Problem:As often noted in lecture, a simple harmonic without particular boundary conditions is highly unphysical- it extends into infinity in space. However, in Lecture 4, we have come to realise how sinusoidal standingwaves serve as components in the Fourier series for more complex solutions to the wave equation.In this Discussion, we will examine the wave packet, a construction of a physical wave with finite spatialrange.(a) First, to capture the full generality of the wave equation solutions, find the complex right-movingsimple harmonic wave.(b) We assert k = . Then, a general wave packet u(x, t) can be constructed as the sum across all possiblek values of the right-moving simple harmonic wave, each with amplitude coefficients A(k). What isthe form of u(x, t)? Note that we do not have the boundary conditions that would produce standingwaves.(c) A(k) is found such that u(x, 0) does not stretch indefinitely in space. What is then an expressionfor A(k) in terms of u(x, 0)? Hint: Consider what would be the complex exponential version of theorthogonality property in finding Fourier coefficients.(d) Let u(x, 0) =e-²+ikoa. Find the wave packet for this known initial state.(e) What is the wave packet from (d) that we actually can see on the medium?

Since you have posted a question with multiple sub-parts, we will solve the first three sub-parts…

Answered: (a) First, to capture the full…

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