A toy boat moves with uniform velocity u across a deep pond (Fig. 16.2). Consider the wave

Question:

A toy boat moves with uniform velocity u across a deep pond (Fig. 16.2). Consider the wave pattern (time-independent in the boat’s frame) produced on the water’s surface at distances large compared to the boat’s size. Both gravity waves and surface-tension (capillary) waves are excited. Show that capillary waves are found both ahead of and behind the boat, whereas gravity waves occur solely inside a trailing wedge. More specifically, do the following.

(a) In the rest frame of the water, the waves’ dispersion relation is Eq. (16.14). Change notation so that ω is the waves’ angular velocity as seen in the boat’s frame, and ω_o in the water’s frame, so the dispersion relation becomes ω_o² = gk + (γ /ρ)k³. Use the Doppler shift (i.e., the transformation between frames) to derive the boat frame dispersion relation ω(k).

(b) The boat radiates a spectrum of waves in all directions. However, only those with vanishing frequency in the boat’s frame, ω = 0, contribute to the time independent (stationary) pattern. As seen in the water’s frame and analyzed in the geometric-optics, these waves are generated by the boat (at points along its horizontal dash-dot trajectory in Fig. 16.2) and travel outward with the group velocity V_go. Regard Fig. 16.2 as a snapshot of the boat and water at a particular moment of time. Consider a wave that was generated at
an earlier time, when the boat was at location P, and that traveled outward from there with speed V_go at an angle ∅ to the boat’s direction of motion. (You may restrict yourself to 0 ≤ ∅ ≤ π/2.) Identify the point Q that this wave has reached, at the time of the snapshot, by the angle θ shown in the figure. Show that θ is given by

where k is determined by the dispersion relation ω₀(k) together with the vanishing ω condition:

Show that

Demonstrate that the capillary-wave pattern is present for all values of θ (including in front of the boat, π/2

(d) Next, specialize to gravity waves, and show that

Demonstrate that the gravity-wave pattern is confined to a trailing wedge with anglesθ gw = sin⁻¹(1/3) = 19.47^o (cf. Fig. 16.2). You might try to reproduce these results experimentally.

Equation 16.14.