where is the zonal velocity ( direction), the meriodional velocity ( direction), is time and is the coriolis frequency. 620x620px|Figure 2: A traveling Kelvin wave is shown for the Northern hemisphere without friction, the coast is always at the right of the direction of motion. The domain is closed on the right hand side, to mimic an enclosed shelf sea. The wave enters the domain on the lower left hand side and travels towards the right. On the right hand side the wave is reflected and travels back towards the left. On the closed side the reflection happens through the creation of Poincare waves which are not modeled here. The upper panel shows the sea surface height and the lower panel the velocity. The colors are indicated in the color bar, the arrows are scaled velocity vectors. Both of these equations take the form of the classical wave equation, where . Which is the same velocity as the tidal wave and thus of a shallow water wave. These preceding equations govern the dynamics of a one-dimensional non-dispersive wave, for which the following general solution exist: where length is the Rossby radius of deformation and is an arbitrary function describing the wave motion. In the most simple form is a cosine or sine function which describes a wave motion in the positive and negative direction. The Rossby radius of deformation is a typical length scale in the ocean and atmosphere that indicates when rotational effects become important. The Rossby radius of deformation is a measure for the trapping distance of a coastal Kelvin wave. The exponential term results in an amplitude that decays away from the coast.  Therefore this set of equations describes a wave that travels along the coast with a maximum amplitude at the coast which declines towards the ocean. These solutions also indicate that a Kelvin wave always travels with the coast on their right hand side in the Northern Hemisphere and with the coast at their left hand side in the Southern Hemisphere. In the limit of no rotation where , the exponential term increase without a bound and the wave will become a simple gravity wave orientated perpendicular to the coast. In the next section, it will be shown how these Kelvin waves behaves when traveling along a coast, in an enclosed shelf seas or in estuaries and basins. Tides in enclosed shelf seas Figure 3 : A traveling Kelvin wave is shown for the Northern hemisphere with  friction, the coast is always at the right of the direction of motion. The domain is closed on the right hand side, to mimic an enclosed shelf sea. The wave enters the domain on the lower left hand side and travels towards the right. On the right hand side the wave is reflected and travels back towards the left. On the closed side the reflection happens through the creation of Poincare waves which are not modeled here. The upper panel shows the sea surface height and the lower panel the velocity. The colors are indicated in the color bar, the arrows are scaled velocity vectors. | 620x620px The expression of tides as a bounded Kelvin wave is well observable in enclosed shelf seas around the world (e.g. the English channel, the North Sea or the Yellow sea). In figure 2, the shape of an enclosed shelf sea is represented as a simple rectangular domain in the Northern Hemisphere which is open on the left hand side and closed on the right hand side. The tidal wave, a Kelvin wave, enters the domain in the lower left corner and travels to the right with the coast on its right. The sea surface height (SSH  ), the tidal elevation, is maximum at the coast and decreases towards the center of the domain. The tidal currents ( lower panel of figure 2 ) are in the direction of wave propagation under the crest and in the opposite direction under the through. They are both maximum under the crest and the trough of the waves and decrease towards the center . This was expected as the equations for and are in phase as they both depend on the same arbitrary function describing the wave motion and exponential decay term. On the enclosed right hand side, the Kelvin wave is reflected and because it always travels with the coast on its right, it will now travel in the opposite direction. The energy of the incoming Kelvin wave is transferred through Poincare waves along the enclosed side of the domain to the outgoing Kelvin wave. The final pattern of  SSH and the tidal currents are made up of the sum of the two Kelvin waves. These two can amplify each other and this amplification is maximum when the length of the shelf sea is a quarter wavelength of the tidal wave. Next to that, the sum of the two Kelvin waves result in several static minima's in the center of the domain which hardly experience any tidal motion, these are called Amphidromic points.  In the real world, the reflected Kelvin wave has a lower amplitude due to energy loss as a result of friction and through the transfer via Poincare waves . Figure 3, shows the Kelvin wave patterns when friction is added. The reflected wave amplitude has decreased and as a result the Amphidromic points shifted towards the top side of the panel .
where is the zonal velocity ( direction), the meridional velocity ( direction), is time and is the Coriolis frequency. Both of these equations take the form of the classical wave equation, where . Which is the same velocity as the tidal wave and thus of a shallow water wave. These preceding equations govern the dynamics of a one-dimensional non-dispersive wave, for which the following general solution exist: where length is the Rossby radius of deformation and is an arbitrary function describing the wave motion. In the most simple form is a cosine or sine function which describes a wave motion in the positive and negative direction. The Rossby radius of deformation is a typical length scale in the ocean and atmosphere that indicates when rotational effects become important. The Rossby radius of deformation is a measure for the trapping distance of a coastal Kelvin wave. The exponential term results in an amplitude that decays away from the coast. The expression of tides as a bounded Kelvin wave is well observable in enclosed shelf seas around the world (e.g. the English channel, the North Sea or the Yellow sea). Animation 1 shows the behaviour of a simplified case of a Kelvin wave in an enclosed shelf sea for the case with (lower panel) and without friction (upper panel). The shape of an enclosed shelf sea is represented as a simple rectangular domain in the Northern Hemisphere which is open on the left hand side and closed on the right hand side. The tidal wave, a Kelvin wave, enters the domain in the lower left corner and travels to the right with the coast on its right. The sea surface height (SSH, left panels of animation 1), the tidal elevation, is maximum at the coast and decreases towards the centre of the domain. The tidal currents (right panels of animation 1) are in the direction of wave propagation under the crest and in the opposite direction under the through. They are both maximum under the crest and the trough of the waves and decrease towards the centre. This was expected as the equations for and are in phase as they both depend on the same arbitrary function describing the wave motion and exponential decay term. Therefore this set of equations describes a wave that travels along the coast with a maximum amplitude at the coast which declines towards the ocean. These solutions also indicate that a Kelvin wave always travels with the coast on their right hand side in the Northern Hemisphere and with the coast at their left hand side in the Southern Hemisphere. In the limit of no rotation where , the exponential term increase without a bound and the wave will become a simple gravity wave orientated perpendicular to the coast. In the next section, it will be shown how these Kelvin waves behaves when traveling along a coast, in an enclosed shelf seas or in estuaries and basins. Tides in enclosed shelf seas 462x462px|Animation 1 : A traveling Kelvin wave is shown for the Northern hemisphere with (lower panel) and without (upper panel) friction, the coast is always at the right of the direction of motion. The domain is closed on the right hand side, to mimic an enclosed shelf sea. The wave enters the domain on the lower left hand side and travels towards the right. On the right hand side the wave is reflected and travels back towards the left. On the closed side the reflection happens through the creation of Poincare waves which are not modelled here. The panels on the left shows the sea surface height and the panels on the right the velocity. The colours are indicated in the colour bar, the arrows are scaled velocity vectors462x462px | Figure 2: The time averaged sea surface height is shown in red shading and the dashed lines are cotidal lines at intervals of roughly an hour. The Amphidromic points are located at the intersect of all lines. The upper panel shows the case for without friction and the lower panel for the case with frictionThe expression of tides as a bounded Kelvin wave is well observable in enclosed shelf seas around the world (e.g. the English channel, the North Sea or the Yellow sea). Animation 1 shows the behaviour of a simplified case of a Kelvin wave in an enclosed shelf sea for the case with (lower panel) and without friction (upper panel). The shape of an enclosed shelf sea is represented as a simple rectangular domain in the Northern Hemisphere which is open on the left hand side and closed on the right hand side. The tidal wave, a Kelvin wave, enters the domain in the lower left corner and travels to the right with the coast on its right. The sea surface height (SSH , left panels of animation 1 ), the tidal elevation, is maximum at the coast and decreases towards the centre of the domain. The tidal currents ( right panels of animation 1 ) are in the direction of wave propagation under the crest and in the opposite direction under the through. They are both maximum under the crest and the trough of the waves and decrease towards the centre . This was expected as the equations for and are in phase as they both depend on the same arbitrary function describing the wave motion and exponential decay term. On the enclosed right hand side, the Kelvin wave is reflected and because it always travels with the coast on its right, it will now travel in the opposite direction. The energy of the incoming Kelvin wave is transferred through Poincare waves along the enclosed side of the domain to the outgoing Kelvin wave. The final pattern of the SSH and the tidal currents is made up of the sum of the two Kelvin waves. These two can amplify each other and this amplification is maximum when the length of the shelf sea is a quarter wavelength of the tidal wave. Next to that, the sum of the two Kelvin waves result in several static minima's in the centre of the domain which hardly experience any tidal motion, these are called Amphidromic points. In the upper panel of figure 2, the absolute time averaged SSH is shown in red shading and the dotted lines show the zero tidal elevation level at roughly hourly intervals, also known as cotidal lines. Where these lines intersect the tidal elevation is zero during a full tidal period and thus this is the location of the Amphidromic points. In the real world, the reflected Kelvin wave has a lower amplitude due to energy loss as a result of friction and through the transfer via Poincare waves (lower left panel of animation 1). The tidal currents are proportional to the wave amplitude and therefore also decrease on the side of the reflected wave (lower right panel of animation 1). Finally, the static minima's are no longer in the centre of the domain as wave amplitude is no longer symmetric. Therefore, the Amphidromic points shift towards the side of the reflected wave (lower panel figure 2) .
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620x620px|Figure 2: A traveling Kelvin wave is shown for the Northern hemisphere without friction, the coast is always at the right of the direction of motion. The domain is closed on the right hand side, to mimic an enclosed shelf sea. The wave enters the domain on the lower left hand side and travels towards the right. On the right hand side the wave is reflected and travels back towards the left. On the closed side the reflection happens through the creation of Poincare waves which are not modeled here. The upper panel shows the sea surface height and the lower panel the velocity. The colors are indicated in the color bar, the arrows are scaled velocity vectors.
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The expression of tides as a bounded Kelvin wave is well observable in enclosed shelf seas around the world (e.g. the English channel, the North Sea or the Yellow sea). Animation 1 shows the behaviour of a simplified case of a Kelvin wave in an enclosed shelf sea for the case with (lower panel) and without friction (upper panel). The shape of an enclosed shelf sea is represented as a simple rectangular domain in the Northern Hemisphere which is open on the left hand side and closed on the right hand side. The tidal wave, a Kelvin wave, enters the domain in the lower left corner and travels to the right with the coast on its right. The sea surface height (SSH, left panels of animation 1), the tidal elevation, is maximum at the coast and decreases towards the centre of the domain. The tidal currents (right panels of animation 1) are in the direction of wave propagation under the crest and in the opposite direction under the through. They are both maximum under the crest and the trough of the waves and decrease towards the centre. This was expected as the equations for and are in phase as they both depend on the same arbitrary function describing the wave motion and exponential decay term.
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Figure 2: The time averaged sea surface height is shown in red shading and the dashed lines are cotidal lines at intervals of roughly an hour. The Amphidromic points are located at the intersect of all lines. The upper panel shows the case for without friction and the lower panel for the case with frictionThe
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