Mathematics of Waves and Fields

PHYS20171 Mathematics of Waves and Fields 10 Semester 1 Undergraduate Level 2 Roger Jones Unit coordinator Department of Physics & Astronomy 1 FHEQ level (Framework for Higher Education Qualifications) ' Middle part of Bachelors ' European Credit Transfer & Accumulation System Rating : 5.0 Mathematics of Waves and Fields Mathematics of Waves and Fields To introduce and develop the mathematical skills and knowledge needed to understand classical fields and quantum mechanics. On completion successful students will be able to:1. Solve partial differential equations using the method of separation of variables. 2. Define the term “orthogonality” as applied to functions, and recognise sets of orthogonal functions which are important in physics (e.g. trigonometric functions and complex exponentials on appropriate intervals, Legendre polynomials, and spherical harmonics). 3. Represent a given function as a linear superposition of orthogonal basis functions (e.g. a Fourier series) using orthogonality to determine the coefficients, 4. State how a Fourier transform differs from a Fourier series, and calculate Fourier transforms of simple functions. 5. Solve eigenvalue problems (differential equations subject to boundary conditions) either in terms of standard functions or as power series. 6. Use partial differential equations to model wave, heat flow and related phenomena. 7. Make basic use of Dirac notation. 1.  Wave problems in one dimensionSeparation of variables Normal modes of a string:  eigenfunctions and eigenvalues General motion of a string 2.  Fourier seriesOrthogonality and completeness of sines and cosines Complex exponential form of Fourier series 3.  Other PDE’s Laplace’s equation The heat-flow equation 4.  Integral transforms Fourier transform Convolutions Wave packets and dispersion 5.  Special functions  Orthogonal sets of eigenfunctions Series solution of differential equations Legendre polynomials and related functions Bessel functions 6.  Problems in two and three dimensions  Normal modes of a square membrane; degeneracy Wave guide Normal modes of circular and spherical systems Heat flow in circular and spherical systems Laplace’s equation:  examples in cartesian and polar coordinates 7.  Dirac notation  Vector spaces Ket notation Inner products and Bras Hilbert spaces   <h4 class="ui header" style="-webkit-text-stroke-width:0px;border-style:none;box-sizing:inherit;color:rgba(0, 0, 0, 0.87);font-family:"Segoe UI", Lato, "Helvetica Neue", Arial, Helvetica, sans-serif;font-size:1.1rem !important;font-style:normal;font-variant-caps:normal;font-variant-ligatures:normal;letter-spacing:normal;line-height:1.28571em;margin:-0.142857em 0px 1rem;orphans:2;padding:0px;scrollbar-color:auto;text-align:start;text-decoration-color:initial;text-decoration-style:initial;text-decoration-thickness:initial;text-indent:0px;text-transform:none;white-space:normal;widows:2;word-spacing:0px;">CUIP Teaching Methods</h4><div class="ui grid" style="-webkit-box-align:stretch;-webkit-box-direction:normal;-webkit-box-orient:horizontal;-webkit-text-stroke-width:0px;align-items:stretch;box-sizing:inherit;color:rgba(0, 0, 0, 0.87);display:flex;flex-flow:wrap;font-family:"Segoe UI", Lato, "Helvetica Neue", Arial, Helvetica, sans-serif;font-size:14px;font-style:normal;font-variant-caps:normal;font-variant-ligatures:normal;font-weight:400;letter-spacing:normal;margin:-1rem;orphans:2;padding:0px;scrollbar-color:auto;text-align:start;text-decoration-color:initial;text-decoration-style:initial;text-decoration-thickness:initial;text-indent:0px;text-transform:none;white-space:normal;widows:2;word-spacing:0px;"><div class="five wide column" style="box-sizing:inherit;display:inline-block;padding:1rem;position:relative;vertical-align:top;width:333.854px;"><figure class="table" style="width:305.854px;"><table class="ui table" style="background-color:rgb(255, 255, 255);border-collapse:separate;border-radius:0.285714rem;border-spacing:0px;border:1px solid rgba(34, 36, 38, 0.15);box-shadow:none;box-sizing:inherit;color:rgba(0, 0, 0, 0.87);font-size:1em;margin:0px;scrollbar-color:auto;text-align:left;vertical-align:middle;" id="teaching_table"><thead class="mobile hidden" style="box-shadow:none;box-sizing:inherit;text-align:inherit;vertical-align:inherit;"><tr style="box-sizing:inherit;"><th style="background-color:rgb(249, 250, 251);border-bottom:1px solid rgba(34, 36, 38, 0.1);border-left-style:none;border-radius:0.285714rem 0px 0px;border-right-color:!important;box-sizing:inherit;color:rgba(0, 0, 0, 0.87);cursor:auto;padding:0.928571em 0.785714em;text-align:inherit;text-transform:none;transition:background 0.1s, color 0.1s;vertical-align:inherit;">Activity</th><th class="right aligned" style="background-color:rgb(249, 250, 251);border-bottom:1px solid rgba(34, 36, 38, 0.1);border-left-style:none;border-radius:0px 0.285714rem 0px 0px;box-sizing:inherit;color:rgba(0, 0, 0, 0.87);cursor:auto;padding:0.928571em 0.785714em;text-align:right;text-transform:none;transition:background 0.1s, color 0.1s;vertical-align:inherit;">Hours</th></tr></thead><tbody style="box-sizing:inherit;counter-reset:numdivcounter 0;text-align:inherit;vertical-align:inherit;"><tr style="box-sizing:inherit;"><td style="border-right-color:!important;border-top-style:none;box-sizing:inherit;padding:0.785714em;text-align:inherit;transition:background 0.1s, color 0.1s;">Tutorials</td><td class="right aligned" style="border-top-style:none;box-sizing:inherit;padding:0.785714em;text-align:right;transition:background 0.1s, color 0.1s;">4</td></tr><tr style="box-sizing:inherit;"><td style="border-right-color:!important;border-top:1px solid rgba(34, 36, 38, 0.1);box-sizing:inherit;padding:0.785714em;text-align:inherit;transition:background 0.1s, color 0.1s;">Lectures</td><td class="right aligned" style="border-top:1px solid rgba(34, 36, 38, 0.1);box-sizing:inherit;padding:0.785714em;text-align:right;transition:background 0.1s, color 0.1s;">22</td></tr></tbody></table></figure></div></div> 0 Other 10% 1 Written exam 90% * Other 10% Tutorial Work/attendance  Students will receive feedback on their work and performance in this module as a component of their weekly tutorial meeting with their academic tutor.  PHYS10071 Mathematics 1 Pre-Requisite Compulsory PHYS10302 Vibrations & Waves Pre-Requisite Compulsory PHYS10372 Mathematics 2 Pre-Requisite Compulsory Pre-requisites:<ul><li><span style="-webkit-text-stroke-width:0px;display:inline !important;float:none;font-family:"Aptos Narrow";font-size:16px;font-style:normal;font-variant-caps:normal;font-variant-ligatures:normal;font-weight:400;letter-spacing:normal;orphans:2;text-align:start;text-decoration-color:initial;text-decoration-style:initial;text-decoration-thickness:initial;text-indent:0px;text-transform:none;white-space:pre-wrap;widows:2;word-spacing:0px;">Vibrations & Waves PHYS10302</li><li><span style="-webkit-text-stroke-width:0px;display:inline !important;float:none;font-family:"Aptos Narrow";font-size:16px;font-style:normal;font-variant-caps:normal;font-variant-ligatures:normal;font-weight:400;letter-spacing:normal;orphans:2;text-align:start;text-decoration-color:initial;text-decoration-style:initial;text-decoration-thickness:initial;text-indent:0px;text-transform:none;white-space:pre-wrap;widows:2;word-spacing:0px;">Mathematics 2 PHYS10372 OR Introduction to Ordinary Differential Equations MATH11412</li></ul> N Boas, M.L. Mathematical Methods for Physical Sciences, 3rd edn. (Wiley, 2006) Martin, B.R. & Shaw, G, Mathematics for Physicists. (Wiley 2015) Riley, K.F. Hobson, M.P. & Bence, S.J. Mathematical Methods for Physics and Engineering, 3rd edn (Cambridge 2006) [Chapters 12 to 19] Stephenson, G. Partial differential equations for scientists and engineers (Imperial College 1996)   Assessment written exam 1.5 Lectures 22 Tutorials 4 0 72.5