 # 1. You may answer as many parts of question 1 as you wish. All wor...

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1. You may answer as many parts of question 1 as you wish. All work you do will be assessed and the marks totaled but note that the maximum total credit for this question will be 20 marks. (a) Consider a wave propagating along a system of two connected strings with impedances Z, and Z2 respectively. i. Can the amplitude of the transmitted wave be greater than the amplitude of the incident wave? If yes, what is the necessary condition? [3 marks] ii. If you answered yes in (i) and stated the condition, is there violation of the conservation of energy then? Explain briefly. [3 marks] (b) State the bandwidth theorem both in (frequency-time) domain and in (wavelength- position) domain and explain its significance. [5 marks] (c) i. Define the three different possible velocities involved in a transverse wave and their direction. [2 marks] ii. What are the conditions for anomalous and normal dispersions of waves? When do we have the case of no dispersion? [2 marks] (d) Verify that the solution x=(A+Bt) satisfies the equation: m x+ ri+sx = 0 when r²/4m=s [6 marks] (e) i. Write down the expressions for the frequency and wave number of the 4th harmonic of a string L, fixed at the both ends. [3 marks] ii. Sketch the behavior of the amplitude of the standing wave of this harmonic along the string. How many nodes do exist? [2 marks] (f) What is the difference between Fraunhofer and Fresnel diffraction? [4 marks] 2. (a) Describe how the method of Fourier Transformations may be used to search for periodic variations in data gathered over a long time. [4 marks] (b) How may a known periodic noise be removed from a set of data using Fourier methods? [4 marks] (c) Describe how deconvolution may be used to sharpen a data set from, for example, a spectrometer. [4 marks) (d) An atom radiates a train of sinusoidal waves for a time interval T defined by If the amplitude of the wave train is represented by A(t) = Ao e(2nfot with fo the frequency, use the Fourier transform to show that this finite wave train may be represented by the superposition of frequency components of amplitude FU) where: F(f) = Ao sin(f-10) w(f-fo) [8 marks]

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