20
Chapter 2
Non-dispersive Waves
We will begin this series of lectures on waves by spending some time discussing
non-dispersive waves. Non-dispersive waves have, by denition, a linear dispersion rela
124
6.3.3
An introduction to weakly nonlinear theory
Weakly nonlinear theory is the study of the dynamics of a system that is only
weakly nonlinear, that is, a system where the amplitude of the pertur
102
Chapter 6
Instabilities I: Convection
In the following chapters, we will study a few of the fundamental instabilities of
uid dynamics, and learn a number of tools to study their properties. The ke
154
7.6
Interfacial shear instabilities
This section is adapted from Chapter 3 of the textbook Introduction to Hydrodynamic stability by Drazin.
In the previous Sections we have studied instabilities
Chapter 7
Shear instabilities
In this nal Chapter, we continue our study of the stability of uid ows by
looking at another very common source of instability, shear. By denition,
shear occurs whenever
6.3. NONLINEAR STABILITY OF RAYLEIGH-BENARD CONVECTION115
6.3
Nonlinear stability of Rayleigh-Bnard cone
vection
In Chapter 1, we saw that linear stability only tells us whether a system is stable
or
Chapter 5
Introduction to wave mean ow interactions
In this nal Chapter related to the study of waves, we will study what happens
to waves that travel in a uid which is not at rest, that is, in a uid
2.2. SOUND WAVES IN AN INHOMOGENEOUS, TIME-DEPENDENT MEDIUM37
2.2
Sound waves in an inhomogeneous, timedependent medium
So far, we have dealt with cases where c was always constant. This, however, is
83
4.2.3
The wave packet approximation
We now derive, as we have done for pressure waves and internal gravity waves,
a wave packet approximation for surface waves. For simplicity, we will neglect
the
Chapter 3
Incompressibility and the
Boussinesq approximation
This Chapter is based on the paper by Spiegel & Veronis, 1960. See also the
textbook Fluid Mechanics by Kundu & Cohen, pages 124-128.
In al
74
4.2
Surface waves
Surface waves are probably the most common example of waves we are visually
exposed to in everyday life from small-scale waves in the bathtub to mediumscale waves approaching a th
58
Chapter 4
Dispersive waves
In this Chapter we will study a much broader class of waves called dispersive
waves, for which the phase speed and group speeds are dierent. Since nondispersive wave must
136
6.3.5
Truncated equations and the Lorentz system
As a nal foray into nonlinear theory, we now explore the idea of using truncated
systems as a tool to study the time-dependent dynamics of convecti
147
7.3.2
Examples of commonly-studied shear ows
Linear shear ows
As seen in the previous section, linear shear ows of the kind u = zex are not
expected to have any growing modes. Lets see this more d
AMS227 Final
December 7, 2013
The two questions are independent of one another. You will get a B for a substantially correct answer
to one of the two questions, and an A for substantially correct answ
Midterm
The two questions are independent of one another. You will get a B for a substantially correct answer
to one of the two questions, and an A for substantially correct answers to both questions.
Chapter 1
Introduction
In this series of lectures we will study waves and instabilities in uid ows, with
particular applications to geophysical and astrophysical systems, and paying
particular attenti
Homework 1
October 3, 2013
1
1.1
Steady-state solutions of the uid equations
Polytropes
As discussed in class, the equation of state usually involves 3 thermodynamic variables. However, there
are cond
Homework 2
October 7, 2013
1
Coursework
Question 1: Show that the Fourier solution and dAlemberts solution are indeed the same, as claimed
below equation (2.24) of the lecture notes. Hint: start with
Homework 2
October 23, 2013
1
Coursework
Consider the governing equation for internal gravity waves, in the case where N is a slowly varying function
of X and . Assume a wave paket solution of the for
Homework 4
November 18, 2013
1
Local analysis
Redo the local stability analysis we did in class, but this time add the dissipation terms we neglected: the
viscous dissipation term in the momentum equa