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1. use knowledge of partial differential equations (PDEs), modelling, the general structure of solutions, and analytic and numerical methods for solutions.
2. formulate physical problems as PDEs using conservation laws.
3. understand analogies between mathematical descriptions of different (wave) phenomena in physics and engineering.
4. classify PDEs, apply analytical methods, and physically interpret the solutions.
5. solve practical PDE problems with finite difference methods, implemented in code, and analyse the consistency, stability and convergence properties of such numerical methods.
6. interpret solutions in a physical context, such as identifying travelling waves, standing waves, and shock waves.

1. Agarwal, R. P. & O'Regan, D. (2009), Ordinary and Partial Differential Equations With Special Functions, Fourier Series, and Boundary Value Problems, Springer.
2. Billingham, J. and  King, A.C. (2000) Wave motion, CUP
3. Haberman, R. (1987), Elementary applied partial differential equations: with Fourier series and boundary value problems, 2ndedn, Prentice--Hall.
4. Kevrekidis, I. G. & Samaey, G. (2009), `Equation-free multiscale computation: Algorithms and applications', Annu.  Rev.Phys.  Chem.  60, 321--44. 
5. Kreyszig, E. (2011), Advanced engineering mathematics, 10th edn, Wiley.
6. Roberts, A. J. (1994), A one-dimensional introduction to continuum mechanics, World Sci.

In more detail, the course includes material from the following.

1. Conservation of mass:   1D, varying cross-section,  application to gas,  physical dimensions of terms, d'Alembert for age structured populations,  
2. Conservation of momentum:   1D, varying cross-section, forces and pressure, physical dimensions, ideal gas law closure, d'Alembert waves,    
3. Separation of variables: constant coeff linear PDE on finite domain, boundary conditions at open\slash closed end of pipe, connect to diffusion PDE, superposition of normal modes, general homogeneous solution, summarise separation of variables,  
4. Wave properties: frequency,  wave number,  dispersion relation, 
5. Sturm--Liouville BVPs: linear and varying coefficients on finite domain, regular Sturm--Liouville, self-adjoint operators, Eigenfunctions are orthogonal, Eigenvalues are real, Eigenfunctions are usually unique, normal modes of vibration, Eigenfunctions expand inhomogeneous solutions, Fredholm alternative,
6. Discretise 1D space:   Taylor's theorem, consistency, varying coefficients, finding equilibria, eigenvalues, symmetric discretisations reflect conservation,   
7. Model shallow water waves: water depth and mean velocity, hydrostatic closure, dispersion relation, varying depth, numerical seiches in 1D,
8. Computational integration: method of lines, explicit time steps, consistency and stability, Lax equivalence theorem, Crank--Nicholson scheme, sparse matrices, numerical dispersion,  
9. General wave systems: deep irrotational water waves, phase and group velocity, energy propagation other dispersion relations, phase and group velocity, wave guide refraction,   
10. PDEs in higher dimension: shallow water in lakes, separation of variables, Sturm--Liouville nature of Helmholtz-like PDEs, finite differences in \(n\)D, consistency, anisotropic dispersion relation,   
11. Classification of PDEs: revisit d'Alembert, characteristics for uni-directional wave, \(xy\)-coordinate transform seeks characteristics, hyperbolic, parabolic, elliptic PDEs,
12. Shocking characteristics: N-wave in 1D gas flow, bore in shallow water with Chezy law, roll wave instability.

 
1. All written assignments are to be submitted to the designated hand-in boxes within the School of Mathematical Sciences with a signed cover sheet attached, or submitted as pdf via MyUni.   
2. Late assignments will not be accepted without a medical certificate.  
3. Assignments normally have a two week turn-around time for feedback to students.