1 BMET2960 Week 9 Tutorial: Fourier Integrals & Transforms In this week's tutorial we will use the Fourier Integralmethod to analytically solve for the diffusion of an aerosol released along an unbounded space – a problem we cannot solve using Fourier series. We will also introduce the Fourier Transformin Matlab to analyse synthetic and real signals in the frequency domain. Part 1 – Fourier Integral Solution to the heat/diffusion equation Dolce & Gabbana have recently set up a new shop mid-way along a long and busy alleyway in Paris. They have a marketing idea for a new women’s fragrance called “B-Me” but aren’t sure how effective the strategy will be. The idea is to release a large quantity of the new fragrance right outside the shop front and have it waft along the alleyway in both directions during the day (Figure 1). D&G want to know how far the fragrance odor will reach along the alleyway, and how many people are therefore likely to come under the influence of their marketing campaign. This will help them determine if it’s a cost effective marketing approach. And they need your help! Assuming no wind, the diffusion of the fragrance along the alleyway is governed by the diffusion equation: )1(xxtDYY=where Yis the mass fraction of the fragrance and Dis the diffusivity. Since the alleyway is long and narrow we will model it as a one dimensional problem. The initial condition describing the fragrance ‘bomb’ released directly outside the shop-front is: )2(elsewherezeroandm1m1for1)()0,(<<−==xxfxYThere are no boundary conditions.