I'm a bit confused wether or not there is a link between harmonic functions (solutions of the Laplace pde) and harmonic oscillating systems? What is the meaning of "harmonic" in these cases? Thanks!
Ok.. So are all the solutions of the Laplace equation (in complex analysis) periodic? By definition, they are harmonic. So is a harmonic function and a periodic function the same thing?mathman said:Harmonic in both physics and math usually refer to things which are periodic. In math it is usually used in descriptions involving sines and cosines. In physics it is usually used in discussing pendulums and such things.
A harmonic function is a function that satisfies the Laplace's equation. In simpler terms, it is a function whose value at any point is equal to the average of its neighboring points.
A harmonic oscillator is a type of mechanical system that follows the laws of simple harmonic motion. It consists of a mass attached to a spring and is characterized by its oscillatory motion around a fixed equilibrium point.
The motion of a harmonic oscillator can be described by a harmonic function. This is because the displacement of the mass from its equilibrium point at any given time can be represented by a sine or cosine function, which are examples of harmonic functions.
Some common examples of harmonic oscillators include a pendulum, a mass-spring system, and a tuning fork. They can also be found in many electronic devices, such as clocks, watches, and radios.
Harmonic functions and harmonic oscillators have many applications in science and engineering. They are used in fields such as physics, mathematics, and electrical engineering to model and study various systems and phenomena, such as vibrations, waves, and resonance.