In the following paper, the author investigates mathematically the wave resistance of a sphere moving parallel to the free surface of a shallow sea. Correcting the errors committed by T. H. Havelock and others, he has obtained a new formula, and has carried out numerical calculations, the results of which are shown in several diagrams.
The new formula is
R=π
A2c2 /ρ∫=
π/2Φ0K3secΦ (1+tanh
Kh)
2/
gsec
2Φ (
c2-
ghsec
2Φ) +
K2c4h (
e-f1K+
e-
fK)
2dΦ.
where
R=the wave resistance
A= ρ
gα
3 α=the radius of the sphere
ρ=the density of the fluid
g=the acceleration due to gravity
c=the velocity of the sphere
h=the depth of the fluid
f1 = the immersion of the sphere
f2=the immersion of the image of the sphere or =2
h-
f1 K and Φ
0, the lower limit of integration, are to satisfy the following relations respectively.
Kc2 =
g sec
2Φtanh
Kh (
c2<
gh sec
2 Φ)
Φ
0=0 when
c2<
ghΦ
0=cos
-1 (
gh/
c2)
1/2 when
c2>
ghFinally, the author makes some discussions on the results obtained, especially on the diagrams, and compares them with certain experimental ones.
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