Gunwale bobbing

We investigate gunwale bobbing, a phenomenon in which a person jumping on the gunwales of a canoe achieves horizontal propulsion by forcing it with vertical oscillations. The canoe moves forward by surfing the resulting wave field. After an initial transient, the canoe achieves a cruising velocity which satisfies a balance between the thrust generated from pushing downwards into the surface gradients of the wave field and the resistance due to a combination of profile drag and wave drag. By superposing the linear wave theories of Havelock [Proc. R. Soc. A 95, 354 (1919)] for steady cruising and Helmholtz for an oscillating source, we demonstrate that such a balance can be sustained. We calculate the optimal parameter values to achieve maximum canoe velocity. We compare our theoretical result to accelerometer data taken from an enthusiastic gunwale bobber and to estimates from videos of other canoeing aficionados. Finally, we discuss the similarities and differences with other examples of macroscopic wave-driven bodies and comment on possible applications to competitive sports.

Figure 1

(a) Gunwale bobbing in action, illustrating the length $L$, width $W$, and draft $D$ of the boat (here a paddle board). (b) Illustration of the thrust and drag forces at play. Thrust originates from pushing into surface gradients producing pitching and heaving forces, $R_{\mathcal{H},\mathcal{P}},R_{\mathcal{P},\mathcal{H}}$, at frequency $\omega$, causing the boat to “surf” on its own wave at cruising speed $U$. Drag originates from wave energy radiation (wave drag, $R_{\mathcal{C},\mathcal{C}}$), skin friction, and vortex separation (profile drag, $R_d$). (c)–(e) Pressure source term (horizontal slice) in the case of heaving motion in Eq. (4), pitching motion in Eq. (5), and heaving-pitching motion in equal proportion ($\varphi =1/2$) and out of phase ($\theta =\pi /2$).

Figure 2

Wave fields at time $t=0$ in the case of cruising motion (a), heaving motion (b), pitching motion (c), and the combination of all three (d). Typical parameter values are chosen for the Froude numbers $\mathrm{Fr}=0.2$, ${\mathrm{Fr}}_{\omega }=0.25$, for which the waves due to cruising are very small, so the color scale in (a) is $\times {10}^{-5}$. (e) Photo of a typical wave field due to gunwale bobbing (see also video in the Supplemental Material [13]).

Figure 3

Horizontal forces due to cruising (a) and oscillating (b). The forces due to cruising $R_{\mathcal{C},\mathcal{C}}+R_d$ are always negative (drag), whereas the summed forces due to oscillating $R_{n,m}$ [see (10)] are either positive (thrust) or negative (drag) depending on the parameters ${\mathrm{Fr}}_{\omega },\theta ,\varphi ,\alpha ,\beta$. Mach limits $\mathrm{Ma}=1$ (13) are indicated with dashed lines.

Figure 4

(a) Solutions to the thrust-drag equation (11), incorporating wave drag (7), profile drag (9), and thrust due to oscillating (8), for different values of the oscillating Froude number ${\mathrm{Fr}}_{\omega }$. Our experimental data point (calculated using an accelerometer) is plotted as a black circle. Other data points taken from video data of gunwale bobbing on canoes and paddle boards of different shapes and sizes are shown in gray (see Appendix pp4 for more details). (b) A sample of the vertical acceleration divided by $g$, measured using an accelerometer while gunwale bobbing.

Figure 5

Canoe shape approximated by two tetrahedrons with total length, width, and draft, $L$, $W$, and $D$, respectively.

Figure 6

Photographs of J.A.N. gunwale bobbing on his canoe of dimensions $L,W,D=4.7,0.94,0.15\mathrm{m}$, taken at intervals of 0.1 s.

Figure 7

Experimental data for the vertical acceleration (normalized by $g$) measured using an accelerometer over five separate trials.