3.3 Simple waves
3.3.3 Surface gravity waves
In addition to the internal gravity waves described above, there is a distinct, and more familiar, type of gravity waves, known, e.g. from the Bay of Aarhus. These are waves at a discontinuity in density.
We consider a liquid at constant density ρ0, with a free surface. Thus the pressure on the surface is assumed to be constant. The layer is infinitely deep. I assume that the liquid is incompressible, so that ρ0 is constant and the density perturbation ρ0 = 0. From the equation of continuity we therefore get
divv= 0. (3.77)
Gravityg is assumed to be uniform, and directed vertically downwards. Since the density perturbation is zero, so is the perturbation to the gravitational potential.
In the interior of the liquid the equations of motion reduce to ρ0∂v
∂t =−∇p0 . (3.78)
The divergence of this equation gives
∇2p0 = 0. (3.79)
We introduce a horizontal coordinate x, and a vertical coordinatez increasing downward, with z= 0 at the free surface. We now seek a solution in the form of a wave propagating along the surface, in the x-direction. Here p0 has the form
p0(x, z, t) =f(z) cos(khx−ωt), (3.80) wheref is a function yet to be determined. By substituting equation (3.80) into equation (3.79) we obtain
d2f
dz2 =k2hf , or
f(z) =aexp(−khz) +bexp(khz). (3.81)
56 CHAPTER 3. A LITTLE HYDRODYNAMICS As the layer is assumed to be infinitely deep,b must be zero.
We must now consider the boundary condition at the free surface. Here the pressure is constant, and therefore the Lagrangian pressure perturbation vanishes (the pressure is constant on the perturbed surface),i.e.,
0 =δp=p0+δδδr· ∇p0 =p0+ξzρ0g0 atz= 0, (3.82) where ξz is thez-component of the displacement. This is obtained from the vertical com-ponent of equation (3.78), for the solution in equation (3.81) withb= 0, as
ξz =− kh
ρ0ω2 p0 . (3.83)
Thus equation (3.82) reduces to
0 =
1− g0kh ω2
p0 , and hence the dispersion relation for the surface waves is
ω2 =g0kh . (3.84)
The frequencies of the surface gravity waves depend only on their wavelength and on gravity, but not on the internal structure of the layer, in particular the density. In this they resemble a pendulum, whose frequency is also independent of its constitution. Indeed, the frequency of a wave with wave numberkh, and wavelengthλ, is the same as the frequency of a mathematical pendulum with length
L= 1 kh = λ
2π . (3.85)
Chapter 4
Equations of linear stellar oscillations
In the present chapter the equations governing small oscillations around a spherical equi-librium state are derived. The general equations were presented in Section 3.2. However, here we make explicit use of the spherical symmetry. These equations describe the general, so-called nonradial oscillations, where spherical symmetry of the perturbations is not as-sumed. The more familiar case ofradial, or spherically symmetric, oscillations, is contained as a special case.
4.1 Mathematical preliminaries
It is convenient to write the solution to the perturbation equations on complex form, with the physically realistic solution being obtained as the real part of the complex solution. To see that this is possible, notice that the general equations can be written as
A∂y
∂t =B(y), (4.1)
where the vectory consists of the perturbation variables (δδδr, p0, ρ0,· · ·),Ais a matrix with real coefficients, and B is a linear matrix operator involving spatial gradients, etc., with real coefficients. Neither A norB depend on time. If y is a complex solution to equation (4.1) then the complex conjugate y∗ is also a solution, since
A∂y∗
∂t =
A∂y
∂t ∗
= [B(y)]∗ =B(y∗), (4.2) and hence, as the system is linear and homogeneous, the real part <(y) = 1/2(y+y∗) is a solution.
Because of the independence of time of the coefficients in equation (4.1), solutions can be found of the form
y(r, t) = ˆy(r) exp(−i ω t). (4.3) This is a solution if theamplitude function yˆ satisfies the eigenvalue equation
−iωA·yˆ =B(ˆy). (4.4)
57
58 CHAPTER 4. EQUATIONS OF LINEAR STELLAR OSCILLATIONS Equations of this form were also considered in Section 3.3 for simple waves. Note that in equations (4.3) and (4.4) the frequencyω must in general be assumed to be complex.
Figure 4.1: The spherical polar coordinate system.
Equation (4.3) is an example of the separability of the solution to a system of linear partial differential equations, when the equations do not depend of one of the coordinates.
As the equilibrium state is spherically symmetric, we may expect a similar separability in spatial coordinates. Specifically I use spherical polar coordinates (r, θ, φ) (cf. Figure 4.1), where r is the distance to the centre, θ is colatitude (i.e., the angle from the polar axis), and φis longitude. Here the equilibrium is independent of θand φ, and the solution must be separable. However, the form of the separated solution depends on the physical nature of the problem, and so must be discussed in the context of the reduction of the equations.
This is done in the next section.
Here I present some relations in spherical polar coordinates that will be needed in the following [see also Appendix 2 of Batchelor (1967)]. Let ar, aθ and aφ be unit vectors in ther,θ andφ directions, letV be a general scalar field, and let
F=Frar+Fθaθ+Fφaφ (4.5)
4.1. MATHEMATICAL PRELIMINARIES 59 be a vector field. Then the gradient ofV is
∇V = ∂V
and consequently the Laplacian ofV is
∇2V = div (∇V) (4.8) Finally, we need the directional derivatives, in the direction, say, of the vector
n=nrar+nθaθ+nφaφ. (4.9) The directional derivative n· ∇V of a scalar is obtained, as would be naively expected, as the scalar product of n with the gradient in equation (4.6). However, in the directional derivatives n· ∇F of a vector field, the change in the unit vectorsar,aθ and aφ must be taken into account. The result is
n· ∇F =
where the directional derivatives ofFr,Fθ and Fφ are the same as for a scalar field.
As the radial direction has a special status, it is convenient to introduce the horizontal (or, properly speaking, tangential) component of the vector F:
Fh=Fθaθ+Fφaφ, (4.11)
and similarly the horizontal components of the gradient, divergence and Laplacian as
∇hV = 1
60 CHAPTER 4. EQUATIONS OF LINEAR STELLAR OSCILLATIONS