This question draws on an old idea but it is different. The original question is, "If everyone in the world jumps at the same time, will the Earth move?" My question is,"If everyone in one hemisphere jumps repeatedly in synch, will the Earth vibrate and, if so can it create a targeted earthquake on the opposite side of the globe?"
The Emperor (may he live forever) lives in a beautiful palace surrounded by parks on the planet he rules. Surrounding the palace are the dwellings of the nobles, outside that are clerks, craftspeople, shopkeepers and everybody who has business living in a city. Next come wealthy farmers. Outside of that are the subsistence farmers and peasants.
Think North Korea for the sort of disparity in wealth.
The difference is that the empire covers the entire globe and at the antipode (yes it is a word) of the location of the Emperor (may he live forever) are the poorest, most exploited masses.
These people are poor but they are not stupid. They hatch a plan to destroy the Emperor's palace without ever going near it.
By leaping up and down in synchrony they intend to create a focused earthquake that occurs at the exact location of the Emperor's palace, and thus destroy it. That is to say that the epicentre of the earthquake is immediately under the palace. The idea is that successive seismic waves will move around and/or through the planet and meet at a single point on the far side.
I have found the following information but I have no idea how or whether it can be applied.
The vibration of plates is a special case of the more general problem of mechanical vibrations. The equations governing the motion of plates are simpler than those for general three-dimensional objects because one of the dimensions of a plate is much smaller than the other two. https://en.wikipedia.org/wiki/Vibration_of_plates
Vibrations of a solid sphere https://www.ucl.ac.uk/~ucapahh/research/atoms/laf3/node7.html
The Physics of Earthquakes https://physicstoday.scitation.org/doi/full/10.1063/1.1387590
It seems very complex so that is why I would be happy for any simplifications to be made. In other words I would be willing to change the makeup of the planet to make the maths easier.
The following seems related but it's beyond my knowledge of maths or physics.
mechanical waves through solids are damped more as the wave frequency increases. https://physics.stackexchange.com/questions/412852/does-a-hollow-sphere-vibrate-longer-than-a-solid-sphere
Given the assumptions listed below: Is it possible for billions of people who are optimally placed and perfectly synchronised in their jumping, to create a focused earthquake at the opposite side of a planet? I am hoping that repeated jumping will set up a resonance and thus amplify the waves massively.
Assumptions (some hand-waving)
There are up to 5 billion human people available ( 5 x 10^9 )
The average mass of a person is 90 kg.
They are arranged however you wish so as to make the plan work. All feet must start at the same level - either ground level or they could all jump off a box up up to 1 m high.
Synchronised jumping can go on for up to 20 minutes.
There is a way of synchronising the jumping and everyone knows exactly what to do and when. (vigorous hand-waving!)
The planet is a sphere (or other plausible shape that makes the calculation easier).
The distribution of the jumpers can take any form and be anywhere in the opposite hemisphere to the palace.
The vibrations from the jumping, end up focused narrowly over a circular area of 10 km diameter or so, precisely where the palace is.
You may make any reasonable adjustments or idealisations to the size, and physical makeup of the planet that may be helpful.
The epicenter, epicentre or epicentrum in seismology is the point on the Earth's surface directly above a hypocenter or focus, the point where an earthquake or an underground explosion originates. https://en.wikipedia.org/wiki/Epicenter
I am hoping that the vibrations travelling through and/or around the world will coincide at a single point.