Beyond the Swing: Turning Rotation into High-Frequency Waves
What happens when we stop watching the wave and start “feeling” the vibration? In this section, we crank up the speed to see how rotation transforms into frequency.
While we previously discovered that the circle’s physical size acts as the “DNA” for wave height in Beyond the Pulse , today we are breaking the speed limit to see how Rotational Velocity transforms a simple swing into a high-frequency vibration.
Lab Results: The Speed-Frequency Transformation
If you look at the “High Frequency” chart, you’ll notice the wave looks like it has been physically squeezed. While the height (Amplitude) remains exactly the same as the slow wave, the number of peaks has multiplied.
In this experiment, we’ve moved from a base frequency \(B=1\) to a high-speed multiplier of 5. This change in Rotational Velocity transforms how the wave occupies space.
The Key Takeaway:
- The Input: The Speed (Velocity) of the rotation.
- The Output: The Frequency (Wavelength) of the sine wave.
By cranking up the speed, we are essentially moving from the world of Static Geometry into the world of Acoustics and Physics. The wave is no longer just a path; it has become a “vibration” that we can practically hear.
The Mathematical Model: Scaling the Pulse
This relationship is a fundamental concept in Trigonometric Projections
and is a classic example of Simple Harmonic Motion.
The horizontal density of the wave is now controlled by the speed multiplier B.
By increasing the speed B to 5, the wave completes five full cycles in the space of one rotation, physically transforming a slow swing into a high-frequency vibration.
The resulting high-frequency wave traces a path in blue.
Name: Source Code: Manim Implementation *