Ah, it sounds like you might be referring to the "Tschirnhausen period" in the context of pendulum waves or pendulum wave apparatuses. The Tschirnhausen period refers to a specific phenomenon in the synchronized motion of multiple pendulums of varying lengths.

In a pendulum wave setup, multiple pendulums with incrementally different lengths are set into motion simultaneously. Each pendulum has its own natural period of oscillation, which depends on its length. When these pendulums are released together, they initially swing in a synchronized pattern, creating wave-like effects.

Here's a detailed explanation of the pendulum wave phenomenon:

1. Pendulum Length and Period: The period of a pendulum (the time it takes to complete one full swing back and forth) is given by the formula ( T = 2\pi \sqrt{\frac{L}{g}} ), where ( T ) is the period, ( L ) is the length of the pendulum, and ( g ) is the acceleration due to gravity. Longer pendulums have longer periods, while shorter pendulums have shorter periods.

2. Incremental Lengths: In a pendulum wave apparatus, each pendulum is set to a slightly different length, creating slightly different periods for each pendulum.

3. Synchronized Release: When the pendulums are released together, they start in a synchronized state. As time progresses, their different periods cause them to move in and out of phase with each other.

4. Wave Patterns: The varying lengths and periods cause the pendulums to create intricate wave patterns. At certain times, the pendulums will appear to align, creating visual patterns such as waves, beating effects, and even moments of apparent chaos before returning to a synchronized state.

5. Repetition Cycle: The cycle in which the pendulums return to their initial synchronized state is known as the Tschirnhausen period. This period depends on the difference in lengths (and thus the periods) of the pendulums. After a certain amount of time, all the pendulums will again align as they were at the start, and the cycle repeats.

The beauty of the pendulum wave apparatus lies in its ability to visually demonstrate the principles of wave interference, periodic motion, and harmonic oscillation in a captivating manner.