How to Calculate Water Wave Speed?

You can calculate the phase speed of water waves by using the wavelength and wave period, applying specific formulas based on physical principles like gravity.

Based on the provided references, here's the method:

1. Calculate the Wavelength (L): The wavelength is the distance between two consecutive wave crests (or troughs). It can be calculated using the wave period (T), which is the time it takes for two successive crests to pass a fixed point.
   The formula provided is:
   L = g * T² / (2 * π)
   Where:

   • L is the wavelength in meters.
   • g is the acceleration due to gravity, approximately 9.8 m/s².
   • T is the wave period in seconds.
   • π (Pi) is a mathematical constant, approximately 3.14159.

2. Calculate the Wave Phase Speed: Once you have the wavelength (L), you can calculate the wave phase speed, which is how fast a specific point on the wave (like the crest) travels across the surface. Just like calculating the speed of a car (distance divided by time), the wave speed is the wavelength divided by the wave period.
   The formula provided is:
   Speed = L / T
   Where:

   • Speed is the wave phase speed (often denoted as c) in meters per second.
   • L is the wavelength in meters.
   • T is the wave period in seconds.

Combining the Formulas for Direct Calculation

You can substitute the formula for L from step 1 directly into the formula for Speed from step 2 to get a formula that calculates speed directly from the wave period:

Speed = (g * T² / (2 * π)) / T

Simplifying this equation gives you:

Speed = g * T / (2 * π)

This means that for waves in deep water (where this formula is most applicable), the speed depends primarily on the wave period. Longer period waves travel faster.

Key Variables and Formulas

Here's a quick summary of the formulas involved:

Variable	Description	Units	Formula Used (from references)
g	Acceleration due to gravity	m/s²	9.8 (constant)
T	Wave Period (time per wave)	seconds	-
L	Wavelength (distance between crests)	meters	L = g * T² / (2 * π)
Speed	Wave Phase Speed	meters/sec	Speed = L / T

Note: The combined formula `Speed = g T / (2 π)` is derived from the provided references.

Example Calculation

Let's calculate the speed of a water wave with a period (T) of 8 seconds.

Using the combined formula Speed = g * T / (2 * π):

• g = 9.8 m/s²
• T = 8 seconds
• π ≈ 3.14159

Speed = (9.8 m/s²) * (8 s) / (2 * 3.14159)
Speed = 78.4 m / 6.28318
Speed ≈ 12.48 m/s

Alternatively, using the two-step process:

1. Calculate L:
   L = g * T² / (2 * π)
L = (9.8 m/s²) * (8 s)² / (2 * 3.14159)
L = (9.8 m/s²) * (64 s²) / 6.28318
L = 627.2 m / 6.28318
L ≈ 99.82 meters

2. Calculate Speed:
   Speed = L / T
Speed ≈ 99.82 m / 8 s
Speed ≈ 12.48 m/s

Both methods yield the same result. The speed of a water wave with an 8-second period is approximately 12.48 meters per second.

This method is accurate for waves in deep water where the water depth is greater than half the wavelength. In shallower water, the depth also affects the wave speed, making the calculation more complex.