Porpoising is the popular name for the high-speed surface piercing motion of dolphins and other species, in which long, ballistic jumps are alternated with sections of swimming close to the surface. The first analysis of this behavior (Au and Weihs, 1980) showed that above a certain “crossover” speed this behavior is energetically advantageous, as the reduction in drag due to movement in the air becomes greater than the added cost of leaping.

Since that publication several studies documented porpoising behavior at high speeds. The observations indicated that the behavior was more complex than previously assumed. The leaps were interspersed with relatively long swimming bouts, of about twice the leap length. In the present paper, the possibility of dolphins using a combination of leaping and burst and coast swimming is examined. A three-phase model is proposed, in which the dolphin leaps out of the water at a speed U_f, which is the final speed obtained at the end of the burst phase of burst and coast swimming. The leap is at constant speed and so the animal returns to the water at U_f, goes to a shallow depth and starts horizontal coasting while losing speed, till it reaches U_i. At that point it starts active swimming, accelerating to U_f. It then starts the next leap. Ranges of speeds for which this three-stage swimming is advantageous are calculated as a function of animal and physical parameters.

Notation

C—Constant defined in ^{equation (12)}

C_D—Coasting drag coefficient

D—Drag

g—Gravitational acceleration

H—Height of jump

J—Energy required for jump

k—Ratio of swim length to jump length

l—Distance

L—Total distance (^{eq. 28})

m—Added mass

M—Animal mass

M₁—Total mass

r—Coefficient defined in ^{eq. (22)}

R—Ratio of energies, for three-phase swimming

R₂—Ratio of energies, for burst and coast swimming

t—Time

T—Thrust

U—Speed

V—Body volume

W—Weight

α—Emergence (=return) angle

β—Swim / coast drag penalty ratio

γ—Surface effects drag ratio

ρ—Density of seawater and cetacean.

Subscripts

a—air

av—Average

b—Burst phase

c—Coast phase

e—Reference (maximal) thrust

f—Final, at end of burst

i—Initial, at start of burst

j—Jump phase

n—Nominal reference thrust

o—Optimal

s—Surface swimming

w—Water