To get started, let's think about a simpler wave first. For a wave on string, the amplitude tells you the displacement from the rest position of the string, $\Delta y$. If a string is strung horizontally parallel to the $x$-axis at height $y_0$ and has a vertical sine-wave propagating along it, the vertical position of each part of the string would be:
$$ y(t,x) = y_0 + \Delta y(t,x),$$
where $\Delta y$ is the wave which acts as a perturbation to the resting string position.
$$\Delta y(t,x) = A_0 \sin(\omega t - kx)$$
A gravitational wave is a perturbation a background spacetime. Spacetime is described by a metric, $g_{\mu\nu}$, that effectively tells us how to measure distances (and time intervals). In the simplest formulation the background is flat, empty space described by the Minkowski metric $\eta_{\mu\nu}$. The total spacetime metric would be the sum of the background spacetime and the wave perturbation:
$$g_{\mu\nu}(t,\vec{x}) = \eta_{\mu\nu} + h_{\mu\nu}(t,\vec{x}),$$
where $h_{\mu\nu}$ is the gravitational wave. For a sinusoidal gravitational wave,
$$h_{\mu\nu}(t,\vec{x}) = A_{\mu\nu} \sin(\omega t - \vec{k}\cdot\vec{x})$$
The wave is a function of time, $t$, and the full 3D location, $\vec{x}$. The $\mu$ and $\nu$ are indices denoting the tensor components, e.g. $(t,x,y,z)$. Since it is 4D and there are two indices there are 16 components in the wave (in the usual choice of coordinate system GR predicts only four of them are non-zero, and in any coordinate system only two are independent). The amplitude of the wave tells us how to measure distances in the spacetime. The wave will stretch and squeeze space time, so as the wave passes the distance between freely floating particles will change. A larger amplitude wave will cause a bigger change in the distance.
Any shaped wave can satisfy the wave equation. To find out which systems make which waves, you need to connect the wave equation to a source term. For a wave a on a string, the source term might describe a driving force which moves one end of the string in a particular pattern. For gravitational waves the source term describes the motion and masses of particles that make up the system.
For a binary system:
- the orbital frequency is proportional to the gravitational wave frequency
- faster speed, i.e. higher orbital frequency, produces higher amplitude waves (for two systems with the same mass)
- higher mass produces higher amplitude (for two systems with the same frequency)
It can get more complicated than this if we include more details about the motion, like the spins of the individual objects or the ellipticity of the orbits.
