Periodic Phenomena · 周期现象
| English | 中文 | Pinyin · 拼音 |
|---|---|---|
| periodic/ˌpɪərɪˈɒdɪk/ | 周期的 | zhōu qī de |
| period/ˈpɪərɪəd/ | 周期 | zhōu qī |
| midline/ˈmɪdlaɪn/ | 中线 | zhōng xiàn |
| amplitude/ˈæmplɪtjuːd/ | 振幅 | zhèn fú |
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.
大自然爱重复
- 潮起潮落,日夜交替,心跳一下又一下。
- 无数真实的量一遍遍循环着相同的值。
- 直线、抛物线或指数都无法刻画这种无尽的重复。
- 我们需要一种新的函数——一种为重复而生的函数。
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.
"周期"是什么意思
- 如果一个函数每隔固定间隔重复相同的模式,它就是周期(periodic)的。
- 一个完整循环的长度就是它的周期(period)。
- 每过一个周期,图像看起来和一个周期之前完全一样。
- 所以知道单独一个循环,就知道了整条无限的图像。
A wave that repeats forever · 一条永远重复的波
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. · 沿这条波滑动,注意它会重复。一次完整重复的水平长度就是它的周期。
A periodic function is one that… · 周期函数是指……
A periodic function cycles through the same values again and again — like tides or a spinning wheel. · 周期函数一遍又一遍地循环相同的值——就像潮汐或旋转的轮子。
The horizontal length of one complete cycle is called the . · 一个完整循环的水平长度叫做。
The period · 周期 is how far along the input axis you go before the pattern repeats. · 周期是你在输入轴上走多远之后,模式开始重复。
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.
中线与振幅
- 周期波围绕一条叫做中线(midline)的中央水平线上下摆动。
- 振幅(amplitude)是它在那条中线之上上升(或之下下沉)的距离。
- 振幅高意味着大幅摆动;振幅小意味着轻微的波纹。
- 中线和振幅一起,确定了波的顶部和底部。
Which of these is a periodic phenomenon? · 下面哪一个是周期现象?
A Ferris wheel returns to the same heights every rotation — a repeating cycle. The others rise or fall once. · 摩天轮每转一圈都回到相同的高度——一个重复的循环。其余的只上升或下降一次。
A periodic wave swings above and below a central midline by a fixed amplitude · 振幅. · 周期波围绕一条中央的中线,以固定的振幅上下摆动。
The midline is the centre level; the amplitude · 振幅 is how far the wave rises above (or dips below) it. · 中线是中央的水平线;振幅是波在它之上上升(或之下下沉)的距离。
Select all · 所有 true statements about periodic functions. · 选出关于周期函数的所有正确说法。
A truly periodic function repeats forever, so it never stops. The other three are correct. · 真正的周期函数永远重复,所以它从不停止。其余三条正确。
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.
读一个循环
- 选任意一个起点,沿波前进,直到模式重新开始。
- 你走过的水平距离就是一个周期。
- 最高点是波峰,最低点是波谷。
- 波峰高度减去中线,就等于振幅。
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.
周期模型为什么重要
- 摩天轮、声波、交流电和四季都是周期性的。
- 一个周期模型能预测下一次高潮或下一次日出。
- 因为模式重复,一个测得的循环就能预报未来的每一个循环。
- 接下来的正弦和余弦函数,正是做这件事的核心工具。
"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.
"周期"意味着模式精确地、永远地重复。一个上升、晃动一次、然后稳定下来的量不是周期性的——真正的周期性需要一个不间断、完全相同的循环。
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.
一个摩天轮座位从底部出发,离地 $2$ 米,升到顶部 $40$ 米。
- 中线:一半高处,在 $\tfrac{2 + 40}{2} = 21$ 米。
- 振幅:从中线到顶部,$40 - 21 = 19$ 米。
- 周期:转一整圈所需的时间——之后高度模式开始重复。
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.
周期函数永远重复相同的循环;一个循环的长度是它的周期。波围绕它的中线以固定的振幅上下摆动。周期模型描述潮汐、声音、旋转和四季——任何会重复的事物。