Constant of integration in Equilibrium Theory of Tides

Question

I am preparing an elementary derivation of Newton/Laplace's equilibrium theory of tides.

The derivation of the tractive force is understood, as is the derivation of the differential equation for change in ocean height (assuming a spherical water covered Earth).

However, I am stuck with this following reasoning on page 14 of the link below.

https://www.uaf.edu/files/sfos/Kowalik/tide_book.pdf

The authors claim that the differential equation is as follows:

$$ \frac{dh}{d \theta} = - \frac{3r}{2} \frac{m}{M} \left( \frac{r}{R} \right) ^3 \sin(2 \theta) $$

[Their I.40]

where:

• $h$ is the change in height of the water
• $r$ and $M$ are the radius and mass of the Earth
• $R$ is the distance from the Earth to the Moon
• $m$ is the mass of the Moon.
• $\theta$ is the angle from the the north pole (ignoring axial tilt, assuming Moon is equatorial)

The authors solve it to give:

$$ h = r \frac{m}{M} \left( \frac{r}{R} \right) ^3 \left(\frac{3}{2} \cos^2\theta + c \right) $$

Here's the trouble: to determine $c$, the authors quote (Proudman, 1953) which I cannot access. They claim that the condition for conservation of mass of water is:

$$\int_{0}^{\pi} \left(\frac{3}{2} \cos^2\theta + c \right)\sin\theta d\theta$$

I don't understand why there is a sine in this integral. What is its physical significance?

Answer 1

The author is integrating over two angular coordinates:

• $\phi$, the azimuthal ("longitude") angle, running from $0$ to $2\pi$, and
• $\theta$, the altitude ("latitude") angle, running from $0$ to $\pi$.

When integrating using non-Euclidean coordinates, one must remember to multiply the integrand by the Jacobian of the transformation from Euclidean coordinates to the ones used in the integrand. In this case, the integrand is in spherical coordinates with fixed radius $R$, so the Jacobian is $R\sin{\theta}$. This is a well-known result whose derivation can easily be found elsewhere. The $R$, and the factor of $2\pi$ obtained from integrating in the $\phi$-direction, are not present presumably because, being constants, they were transferred out of the integral and brought to the other side of the equation representing conservation of mass (only half of which is present at the time of posting).

The physical meaning of this is geometric: imagine integrating across the surface area of a sphere using strips parallel to the latitude lines. Note that the area of these strips decreases as you get closer to the poles. The factor of $\sin{\theta}$ describes the ratio of the area of a strip at latitude $\theta$ to the area of the strip at $\theta=\pi/2$.