(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.
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Transcribed Image Text: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)? Hint: Consider what would be the complex exponential version of the
orthogonality 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?
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