Level:Part II (yr 2)
Course Convenor:Dr J McDonald
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Prior to PHYS213, the student must have successfully completed:
The student must take the following modules:
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CMod descriptionback to top
Fourier series representation of periodic
functions. Real and complex Fourier series. Expansion of saw-tooth and step
functions into Fourier series. Use of Kronekker delta-symbol in calculations.
Application of Fourier series to linear ODEs and charge oscillations in
circuits, evaluation of dissipative losses in an oscillator driven by a
periodic AC source. Phenomenon of the resonance.
Fourier integral and the Fourier transform.
Delta-function and the use of its properties. Wave equation on an infinite
line with initial conditions, dAlemberts solution. Solution using the
Fourier transformation method and comparison to dAlemberts solution..
Relation between integral and differential formulation of physics laws
(examples from electro- and magneto-statics), revision of integral theorems,
operations of grad and div. Multi-dimensional Fourier transform, transformation
of the operator s and Laplace operator.
Method of Fourier transform in electrostatic problems. Potential of a point
charge. Potential created by a periodic charge localised on a 2D plane.
[Screening in metals, analysis using Fourier transform. Plasma waves.]
equation. Operations of grad and div in the formulation of current
conservation and the derivation of diffusion equation (and/or heat transfer
equation). Boundary conditions for DE, analysis of its stationary solutions.
Separation of variables in the 3D stationary diffusion problem. Solution of a
non-stationary diffusion problem using the Fourier transform: diffusive
spreading of a δ-function source. Green functions introduction and a brief
Curriculum Design: Outline Syllabusback to top
Fourier series representation of periodic functions: Real and complex Fourier series. Examples of Fourier expansion of periodic functions. Application of Fourier series to physical systems with forced oscillations. Charge oscillations in AC circuits. Evaluation of dissipative losses in an oscillator driven by a periodic AC source. Parseval's theorem.
Fourier integral and the Fourier transform: Expression of a function as a Fourier integral. Definition of the Fourier transform and its inverse. The Dirac delta-function and its integral representation. General solution of the wave equation using Fourier transforms.
1-D wave equation with initial conditions - d'Alembert's solution. An example of the Fourier transform in Quantum Mechanics - the Uncertainty Principle. The convolution and its use in physics. The Convolution Theorem. Multi-dimensional Fourier transform.
Diffusion equation: Derivation of the diffusion equation. Time-dependent diffusion problem. Diffusion of a point source. Laplace's equation and the Uniqueness Theorem. Separation of variables in steady-state diffusion problems with boundary conditions. The heat equation. Seperation of variables in time-dependent diffusion problems.
Electrostatics: Revision of vector analysis and integral theorems, operations of grad and div. Relation between integral and differential formulation of physical laws; examples from electrodynamics. Method of Fourier transform in electrostatic problems.
Boundary condition problems and Laplace's equation in electrostatics. Potential created by a periodic charge localised on a 2D plane.
Educational Aims: Subject Specific: Knowledge, Understanding and Skillsback to top
The LTA strategy is fourfold. Each week the core physics material is developed in the lectures. Students are expected to reinforce and extend the lecture material by private study of the course textbook and other sources. Student's understanding is consolidated and assessed via the fortnightly work sheet, which is completed by students independently, then marked and discussed by the lecturer at the seminar.
Learning Outcomes: Subject Specific: Knowledge, Understanding and Skillsback to top
On completion of the module, students should be able to:
- express a periodic function as a Fourier series,
- find the Fourier transform of a function,
- solve linear ODE's and PDE's using Fourier techniques,
- solve the diffusion equation with initial conditions and/or spatial boundary conditions,
- solve problems in electrostatics using Fourier methods.
Curriculum Design: Select Bibliographyback to top
(R) Mathematical Techniques, Jordan and Smith (Oxford UP)
(B) Advanced Engineering Mathematics, Kreyszig (Wiley)