Lecture 18 3D Cartesian Systems

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Lecture 183D Cartesian Systems

• Only 4 lectures left 1 revision session
• Come to see me before the end of term
• Ive put more sample questions and answers in
  Phils Problems
• Past exam papers
• Complete solution from last lecture
• Have a look at homework 2 (due in on 12/12/08)

Remember Phils Problems and your notes
everything
http//www.hep.shef.ac.uk/Phil/PHY226.htm
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Introduction to PDEs
In many physical situations we encounter
quantities which depend on two or more variables,
for example the displacement of a string varies
with space and time y(x, t). Handing such
functions mathematically involves partial
differentiation and partial differential
equations (PDEs).

Elastic waves, sound waves, electromagnetic
waves, etc.
Wave equation
Quantum mechanics
Schrödingers equation
Heat flow, chemical diffusion, etc.
Diffusion equation
Electromagnetism, gravitation, hydrodynamics,
heat flow.
Laplaces equation

Poissons equation
As (4) in regions containing mass, charge,
sources of heat, etc.
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3D Coordinate Systems
We can describe all space using coordinates (x,
y, z), each one ranging from -8 to 8.
1. PDEs in 3D Cartesian Coordinates Consider
the wave equation. In one dimensional space we
had
In 3D equation becomes
,
which may be written in shorthand as
Let us look for a solution of the form , i.e. we
substitute and separate the variables, as done in
1D .
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3D Coordinate Systems
Substituting these into PDE then dividing both
sides by
()
gives

As before for 1D case we need solutions that can
be zero more than once to fulfil boundary
conditions, so we choose each term to equal a
negative constant to ensure we get LHO style
solutions.
Let
Comparing with () the defined constants, w, kx,
ky, kz are related by
Each of the ODEs above has the normal harmonic
solutions, which we can write in terms of sines
and cosines below.
.
Y(y)
Z(z)
T(t)
X(x)
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3D Coordinate Systems
Y(y)
Z(z)
T(t)
X(x)

Giving special solutions of the form
www.falstad.com/mathphysics.html
Or sometimes it is more convenient to use complex
exponentials,
Then depending on the boundary conditions we can
get special solutions such as
where
and
As we might have expected, these solutions are
plane waves with wavevector k (which is also the
direction of travel of the wave).
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3D Coordinate Systems

A general solution can then be written as a sum
over all special solutions. By applying boundary
conditions we can then determine which terms
contribute and the allowed values of kx, ky, kz
as before in 1D examples.
For example, suppose we have a box with
dimensions L1, L2, L3 in the x, y, z directions
respectively and know that Y must vanish at the
walls and that it is zero at t 0. Then the
special solutions after these boundary conditions
have been applied will be
where
So each special solution, or mode will be
characterized by three integers, n1, n2, n3. And
this mode will have angular frequency
A common question is to deduce how many different
modes (i.e. unique combinations of integers n1,
n2, n3) exist in a given frequency range w to w
dw ? e.g. Plancks Law for blackbody radiation.
www.falstad.com/mathphysics.html
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3D Coordinate Systems
2. Integrals in 3D Cartesian Coordinates
We have dV dx dy dz, and must perform a triple
integral over x, y and z. Normally we will only
work in Cartesians if the region over which we
are to integrate is cuboid.

Example 1 Find the 3D Fourier transform,

if
and
and
The integral is just the product of three 1D
integrals, and is thus easily evaluated
Just integrating over x gives
Mistake in notes
This is therefore a product of three sinc
functions, i.e.
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Corrections to notes- sorry!!
Homework 2

Lecture 17