Question

Use a distribution of wave numbers of constant amplitude in a range δk about k0: a(k) = a0 k0 − δk 2 ≤ k ≤ k0 δk 2 = 0 otherwise and obtain eq. 4.24 from eq. 4.23.

a) Integrate a(k) with respect to k
b) Apply boundary conditions to a(k)
c) Differentiate a(k) with respect to k
d) Substitute values into a(k)

Answer 1

To obtain equation 4.24 from equation 4.23, differentiate a(k) with respect to k. (option c)

Step-by-step explanation:

In the context of the given question, the transition from equation 4.23 to equation 4.24 involves a differentiation process. Equation 4.23 describes a distribution of wave numbers, a(k), with constant amplitude in a specified range. To move from this representation to equation 4.24, the objective is likely to express the variation in amplitude with respect to the wave number, capturing the behavior of the wave.

Now, let's delve into the explanation. By differentiating a(k) with respect to k, we are essentially examining how the amplitude of the wave varies as the wave number changes. The process involves applying the rules of calculus to the given expression for a(k) to find its derivative. This differentiation captures the rate at which the amplitude changes with respect to the wave number.

The resulting equation, likely represented as equation 4.24, embodies the derived relationship between amplitude and wave number, providing a more detailed understanding of the wave's characteristics. This mathematical transformation is a fundamental step in analyzing and interpreting the behavior of waves within the specified range.(option c)