Dispersion (water Waves)

In fluid dynamics, dispersion of water waves generally refers to frequency dispersion, which means that waves of different wavelengths travel at different phase speeds. Water waves, in this context, are waves propagating on the water surface, with gravity and surface tension as the restoring forces. As a result, water with a free surface is generally considered to be a dispersive medium. For a certain water depth, surface gravity waves - i.e. waves occurring at the air-water interface and gravity as the only force restoring it to flatness - propagate faster with increasing wavelength. On the other hand, for a given (fixed) wavelength, gravity waves in deeper water have a larger phase speed than in shallower water. In contrast with the behavior of gravity waves, capillary waves (i.e. only forced by surface tension) propagate faster for shorter wavelengths. Besides frequency dispersion, water waves also exhibit amplitude dispersion. This is a nonlinear effect, by which waves of larger amplitude have a different phase speed from small-amplitude waves.

Dispersion (water Waves) 1

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Example: Deep water waves

The Stokes drift was formulated for water waves by George Gabriel Stokes in 1847. For simplicity, the case of infinite-deep water is considered, with linear wave propagation of a sinusoidal wave on the free surface of a fluid layer: = a cos ( k x t ) , displaystyle eta ,=,a,cos ,left(kx-omega t

ight), where is the elevation of the free surface in the z-direction (meters), a is the wave amplitude (meters), k is the wave number: k = 2 / (radians per meter), is the angular frequency: = 2 / T (radians per second), x is the horizontal coordinate and the wave propagation direction (meters), z is the vertical coordinate, with the positive z direction pointing out of the fluid layer (meters), is the wave length (meters), and T is the wave period (seconds).As derived below, the horizontal component S(z) of the Stokes drift velocity for deep-water waves is approximately: u S k a 2 e 2 k z = 4 2 a 2 T e 4 z / . displaystyle overline u_S,approx ,omega ,k,a^2,texte^2kz,=,frac 4pi ^2,a^2lambda ,T,texte^4pi ,z/lambda . As can be seen, the Stokes drift velocity S is a nonlinear quantity in terms of the wave amplitude a. Further, the Stokes drift velocity decays exponentially with depth: at a depth of a quarter wavelength, z = - , it is about 4% of its value at the mean free surface, z = 0. DerivationIt is assumed that the waves are of infinitesimal amplitude and the free surface oscillates around the mean level z = 0. The waves propagate under the action of gravity, with a constant acceleration vector by gravity (pointing downward in the negative z-direction). Further the fluid is assumed to be inviscid and incompressible, with a constant mass density. The fluid flow is irrotational. At infinite depth, the fluid is taken to be at rest. Now the flow may be represented by a velocity potential , satisfying the Laplace equation and = k a e k z sin ( k x t ) . displaystyle varphi ,=,frac omega k,a;texte^kz,sin ,left(kx-omega t

ight). In order to have non-trivial solutions for this eigenvalue problem, the wave length and wave period may not be chosen arbitrarily, but must satisfy the deep-water dispersion relation: 2 = g k . displaystyle omega ^2,=,g,k. with g the acceleration by gravity in (m / s2). Within the framework of linear theory, the horizontal and vertical components, x and z respectively, of the Lagrangian position are: x = x x d t = x a e k z sin ( k x t ) , z = z z d t = z a e k z cos ( k x t ) . displaystyle beginalignedxi _x,&=,x,,int ,frac partial varphi partial x;textdt,=,x,-,a,texte^kz,sin ,left(kx-omega t

ight),xi _z,&=,z,,int ,frac partial varphi partial z;textdt,=,z,,a,texte^kz,cos ,left(kx-omega t

ight).endaligned The horizontal component S of the Stokes drift velocity is estimated by using a Taylor expansion around x of the Eulerian horizontal-velocity component ux = x / t at the position : u S = u x ( , t ) u x ( x , t ) = [ u x ( x , t ) ( x x ) u x ( x , t ) x ( z z ) u x ( x , t ) z ] u x ( x , t ) ( x x ) 2 x x t ( z z ) 2 x z t = [ a e k z sin ( k x t ) ] [ k a e k z sin ( k x t ) ] [ a e k z cos ( k x t ) ] [ k a e k z cos ( k x t ) ] = k a 2 e 2 k z [ sin 2 ( k x t ) cos 2 ( k x t ) ] = k a 2 e 2 k z . displaystyle beginalignedoverline u_S,&=,overline u_x(boldsymbol xi ,t),-,overline u_x(boldsymbol x,t),&=,overline left[u_x(boldsymbol x,t),,left(xi _x-x

ight),frac partial u_x(boldsymbol x,t)partial x,,left(xi _z-z

ight),frac partial u_x(boldsymbol x,t)partial z,,cdots

ight]-,overline u_x(boldsymbol x,t)&approx ,overline left(xi _x-x

ight),frac partial ^2xi _xpartial x,partial t,,overline left(xi _z-z

ight),frac partial ^2xi _xpartial z,partial t&=,overline bigg [-a,texte^kz,sin ,left(kx-omega t

ight)bigg ],bigg [-omega ,k,a,texte^kz,sin ,left(kx-omega t

ight)bigg ],&,overline bigg [a,texte^kz,cos ,left(kx-omega t

ight)bigg ],bigg [omega ,k,a,texte^kz,cos ,left(kx-omega t

ight)bigg ],&=,overline omega ,k,a^2,texte^2kz,bigg [sin ^2,left(kx-omega t

ight)cos ^2,left(kx-omega t

ight)bigg ]&=,omega ,k,a^2,texte^2kz.endaligned

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