[Ex] Consider a deep-water wave train with a Gaussian envelope that resides near \( x = 0 \) at \( t = 0 \) and travels in the positive-\(x\) direction. The surface shape at any time is a Fourier superposition of waves with all possible wave numbers: \[ \eta(x,t) = \int_{-\infty}^{+\infty} \tilde{\eta}(k) \exp\!\left[ i\left( kx - (g|k|)^{1/2} t \right) \right] dk \tag{†} \] where \( \tilde{\eta}(k) \) is the amplitude of the wave component with wave number \(k\), and the dispersion relation is \( \omega = (gk)^{1/2} \). For the following items assume the surface shape at \( t = 0 \) is: \[ \eta(x,0) = \frac{a}{\sqrt{2\pi}\,\alpha} \exp\!\left\{ -\frac{x^2}{2\alpha^2} + i k_d x \right\} \] Here, \( k_d > 0 \) is the dominant wave number, and \( \alpha \) sets the initial horizontal extent of the wave train, with larger \( \alpha \) producing a longer wave train.

(1) Plot \( \Re[\eta(x,0)] \) for \( |x| \le 40 \,\text{m} \) when \( \alpha = 10 \,\text{m} \) and \( k_d = 2\pi/\lambda_d = 2\pi/10 \,\text{m}^{-1} \).

(2) Use the inverse Fourier transform at \( t = 0 \), \[ \tilde{\eta}(k) = \frac{1}{2\pi} \int_{-\infty}^{+\infty} \eta(x,0)\, e^{-ikx}\, dx, \] to find the wave amplitude distribution: \[ \tilde{\eta}(k) = \frac{1}{2\pi} \exp\!\left\{ -\frac{1}{2}(k - k_d)^2 \alpha^2 \right\}, \] and plot this function for \( 0 < k < 2k_d \) using the numerical values from question (1). Does the dominant contribution to the wave activity come from wave numbers near \( k_d \) for the question (1) values?

(3) For large \(x\) and \(t\), the integrand of (†) happens to be constant. Thus, for any \(x\) and \(t\), the primary contribution to \( \eta \) will come from the region where the phase in (†) does not depend on \(k\). Thus, set \( d\Phi/dk = 0 \), and solve for \( k_s \) (the wave number where the phase is independent of \(k\)) in terms of \(x\), \(t\), and \(g\).

(4) Using the result of question (2), set \( k_s = k_d \) to find the \(x\)-location where the dominant portion of the wave activity occurs. At this location, the ratio \(x/t\) is the propagation speed of the dominant portion of the wave activity. Is this propagation speed the phase speed, the group speed, or another speed altogether?