The top surface of the water is often called the free surface since it is free to change its shape as the wave passes. It is on and around this surface where water waves live. Deep under the surface, there is no restoring force trying to realign water parcels should they be disturbed, and without a restoring force, there can be no waves. On the surface, gravity pulls water parcels that are above the average water height back down. Parts of the surface that are below average get pushed up by gravity induced buoyant forces. It is this restoring force along with the momentum of the water in a wave that makes for wave motion. We can view the surface as an elastic film that has a strong gravitational force to restore it to a perfectly flat condition. At the same time, it is a quite massive film, that once waves get started, has a lot of momentum. We color the "film" dark blue in the above animation (mouse over it to see the animation). Note that the water is not flowing, but instead the water moves as indicated by the fishing float, back and forth and up and down in a cyclic manner. Only the waves (the pattern) move left to right.
The animation perhaps implies that there is a sharp cutoff to the active region of the wave. This is not the case. A tiny fraction of wave motion exists quite far below the surface. However, because the wave motion falls off exponentially with the depth (for deep bodies of water), most of the wave activity occurs within one third of a wavelength of the surface. At that depth, the wave motion is only 15% of what it is at the surface, at one half a wavelength, the motion is 4% its surface strength. It is proportional to exp(−κ d) where d is the depth below the surface and κ = 2π/λ and λ is the wavelength.
In the previous posting we derived partial solutions for traveling water waves. These are summarized in Fig. 1 below. These solutions have to do with the water in and below the surface, how it must behave in order to remain uncompressed and without rotation, given that there are periodic disturbances on its surface. In other words, these solutions are just the reaction of an incompressible and irrotational fluid to periodic disturbances on its surface. The water surface, on the other hand, causes or powers these motions. The surface dynamics makes a disturbance in one water parcel carry over to its neighbors and propagate as wave motion. These equations do not reflect these surface dynamics (i.e. F=ma). In fact, the equations in Fig. 1 do not include the requirement that Newton's Second Law F=ma by obeyed anywhere in the fluid. We will work to correct this shortcoming in the material following Figure 1.
|Fig. 1. Equations for a traveling wave from part 1|
These equations all concern periodic disturbances. A person may wonder if non-periodic disturbances can be modeled. The answer is yes. At this moment, we are not particularly interested in this type of solution; however, one standard mathematical approach to doing this is to build on the periodic or sinusoidal solutions that we are developing here and consider other types of water flow patterns as sums of sinusoidal solutions, using many Fourier concepts. We hope to delve into these at some point in the future.
On to the dynamics of the water surface. For this we will use Newton's second law of motion. Newton's second law of motion, F = ma , applies to all water parcels (or blobs) in the system. To use this equation with a fluid, we replace the mass m in the equation with the mass density ρ (mass per unit volume) of water and the force with the gravitational force per unit volume −ρg in the y direction. We also need to include the force created by the pressure. This force (also per unit volume) equals minus the gradient of the pressure (the force that will move a parcel comes from the unbalance, i.e. gradient, in the pressure, not from the pressure itself):
where is the unit vector in the y or upwards direction and the total derivative, Dv/Dt, is the time rate of change of the velocity of a particular water parcel and is given by:
The first term on the right side of (5) tells us how much the fluid velocity is changing in time at a fixed location. The last two terms tell us how much velocity change a fluid parcel that moves through that point is experiencing because the velocities vary as a function of position. Thus, the distinction between the total derivative Dv/Dt and the partial derivative ∂v/∂t can be considered a difference between reference frames of the observer, i.e. between Lagrangian and Eulerian reference frames. In the Lagrangian frame you are moving with the fluid and in the Eulerian frame you are stationary. Consider the situation where water flow is accelerating, say due to a steepening slope of a stream bed, so it is slow on the left side of the view, but then the water speeds up as it flows to the right side. Further suppose this flow pattern is constant in time as you might see in many streams. In this case ∂v/∂t would be zero everywhere, since everywhere the flow is constant in time at that particular spot. On the other hand Dv/Dt would be positive, since if we follow a water parcel flowing from left to right, we see increasing velocity as we move from left to right. Put another way, Dv/Dt would indicate the change in velocity that a float would experience as it were swept down the stream, whereas ∂v/∂t would monitor the change you would see on a velocity meter put at a fixed location in a stream.
Equation (4) is a simplified version of the famous Navier-Stokes Equation that governs fluid flow. The complete version also accounts for viscosity.
For small amplitude waves, we ignore the last two terms, arguing that each of them involves the products of two factors. Each factor is proportional to the amplitude A and therefore is smaller than the first term that is proportional only to a single factor of A, i.e. first order in A as opposed to second order in A.
|Fig. 3. Differential motion of the water at the surface.|
One problem with using equation (4) above is that we do not have an equation for the pressure P as a function of position. We can avoid this pressure question by applying this equation to the water at the free surface in the direction parallel to the surface of the water. At the surface, the pressure is constant and is equal to the atmospheric pressure. A constant pressure (or any scalar quantity that is constant) will have a zero gradient. Thus there is no force due to pressure in the direction parallel to the surface. This is only one of the important components of the restoring force of the surface, but it is the one that we can easily get a hold of and solve, so we choose to start here.
We therefore write the above equation for the component of force and motion of a water parcel at the free surface in a direction parallel to the free surface, as shown in Fig. 3. The component of gravity in the positive (or right pointing) direction of the water surface is −ρgsinα, where α is the angle the free surface makes with the horizontal, as shown in Fig. 3. Thus we write:
For small amplitude waves, α is small, allowing us to make the approximation sinα ≅ tanα = d(Δy)/dx, i.e. that sinα equals the derivative or slope of the surface (remembering that Δy is the amount the surface deviates from the equilibrium, from a flat surface of zero slope). Thus (6) becomes:
where we have also made the approximation that the velocity of water parallel to the surface is approximately equal to vx for small amplitude waves. This last assumption may be confusing: why worry about the correct component of the gravitational force (and use sinα) and make a crude approximation for the velocity? We treat the gravitational force more carefully, because there is no horizontal component of it, so we use the sinα to tease out the only restoring force there is. This restoring force is essential to wave dynamics. The change in velocity is more robust and will not change much by our approximation (being proportional to cosα) and so we can do our approximation.
We now cancel the ρ to get:
We next substitute our solutions (2a) and (3b) into (8). The left hand and right hand sides become, respectively:
Equating these two sides and canceling common factors, we have:
This can be solved for ω:
where we have substituted h in for y, since this equation is valid for the surface of the waves where y = h. We can use this to calculate the velocity (the phase velocity) of small amplitude waves:
The last two equations are plotted below. The phase velocity is plotted versus the wavelength at the end of a previous posting (remember that the wavelength λ and wavenumber κ are related by: κ = 2π/λ .)
|Fig. 4. A log-log plotting of (11), i.e. the angular frequency of the wave versus the wavenumber (ω versus κ) for various depths, h.||Fig. 5. A log-log plotting of (12), i.e. the phase velocity of water waves versus wavenumber (c versus κ) for various depths, h.|
By analyizing the forces on the surface we have derived a relationship (12) between wave number and frequency for water waves. Because the wave velocity varies with wavenumber, water waves are considered dispersive. The relationship (11) between ω and κ (angular frequency and wavenumber) is called the dispersion relation.
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