The Growing Vibration: When Waves Catch Fire
What happens when you push a system at exactly the right moment? In this section, we explore Resonance—the geometric phenomenon where energy builds upon itself until the wave grows beyond its original boundaries.
In The Geometry of Decay, we saw how waves gradually lose energy through friction and resistance.
Resonance represents the opposite phenomenon. When an external force matches the natural frequency of a system, energy is added at exactly the right rhythm. Instead of shrinking, the wave’s amplitude begins to grow.
This process, known as driven oscillation, creates an expanding envelope of motion and explains why structures like bridges can begin to sway—or why a glass can shatter—when exposed to the perfect resonant frequency.
Lab Results: The Resonant Explosion
If you look at the Resonance chart, you’ll notice the wave begins almost invisible. However, as time progresses, the peaks reach further and further from the center line. Unlike our previous waves, this one is no longer “contained” by a static boundary.
In this experiment, we’ve applied a growth factor. The wave is essentially “stealing” energy from its environment to increase its physical presence.
The Key Takeaway:
- The Input: Synchronized Timing (The Push).
- The Output: Infinite Growth (The Explosion).
Note: This relationship is a fundamental concept in Trigonometric Projections and is a classic example of Driven Harmonic Motion.
The Mathematical Model: The Shrinking Envelope
In these experiments, we explore the two extremes of harmonic motion: Damped Harmonic Motion (Energy Loss) and Driven Resonance (Energy Gain). Both are governed by an external “envelope” function A(t) that scales the sine wave over time.
The Geometry of Decay
We define the shrinking boundary using the pattern A(t) = e-at. As time increases, the amplitude asymptotically approaches zero.
Observation: By applying the shift from Phase Shifts and Timing (where φ = π/2), the wave begins at its maximum possible displacement before being “suffocated” by the exponential curve.
The Growing Vibration (Energy Gain)
In contrast to decay, we define the growth boundary using the pattern A(t) = G ⋅ t, where the phase alignment ensures the growth starts from the equilibrium point.
Observation: Here, the gain factor G acts as a linear multiplier that expands a “V-shaped” envelope. At the start of the timeline, the system is at rest; by the end, the amplitude has grown to fill the entire geometry of the viewport.
Name: Source Code: Manim Implementation *
This is incredible — it really makes you want to dive deeper and explore all the details behind it!