The Elastic Wave: Squeezing and Stretching Time
What happens when the frequency itself starts to vibrate? In this section, we explore Frequency Modulation (FM)—the geometric foundation of how data and music are carried through the air.
Until now, our waves have had a constant “heartbeat.” But in nature and technology, frequencies often shift dynamically. Frequency Modulation occurs when a secondary “modulator” wave controls the speed of our primary wave. This creates a fascinating visual where the wave appears to bunch up like an accordion in some areas and stretch thin in others.
Lab Results: The Accordion Effect
If you look at the Elastic Wave chart, you’ll notice that unlike our previous experiments, the distance between peaks is no longer uniform. In the “bunched” sections, the frequency is at its highest; in the “stretched” sections, the frequency drops.
In this experiment, we aren’t just changing the wave’s height; we are warping its Temporal Density.
The Key Takeaway:
- The Input: A Modulator Wave (The Instruction).
- The Output: Variable Pitch/Frequency (The Result).
Note: This relationship is a fundamental concept in Trigonometric Projections and is the core technology behind FM Radio and Digital Synthesis.
The Mathematical Model: The Shrinking Envelope
To visualize the Moving Target, we move away from static values. We define the frequency swing using the pattern f(t) = fc + Δf ⋅ sin(2πfmt), where the timing is governed by the modulator’s phase.
This results in the displacement formula:
In this dynamic model, the frequency is no longer a static number—it is a value that changes at every single point along the x-axis:
- • The Carrier (B): Represents the base frequency fc. Without modulation, the wave would be a perfect, unchanging sine.
- • The Frequency Swing (Δf): Managed by the modulation index M. This determines how far the wave stretches and squeezes from its center point.
- • Instantaneous Change: Because f(t) is integrated into the sine argument, the wavelength is “elastic,” creating the accordion-like compression seen in the visual.
Name: Source Code: Manim Implementation *