Yes, there is a mathematical formula that describes the increase in amplitude when resonance occurs in a driven harmonic motion system.
Let's consider a simple harmonic oscillator, such as a mass-spring system. The equation of motion for this system is given by:
m * d^2x/dt^2 + k * x = F0 * cos(ωt)
where m is the mass of the object, x is the displacement from the equilibrium position, t is time, k is the spring constant, F0 is the amplitude of the driving force, ω is the angular frequency of the driving force, and cos(ωt) represents the driving force as a function of time.
In this equation, the term on the left represents the restoring force due to the spring, and the term on the right represents the driving force. When the driving force matches the natural frequency of the system, resonance occurs.
The natural frequency of the system, denoted as ω0, is related to the mass and spring constant as:
ω0 = √(k/m)
When resonance occurs, ω = ω0, and we can rewrite the equation of motion as:
m * d^2x/dt^2 + k * x = F0 * cos(ω0 * t)
To find the solution for the amplitude of the system's response, we assume a solution of the form:
x(t) = A * cos(ω0 * t + φ)
where A is the amplitude of the system's response, and φ is the phase angle.
Substituting this solution into the equation of motion:
-m * ω0^2 * A * cos(ω0 * t + φ) + k * A * cos(ω0 * t + φ) = F0 * cos(ω0 * t)
Simplifying:
(A * (k - m * ω0^2)) * cos(ω0 * t + φ) = F0 * cos(ω0 * t)
Since cosine functions are equal only if their arguments are equal, we have:
A * (k - m * ω0^2) = F0
Solving for the amplitude A:
A = F0 / (k - m * ω0^2)
From this formula, we can observe that when the driving frequency matches the natural frequency (resonance), the denominator (k - m * ω0^2) approaches zero. As a result, the amplitude A tends to infinity, indicating a large increase in the amplitude of the system's response.
This phenomenon demonstrates how resonance, occurring when the driving frequency matches the natural frequency of a system, leads to a significant increase in the amplitude of the system's motion.