Question

the angular frequency of the surface waves in a liquid is given in terms of wave number k by omega=ROOt of gk+tk^3÷rho ehere g is acceleraion due to gravity rho is the density of liquid and T is the surface tension (which gives an upward force on element of rhe surgave liquid).Find phase and group velocities for limitimg cases when the surface wa es have very large wave length and very small wavelengrhs​

Answer 1

Answer:

The type of wave motion which most people are familiar with are waves that occur on the free surface of water. For example, the ripples that occur when a small rock is dropped into the water or the waves that can be seen breaking on a beach (Pinery Provincial Park on Lake Huron, below).

Pinery Provincial Park on Lake Huron

In this type of wave motion the restoring force is gravity (sometimes surface tension needs to be considered as well, and in some situations surface tension can dominate gravity) and therefore these waves are known as surface gravity waves. We would like to create a mathematical formulation of this problem. To begin we will make some simplifying assumptions:

Inviscid: nu = 0

Constant density: rho = const => grad u = 0

Irrotational: vortcity = 0

The mean depth of the fluid is constant: H = const

The fluid is invariant with y (the in-page direction): derivative with respect to y -> 0

Surface wave diagram

 

The displacement of the free surface will be denoted by,

 

eq1

 

Thus, our surface is found at,

 

eq2

 

From the irrotational assumption we can deduce that,

 

eq3

 

Then using our incompressibility (constant density) assumption we get our first governing equation,

 

eq4

 

 

To get the remainder of the equations we now need to consider some boundary conditions. First we will look at the bottom boundary. The bottom is solid and hence there is no flow normal to this boundary. Mathematically this gives us w=0 at z =-H , assuming we have a flat bottom. This is the kinematic bottom condition.

We have considered the bottom boundary so next we will look at the surface. At the surface we want to ensure that particles that start off on the surface always remain on the surface. This condition is known as the kinematic surface condition. Following a particle which is on the surface the rate of change of the difference of the z position and the displacement of the surface is zero,

eq5

Mathematically, it gives the following condition, note that this condition is nonlinear however it does assume there is no wave breaking:

eq6

The final boundary condition is the balance of forces on at the free surface. Since the fluid is inviscid, the forces at the surface are the pressures above and below the surface (a surface force from the stress tensor) and the surface tension (a line force that occurs only at the surface, and is chemical in nature).

Surface tension diagram

After some work the balance of forces gives us the following equation for pressure at the surface:

eq7

The atmospheric pressure is often taken to be approximately zero (relative to the pressure in the water). Since, we have inviscid and irrotational flow we can use Bernoulli's Equation,

eq8

and without loss of generality we can take B(t) = 0 this is a tricky point that you should think about) which gives our last boundary condition, the dynamic surface condition.

eq9

To summarize what we have deduced so far:

eq10

The above problem is fully nonlinear and still remains unsolved. The next thing to consider are waves that have a relatively small amplitude, a << 1. For this case we will set,

eq11

and,

eq12

Since, a​2​ << a we can use this to linearize our problem, to first order this gives us the linearized wave equations:

eq13

Solving the linearized wave equation

To solve the linearized wave equation we start by looking for normal mode solutions of the from,

eq14

It is sufficient to only consider solutions of this form since any continuous function can be written as a sum of sines and cosines.

From this starting point and using the kinematic boundary condition at the bottom and the dynamic boundary condition at the surface one can deduce that we should look for solutions of the following form,

At an air water interface at temperatures of about 20 degrees Celsius we have the following values for our parameters, sigma = 0.0728 N/m and p = 1000 kg/m3​. Which gives k​2​ >> 1.35x105​ m-2​  for capillary waves or lambda << 1.7cm ~ 2cm. This tells us for waves with wave length much less then 2cm gravity is negligible and surface tension is the dominating force, giving us ca