing period, and the energy period. For most engineering applications, the peak period or significant wave period is the most useful descriptor of the sea state.
How is wave period related to wavelength?
The relationship between wave period and wavelength depends on water depth through the dispersion relation. In deep water, wavelength equals g times the period squared divided by two pi, giving L = 1.56 * T^2 in meters when T is in seconds. This means a 10-second wave has a wavelength of about 156 meters. In shallow water, wavelength equals the period times the square root of g times depth, so it depends on both period and depth. The dispersion relation also shows that in deep water, longer-period waves have longer wavelengths and travel faster, a property called dispersion. In shallow water, all wavelengths travel at the same speed determined only by depth, so waves are non-dispersive. This fundamental relationship is essential for converting between period and wavelength in wave calculations.
What determines the wave period generated by wind?
Wind-generated wave period depends primarily on three factors: wind speed, fetch length (the distance over which wind blows across open water), and wind duration. Higher wind speeds over longer fetches for longer durations produce waves with longer periods. The relationship is nonlinear because wave period grows more slowly than wave height with increasing fetch. Fully developed seas, where waves have reached equilibrium with the wind, require very long fetches and durations that are rarely achieved in practice. For example, a 20 m/s wind needs about 1,500 km of fetch and 23 hours of sustained blowing to produce a fully developed sea with a peak period of about 13 seconds. In coastal waters with limited fetch, wave periods are typically 3 to 8 seconds, while open ocean swell can have periods of 12 to 20 seconds.
What is the difference between sea and swell in terms of wave period?
Sea waves are locally generated by current wind conditions and typically have short periods of 3 to 8 seconds, irregular shapes, and steep profiles. Swell waves are generated by distant storms and have traveled long distances across the ocean, resulting in longer periods of 8 to 20 seconds, smooth regular shapes, and gentle slopes. During propagation, shorter-period waves lose energy faster due to dispersion and dissipation, so only the longer-period components survive long transoceanic journeys. Wave period is the primary indicator for distinguishing sea from swell: periods under about 8 seconds suggest local wind waves, while periods above 10 seconds indicate swell from distant sources. Most real sea states contain a mix of sea and swell components, creating complex wave spectra with multiple peaks.
How does wave period affect coastal processes?
Wave period profoundly influences coastal processes because it determines wave energy, orbital velocities at the seabed, and the depth to which waves can move sediment. Longer-period waves carry more energy per unit height because energy flux is proportional to both height squared and period. Longer-period waves also penetrate deeper into the water column, generating stronger orbital velocities at the seabed that can mobilize larger sediment particles. The wave period controls whether waves will break as plunging or spilling breakers through the surf similarity parameter (Iribarren number). Longer-period waves on steep beaches tend to produce plunging breakers that create more energetic swash and greater beach erosion. Harbor resonance is also period-dependent, with certain periods matching the natural oscillation frequencies of enclosed basins.
What is wave steepness and why does it matter?
Wave steepness is the ratio of wave height to wavelength (H/L) and is a dimensionless parameter that describes the shape of a wave. The maximum theoretical steepness for a stable deep water wave is approximately 1/7 or 0.143, above which the wave becomes unstable and breaks. Typical ocean swell has steepness values of 0.01 to 0.04, while locally generated wind seas may reach steepness of 0.05 to 0.10. Wave steepness affects wave-structure interaction, as steeper waves exert larger forces on coastal structures and are more likely to cause wave overtopping. In ship design, wave steepness determines the likelihood of slamming and green water on deck. Wave steepness also influences radar and satellite measurements of sea state because steeper waves create more radar backscatter.