Finding the speed of a deep water wave depends on properties like its wavelength or period, influenced primarily by gravity.
In oceanography, a wave is considered a "deep water" wave when the water depth is greater than half of the wave's wavelength. In this condition, the wave's speed is determined by its own characteristics (wavelength or period) and the acceleration due to gravity, rather than the depth of the water.
You can calculate the speed of a deep water wave using one of two primary formulas:
Calculating Speed Using Wavelength
The speed ($v$) of a deep water wave can be found using its wavelength ($\lambda$):
$$ v = \sqrt{\frac{g\lambda}{2\pi}} $$
Where:
- $v$ = wave speed (commonly in meters per second, m/s)
- $g$ = acceleration due to gravity (approximately 9.81 m/s² on Earth)
- $\lambda$ = wavelength (the distance between consecutive wave crests or troughs, in meters)
- $\pi$ = the mathematical constant pi (approximately 3.14159)
This formula shows that in deep water, wave speed increases with the square root of the wavelength.
Calculating Speed Using Period
Alternatively, the speed ($v$) of a deep water wave can be calculated using its period ($T$), which is the time it takes for two consecutive wave crests (or troughs) to pass a fixed point:
$$ v = \frac{gT}{2\pi} $$
Where:
- $v$ = wave speed (m/s)
- $g$ = acceleration due to gravity (approximately 9.81 m/s²)
- $T$ = wave period (in seconds)
- $\pi$ = the mathematical constant pi
This formula shows that in deep water, wave speed is directly proportional to the wave's period.
The Wave Speed Equation and Wavelength Relationship
In the context of the general Wave Speed Equation, which relates wave speed ($v$), frequency ($f$), and wavelength ($\lambda$) as $v = f\lambda$, a fundamental relationship exists. As noted in the reference provided, "[A]nd wavelength are inversely proportional to each other." This statement typically refers to the relationship between wavelength ($\lambda$) and frequency ($f$) or period ($T$) when the wave speed ($v$) is held constant. Since frequency is the inverse of the period ($f = 1/T$), for a constant speed, wavelength ($\lambda = v/f = vT$) is inversely proportional to frequency and directly proportional to the period.
It is important to note that while this inverse relationship between wavelength and frequency/period for a given speed is a general property discussed with the wave speed equation, the specific deep water formulas above ($v = \sqrt{g\lambda/2\pi}$ and $v = gT/2\pi$) show how speed relates to wavelength and period in deep water conditions, where speed is not constant but depends on $\lambda$ or $T$.
Key Takeaways
- Deep water wave speed is primarily determined by gravity and either the wave's wavelength or period.
- Use $v = \sqrt{g\lambda / 2\pi}$ if you know the wavelength.
- Use $v = gT / 2\pi$ if you know the period.
- The general relationship where wavelength and frequency/period are inversely proportional (as mentioned in the reference) applies to the wave speed equation $v=f\lambda$, highlighting a fundamental wave property.
