Ever wondered how waves behave on muddy beaches?
Ocean waves are one of the most fascinating things in nature. Most of them are generated by the wind that blows over the surface of the ocean. Once they are born, the waves can travel miles away, crossing entire oceans, until they approach the shore and ... break? That is probably the first thing that most of the beach-users think, right? Well, waves do break as they enter shallow water but that is not always the case.
Let's first try to understand why and how waves break.
The life cycle of the wind generated waves involves a dramatic transformation from the chaotic area of generation in a storm offshore, the propagation of the smooth swell until the breaking at the beach. As the swell approaches a shore, it is transformed, changing its shape, speed and wavelength, height. The period of propagation is the only wave parameter assumed as constant.
In a water depth smaller than half of its wavelength, the wave celerity is a direct function of the water depth, decreasing when the depth decrease. The velocity of each small portion of a wave eta propagates with the long-wave velocity [g(eta+d)]¹/², where d is the water depth and g is the acceleration due to gravity. Thus the higher parts of the wave has higher velocity.
The small amplitude wave theory is the only way to solve the problem analytically by taking in account the first nonlinear term resulting in a near-linear approximation and resulting in the Boussinesq equations.
Another way to see the problem is by the wave energy conservation. The theory developed by Airy (1845) shows that the energy transport rate depends on the square of the wave height and on its speed. As shown above, when the waves travels into shallow waters, its velocity decreases. So, if the energy transport rate remains the same and the waves slow down, the wave height must increase.
Okay, but when the waves finally breaks? It is known that when the wave steepness [usually estimated by the ratio of the wave height over its wavelength (H/L)] reaches the value of 1/7, the waves becomes unstable, which mean that a wave with one meter of height and 7 meters of wavelength is unstable and as the waves keep propagating towards shallower waters , the wave height increase and the waves eventually breaks.
Sandy white beaches are very attractive for people of all taste.