Periodic Phenomena
Introduced| English | Meaning |
|---|---|
| periodic/ˌpɪərɪˈɒdɪk/ | Repeating the same pattern after a fixed interval of the independent variable. |
| period/ˈpɪərɪəd/ | The time taken for one complete oscillation or cycle. |
| cycle/ˈsaɪkl/ | One complete repetition of a periodic graph. |
| midline/ˈmɪdlaɪn/ | A reference line halfway between the maximum and minimum values of an oscillating graph. |
| amplitude/ˈæmplɪtjuːd/ | The maximum displacement from an equilibrium position. |
| frequency/ˈfriːkwənsi/ | The number of complete cycles per unit time. |
| sinusoidal/ˌsaɪnəˈsɔɪdl/ | sinusoidal |
Follow a rotating seat
- A Ferris-wheel seat is 2 m above the ground at the bottom and 40 m at the top. Suppose the wheel turns steadily once every 8 minutes, without stopping. Its height pattern then repeats every 8 minutes.
- This is an ideal model of a periodic 周期的 quantity. A real ride may stop, speed up or slow down; repeating motion is evidence for a model, not a promise that every future ride is identical.
Define a complete repeat
- For an ideal repeating function defined on all real inputs, a positive number $P$ is a period 周期 if $f(t+P)=f(t)$ for every real $t$. A cycle 周期段 is one complete copy of the pattern.
- For the nonconstant waves in this lesson, “the period” means the smallest positive repeat interval. The wheel has fundamental period 8 minutes; 16 minutes is also a repeat interval, but contains two cycles.
For a permitted positive interval $P$, a periodic pattern repeats when…
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A period matches the whole output pattern: f(t+P)=f(t) at every permitted pair of inputs, not just at one pair of points.
A wave has fundamental period 8 minutes. What does a 16-minute repeat contain?
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Sixteen minutes spans two fundamental cycles; it is also a valid repeat interval.
Match the state of the pattern
- The wheel starts at the bottom. At times $0,2,4,6,8$ minutes, its model heights are $2,21,40,21,2$ m. The two heights of 21 m occur on the way up and on the way down.
- Returning to a height is not enough to identify a period. Match the pattern around the points: bottom to the next bottom, or an upward crossing of 21 m to its next upward crossing. Peak to trough is half a cycle for this symmetric wheel model.
Which situation supports a nonconstant periodic height model while the stated conditions continue?
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A wheel rotating steadily repeats its height and motion direction each turn. Changes in speed or stops can limit that model.
If a wheel seat has the same height at two times, their time difference must be a period.
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It can pass a height while rising and again while falling. A period must match the full pattern.
Measure a period from a graph
- If consecutive peaks of a repeating wave occur at times 3 and 11 minutes, the period is $11-3=8$ minutes. Use input differences, not the second time alone. Peaks at 3 and 19 are two periods apart if another peak occurs at 11.
- When a graph is cut off, trace enough of the repeating shape to justify a complete cycle. A short list of measurements can suggest a period, but it cannot prove what happens between measurements or beyond the observed window.
The smallest positive repeat interval of a nonconstant wave in this lesson is its fundamental ____.
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A fundamental period is one complete repeat. Larger integer multiples repeat the pattern too, but contain more cycles.
Consecutive peaks occur at 3 and 11 minutes. What is the period in minutes?
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Use the difference 11−3=8, not the second input alone.
Find the midline and amplitude
- For a bounded wave with minimum $L$ and maximum $H$, its central height is the midline 中线 $M=(H+L)/2$. Its amplitude 振幅 is the half-range $A=(H-L)/2$, so the extremes are $M+A$ and $M-A$.
- For the wheel, $M=(40+2)/2=21$ m and $A=(40-2)/2=19$ m. The amplitude is not the full 38 m height change. These vertical measurements do not determine how long a cycle takes.
For a bounded wave with minimum L and maximum H, the midline is $(H+L)/2$ and the amplitude is $(H-L)/2$.
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The midline is halfway between the extremes; the amplitude is half their difference. They describe vertical size, not timing.
A wheel seat ranges from 2 m to 40 m. What is its amplitude in metres?
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Half the range is(40−2)/2=19 m; the midline is 21 m.
Separate height from timing
- A second wheel can have the same minimum and maximum but take 12 minutes per turn. Its midline and amplitude stay 21 m and 19 m, while its period changes.
- Adding 5 m to every model height moves the midline up by 5 m and leaves the amplitude and period unchanged. A horizontal shift changes when the peaks occur, without changing their spacing.
Adding 5 m to every height changes which quantities?
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A vertical translation changes both extremes and their midpoint equally; their separation and timing stay the same.
Count cycles per time unit
- Frequency 频率 is the number of complete cycles per time unit. For a fixed period $T$, $f=1/T$. A period of 8 minutes means frequency $1/8$ cycle per minute, or 7.5 cycles per hour.
- Keep units attached: 8 minutes per cycle is not 8 cycles per minute. A 12-minute period gives 5 cycles per hour. A longer period means lower frequency when the time unit is the same.
A cycle takes 8 minutes. What is its frequency in cycles per hour?
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Convert the time unit:60/8=7.5 cycles per hour.
Predict by reducing to one cycle
- In the ideal wheel model, time 18 minutes is two full 8-minute cycles plus 2 minutes. Its height therefore equals the height at 2 minutes: 21 m, on the way up.
- Use $f(t+8n)=f(t)$ for integer $n$ where both times are in the model's domain. A model used only while the ride runs need not describe negative times or times after it stops. Repeated extension is a mathematical assumption.
Compare a complete cycle with a repeated height
Match times separated by a proposed repeat interval. Compare a wheel, a triangular wave, changing patterns and a constant.
An ideal wheel has period 8 minutes and height 21 m at time 2 minutes. What is its height at 18 minutes?
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Eighteen is two periods after 2, so the height is unchanged at 21 m.
Periodic does not mean sinusoidal
- A repeating triangular pattern or on/off signal can be periodic without being a smooth sine curve. A sinusoidal 正弦型的 model is a particular smooth periodic shape, suited to a steadily rotating wheel's vertical height.
- Periodicity specifies repetition, not continuity, a unique peak per cycle or a particular shape. A midline computed halfway between the extremes is not necessarily the time-average of an uneven on/off signal.
A repeating on/off signal must be a smooth sine curve to be periodic.
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Repetition does not require a smooth shape. An on/off pattern can repeat exactly while having jumps.
Recognise a changing pattern
- A rising trend with repeated ripples is not exactly periodic if the whole graph shifts upward each time. A damped vibration can repeat its crossing times while its peaks shrink; the complete output pattern does not repeat.
- Tides and seasonal temperatures can often be approximated by periodic models over a useful window, but weather and other influences produce deviations. Check later cycles, timing, heights and the context before extending a fitted pattern.
Select all true statements about periodic patterns and their models.
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Periodic functions repeat their entire pattern. A sinusoidal model is one possible shape; real cycles can depart from an ideal model.
Peaks occur at equally spaced times but their heights keep shrinking. What follows?
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Regular event timing alone does not make the changing output pattern exactly periodic.
Handle a constant separately
- A constant function satisfies $f(t+P)=f(t)$ for every positive $P$ on its domain. It has no smallest positive period, so it has no unique fundamental cycle or cycle frequency.
- The constant case is why a definition of “the period” needs care. In this lesson, period measurements refer to nonconstant patterns with a smallest positive repeat interval. A repeated value at a few sampled times alone does not establish such a pattern.
Identify a complete repeated pattern before measuring a period. Keep vertical quantities separate from time quantities, attach units to frequency, and distinguish an exact periodic function from a limited model of real observations.
What is special about the constant function f(t)=5 on all real inputs?
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Every positive shift leaves it unchanged, so there is no smallest positive period and no unique cycle frequency.
Explain the extremes and a repeat
- A bounded periodic wave has midline 5, amplitude 3 and fundamental period 8 minutes. Find its maximum and minimum, and explain what the period says about values 8 minutes apart.
- Decide whether a height at time 3 minutes must equal the height at time 7 minutes. Explain why midline and amplitude alone do not settle that comparison.
A bounded wave has midline 5, amplitude 3 and fundamental period 8 minutes. Find its extremes and explain what can be inferred about heights at times 3, 7 and 11 minutes.
The extremes come from midline ± amplitude; an eight-minute shift matches the whole pattern. A shorter gap may match selected values without being a period.
A bounded wave has midline 5 and amplitude 3. What is its maximum?
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Maximum = midline + amplitude = 5+3=8; the minimum is 2.