Prove that the spherical wave given by: ?(?,?) = (A ei k (r-vt))/r is indeed a solution of the three-dimensional wave equation Hint: Write the wave equation in spherical coordinates (therefore, you should use the Laplacian written in terms of r, ?, ?, and insert the solution given above into the wave equation.

Prove that the spherical wave given by: ?(?,?) = (A ei k (r-vt))/r is indeed a solution of the three-dimensional wave equation Hint: Write the wave equation in spherical coordinates (therefore, you should use the Laplacian written in terms of r, ?, ?, and insert the solution given above into the wave equation.

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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 standing waves 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 spatial range. (a) First, to capture the full generality of the wave equation solutions, find the complex right-moving simple harmonic wave. (b) We assert k = . Then, a general wave packet u(x, t) can be constructed as the sum across all possible k values of the right-moving simple harmonic wave, each with amplitude coefficients A(k). What is the form of u(x, t)? Note that we do not have the boundary conditions that would produce standing waves. (c) A(k) is found such that u(x, 0) does not stretch indefinitely in space. What is then an expression for A(k) in terms of u(x, 0)?…
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