(a) Show that in a gravity wave in water of arbitrary depth (deep, shallow, or in between), each fluid element undergoes forward-rolling elliptical motion as shown in Fig. 16.1. (Assume that the amplitude of the water’s displacement is small compared to a wavelength.)

(b) Calculate the longitudinal diameter of the motion’s ellipse, and the ratio of vertical to longitudinal diameters, as functions of depth.

(c) Show that for a deep-water wave, kh_o ≫ 1, the ellipses are all circles with diameters that die out exponentially with depth.

(d) We normally think of a circular motion of fluid as entailing vorticity, but a gravity wave in water has vanishing vorticity. How can this vanishing vorticity be compatible with the circular motion of fluid elements?

(e) Show that for a shallow-water wave, kh_o ≪ 1, the motion is (nearly) horizontal and is independent of height z.

(f) Compute the fluid’s pressure perturbation δP (x, z, t) inside the fluid for arbitrary depth. Show that, for a shallow-water wave, the pressure is determined by the need to balance the weight of the overlying fluid, but for greater depth, vertical fluid accelerations alter this condition of weight balance.

Figure 16.1.

Transcribed Image Text:

h₂ -2π/k- ZA x T///////////// O O - § (x, t)