(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)? Hint: Consider what would be the complex exponential version of the orthogonality property in finding Fourier coefficients.

(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)? Hint: Consider what would be the complex exponential version of the orthogonality property in finding Fourier coefficients.