Periodic Phenomena
Introduced| English |
|---|
| periodic/ˌpɪərɪˈɒdɪk/ |
| period/ˈpɪərɪəd/ |
| midline/ˈmɪdlaɪn/ |
| amplitude/ˈæmplɪtjuːd/ |
Nature loves to repeat
- Tides rise and fall, days follow nights, a heartbeat beats again.
- Countless real quantities cycle through the same values over and over.
- No line, parabola, or exponential can capture this endless repetition.
- We need a new kind of function — one built to repeat.
What "periodic" means
- A function is periodic 周期的 if it repeats the same pattern at regular intervals.
- The length of one complete cycle is its period 周期.
- After each period, the graph looks exactly as it did one period earlier.
- So knowing a single cycle tells you the whole infinite graph.
A wave that repeats forever
y = a·sin(b(x − c)) + d
Slide along this wave and notice it repeats. The horizontal length of one full repeat is its period.
A periodic function is one that…
A periodic function cycles through the same values again and again — like tides or a spinning wheel.
The horizontal length of one complete cycle is called the ____.
The period is how far along the input axis you go before the pattern repeats.
Midline and amplitude
- A periodic wave swings above and below a central level called the midline 中线.
- The amplitude 振幅 is how far it rises above (or dips below) that midline.
- A tall amplitude means big swings; a small one means gentle ripples.
- Together, midline and amplitude fix the top and bottom of the wave.
Which of these is a periodic phenomenon?
A Ferris wheel returns to the same heights every rotation — a repeating cycle. The others rise or fall once.
A periodic wave swings above and below a central midline by a fixed amplitude.
The midline is the centre level; the amplitude is how far the wave rises above (or dips below) it.
Select all true statements about periodic functions.
A truly periodic function repeats forever, so it never stops. The other three are correct.
Reading a cycle
- Pick any starting point and follow the wave until the pattern begins again.
- The horizontal distance you travelled is one period.
- The highest point is the peak, the lowest is the trough.
- Peak height minus midline equals the amplitude.
Why periodic models matter
- Ferris wheels, sound waves, alternating current, and seasons are all periodic.
- A periodic model predicts the next high tide or the next sunrise.
- Because the pattern repeats, one measured cycle forecasts every future cycle.
- The sine and cosine functions, coming next, are the master tools for this.
"Periodic" means the pattern repeats exactly and forever. A quantity that rises, wobbles once, and then settles is not periodic — true periodicity requires an unending, identical cycle.
A Ferris wheel seat starts at the bottom, $2$ m above the ground, and rises to $40$ m at the top.
- Midline: halfway up, at $\tfrac{2 + 40}{2} = 21$ m.
- Amplitude: from midline to top, $40 - 21 = 19$ m.
- Period: the time for one full rotation — after which the height pattern repeats.
A periodic function repeats the same cycle forever; the length of one cycle is its period. The wave swings a fixed amplitude above and below its midline. Periodic models describe tides, sound, rotations, and seasons — anything that repeats.