"However, spacetime is incredibly stiff, and I think all the known real-world sources produce pretty smooth waves."
In English, why does stiffness correlate to smooth waves? What does stiff spacetime mean? I'd have thought a square wave would be "stiff" as it's quite the opposite of smooth.
In order for a square wave to travel through a material, it would need to allow for infinitely high frequencies (see the animation on wikipedia[0] for a visual demonstation). The lower the maximum frequency possible, the more every wave will resemble a sine wave because it's the most basic shape: any non-smooth wave has higher-frequency components to give it its shape. Filter out those higher frequencies, and you're left with the basic smooth sine wave again.
A stiff material tends to dampen high frequencies, simply because it cannot deform fast enough to follow the wave's shape. In a way, the medium acts as a low-pass filter; compare, fow example, how fast you can clap your hands in air vs in water: the stiffness of water slows down your movements so you cannot reach high clapping frequencies.
> A stiff material tends to dampen high frequencies
All materials roll off their frequency response. But this is backwards - stiffer materials have higher resonant frequencies given equal density. It’s basically a word to describe a high spring constant.
I think what the GP was stating here were two independent reasons why we would find it hard to know what would happen if a gravitational square-ish wave interacted with spacetime: 1. spacetime is "stiff", and 2. there are no gravitational square-ish wave generators around to observe.
The two things are kind of related, though. One "natural" way to create a square wave in nature, is to "interrupt" a material transmitting a sinusoid wave, at the peak of its transmittance. And one way to do that, is to break through the modulus of elasticity of the material transmitting the wave, such that it switches from the elastic-deformation domain (transmitting the wave) to the plastic-deformation domain (ripping apart.)
Imagine a speaker cone tearing at the peak of a high-amplitude drum beat. The cone pushes "out" — and then doesn't push back "in", because instead the air behind it rips forward through it. The air created by the speaker cone wants to rush back "in", but now there's no longer a speaker cone acting as a waveguide for the inward flow, so the natural turbulence cancels out much of the "falling" energy of the wave, making it look much more like a square-wave drop.
I believe that the GP is saying that, because spacetime has such a high effective "modulus of elasticity", we haven't yet observed any practical way to perturb so as to create the conditions that would generate a gravitational square wave.
Imagine using a flick of your wrist to create a traveling transverse square wave in a string vs. in a long metal dowel. It's actually not so hard to create a smooth sinusoidal wave in a metal dowel. (It will be long wavelength, but it will be easily visible if you put your eye near one end and look down it.) But it would be impossible for you to make a square wave.
In English, why does stiffness correlate to smooth waves? What does stiff spacetime mean? I'd have thought a square wave would be "stiff" as it's quite the opposite of smooth.