One stage in becoming an adult is the realization that you can skim (or even skip) sections of a book or paper and still look yourself in the mirror the next morning. Thus, if you don't like equations try skipping along to section below.

In the equation to the right, velocity of a wave is represented by c (not v) just because most people do that. Since the expression to the right of the equal sign is the value for c squared, when you want c (when wouldn't you) take the square root.

"tanh" is the hyperbolic tangent. Not the most familiar function, but common enough to be in the set of functions supplied in MicroSoft's Excel spread sheet application.

The international system of units is meter-second-kilogram (MSK)

c: velocity (m/s)
g: acceleration of gravity = 9.8 m/s²
L: wavelength (m)
2pi: 2 times pi = 6.283
h: depth of water (m)

Waves have been of interest to physicists and applied mathematicians for centuries, so there is a rich literature. A classic treatment is Sir Horace Lamb's Hydrodynamics, 1st published in 1879. The 6th edition (1932) is available in a Dover paperback. A more modern and approachable exposition is Blair Kinsman's Wind Waves (1984), also available as a Dover paperback.

If the depth of the water is greater than 1/4 of the wave length you can forget the "tanh" expression (it is then almost equal to 1). The speed then is just proportional to the square root of the wave length.

A simple formula in this case is: c = 1.340 times the square root of wavelength
where c is in knots and wavelength is in feet.

The above formulas describe a single wave. In a simple situation wind waves moving on a large open body of water may just be a series of very similar waves, with parallel crests, moving with equal speed. These formulas give the velocity of each of these waves; they all have the same velocity.

However, sometimes a discrete group of waves is generated, e.g. when a rock is thrown into water or a boat passes leaving a bow wave. The profiles on the right could be a simple bow wave, a series of smooth waves, equally spaced, with the highest in the middle. If we follow the progress of wave P, we see that it becomes smaller as it moves ahead of the group and then eventually disappears However, the wave just behind P at first becomes larger until it is as large as P was, and thus becomes the middle wave of the group. Then it becomes progressively smaller also as it moves out from the group and disappears The speed of each single wave is called the phase speed (because the relative position of a sine wave is called its phase), and it is clearly different from the speed of the group: the phase speed is twice the group speed.

In many cases the wave generated by a passing boat will not be a simple wave or group of waves with one wavelength. An extreme and hypothetical example is the wave in the top panel on the right. It is a square wave. In the next two panels the wave moves out away from the boat.

The first thing to realize is that the square wave can be represented as a sum of a series of simple, smooth waves (called sine waves) with short to long wavelengths. You may wonder why this is an important issue, why doesn't the square wave just move away from the boat. Remember that the speed of a wave increases with wavelength (the change of speed with wavelength is called dispersion). Dispersion causes long wavelength waves in the square wave to move faster than the short wavelength waves. As time progresses the square wave is thus decomposed into a spreading series of waves of different wavelengths, as seen in the next two panels.

The bow wave from a boat that makes a simple wave will travel for a long distance without decreasing in amplitude, while a complex bow wave will quickly spread out so the amplitude of any one component at any position is much smaller. The character of the bow wave depends on the shape of the hull, speed, etc.

Many kinds of waves show little or no dispersion. Sound and light waves moving through air exhibit only slight dispersion. Light waves moving through glass show some dispersion, which is why a glass prism can separate out the colors in white light and generate a spectrum (or rainbow) of the colors.