# Lecture3 - Lecture 3 Notes 95.658 Spring 2012 Electromagnetic Theory II Dr Christopher S Baird UMass Lowell 1 Superposition of Waves Up till now we

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Lecture 3 Notes, 95.658, Spring 2012, Electromagnetic Theory II Dr. Christopher S. Baird, UMass Lowell 1. Superposition of Waves - Up till now we have only looked at monochromatic waves and how they behave in materials and hinted at how a frequency-dependent permittivity would lead to dispersion of a wave packet made up of many monochromatic waves. Let us look in more detail at what this really means. - There is no such thing as a truly monochromatic wave. - A wave cannot always have been generated in the infinite past and continue to be generated in the infinite future. Every electromagnetic wave therefore has a definite spatial beginning point and spatial endpoint. A truly monochromatic wave cannot exist because it would require an infinite extent and infinite existence. - In this sense, every wave is actually a wave packet or pulse with finite extent. - Many frequencies (or wavenumbers) are required in order to build up a wave packet, so every real wave actually has a range of frequencies present. - Wave packets that have a very large width require a narrower range of superimposed frequencies to be mathematically constructed, and may be approximated as monochromatic waves. - For a nearly monochromatic wave, the width of the wave's range of frequencies is known as the “spectral linewidth”. - The linewidth of a wave may also experience broadening due to changes in the source or due to propagation phenomena. - Because a dispersive material has a permittivity that depends on frequency, any wave can be thought of as a superposition of monochromatic plane waves which each interact with the material independently of the others, dependent on its frequency. - For simplicity, we will investigate the behavior of one vector component of the electric field and label it u ( x , t ) so that for instance E x ( x , t ) = u ( x , t ). Because the components of a vector are independent, the end solution is just the sum of each vector component after being solved separately. - As found previously, the principal solution of the wave equation is (showing now only one component): u x ,t = Ae i k x − t where k  - We will treat the wave number k as the independent variable so that the frequency ω is a function of the wave number k according to: ω = ω ( k ). This is actually the more common form of the dispersion relation. - Assume for now that ω and k are real. - The general solution is just the superposition of all possible solutions: u ( x ,t )= 1 2 π −∞ A ( k ) e i ( k x −ω( k ) t ) dk General solution to the Wave Equation - The coefficient function A ( k ) in the general solution is what uniquely determines the solution to a particular problem. - This is just a Fourier transform and we can apply Fourier Theory to find the coefficients.

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## This note was uploaded on 02/13/2012 for the course PHYSICS 95.658 taught by Professor Staff during the Spring '11 term at UMass Lowell.

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Lecture3 - Lecture 3 Notes 95.658 Spring 2012 Electromagnetic Theory II Dr Christopher S Baird UMass Lowell 1 Superposition of Waves Up till now we

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