This article explains how wave motion is modified in shallow water by deriving the speed of travel of a wave through the variables amplitude, horizontal distance, and wavelength, and analyzing the acceleration and velocity patterns of water particles.
We’ll derive the speed of travel of a wave through three variables: amplitude, horizontal distance, and wavelength. Before we get started, let’s define the terms. Amplitude, which is wave amplitude in English, is the distance that water particles on the surface rise and fall from their mean height. In other words, it’s the displacement from the mean height and is directly related to the energy of the wave. The larger the amplitude, the more energy the wave has, and the more powerful it is.
Next, horizontal distance refers to the horizontal distance that water particles on a surface move forward or backward from their average position. This concept plays an important role in understanding how particles move in the direction of a wave’s travel: the horizontal distance indicates the directionality of the energy that the wave transmits to the particles, which is closely related to the wave’s propagation speed.
Wavelength is the distance from neighboring goal to goal or floor to floor, which represents one period of the wave. Wavelength is an important factor in describing the periodic nature of waves, which affects their frequency and speed. Wavelength also describes the physical size of a wave and is an essential variable for understanding the behavior of a wave in a particular environment.
A simple harmonic motion is one in which the start and end values are the same. For example, a sin curve is a typical example of such a motion, and is often used in the mathematical modeling of waves. Waves are based on periodic motion, and it is this periodicity that gives them the property of transferring energy at a constant rate.
Next, let’s talk about the acceleration pattern of waves: The “downward direction” we think of is the direction of the combined force, which generally coincides with the direction of gravity. The important thing to note here is that gravity provides a constant acceleration that acts on all objects. Imagine we are on a rapidly spinning object, such as a roller coaster. If there is a fishbowl of water around us, which way is the surface of the water facing? What is the direction we ‘feel’ we are going down? The ‘horizon’ is the calm, wave-free surface of the ocean. The normal of the horizon corresponds to the direction of gravity. However, if waves are present, the water surface deviates from the horizon. The observer’s perception of “down” may be related to the direction of the wave’s travel.
The shape of the wavefront is determined by gravity and the acceleration of the water particles. Water particles move along a wavefront, which in turn indicates the direction of the acceleration they are experiencing. So by observing the direction of the wavefront, we can infer the direction of the acceleration the particles are experiencing. In this process, we can say that the direction of travel of the wave, i.e., the direction of progression, does not matter. This is because even though the wave is traveling in a certain direction, the particles experience separate accelerations based on their motion.
Next, we’ll talk about the velocity patterns of waves. When a wave is present, the slope of the ocean surface shows the acceleration of the water particles. This slope is an important factor in determining the speed at which water particles rise or fall along the wave front. From the acceleration pattern, we can get an idea of what the velocity pattern of the particles is like. All water particles share the same pattern of motion: circular motion. It’s just that the timing of that particular motion is different depending on their location. Here, we assume that the wave is traveling from left to right, so if we consider the direction of travel of the wave, the particle between the two points in time will have reached its maximum horizontal velocity in the forward direction, and we know that this is when the floor of the wave passes this particle. This same discussion applies to the goal of the wave. As the wave passes, the particle will have a maximum horizontal velocity in the backward direction.
This velocity pattern is obtained by simply integrating the acceleration pattern, where the acceleration only dictates the increase or decrease in velocity. We have simply assumed the integration constant to be zero, but it is worth further thought whether it is safe to assume the integration constant to be zero, or whether the above discussion would still hold if we introduced an arbitrary integration constant.
Finally, let’s talk about the behavior of waves in shallow water. Suppose a wave is traveling through very shallow water. If the wave is traveling in the right direction, then at every point between the crest and the goal it follows, the water particles on the surface will move downward. The path of the water is strongly influenced by the shallowness of the water, meaning that the water particles have a greater horizontal displacement the shallower the water, which makes the path of the particles a horizontally elongated ellipse. The ratio of the long radius to the short radius of this ellipse is determined by the relationship between the depth of the water and the wavelength.
So the longer the wavelength, the shallower the water, the more elongated the ellipse, which means that the horizontal velocity of the particle is greater than its vertical velocity. This property of waves is an important clue to understanding wave behavior in shallow water. At shallower depths, the energy in a wave is primarily transferred horizontally, which goes a long way toward explaining how waves deform near the shore and how they transfer energy.
