Superposition and stationary waves · 叠加与驻波
| English | 中文 | Pinyin · 拼音 |
|---|---|---|
| superposition/ˌsuːpəpəˈzɪʃn/ | 叠加 | dié jiā |
| constructive/kənˈstrʌktɪv/ | 相长 | xiāng zhǎng |
| destructive/dɪˈstrʌktɪv/ | 相消 | xiāng xiāo |
| in phase/ɪn feɪz/ | 同相 | tóng xiāng |
| stationary wave/ˈsteɪʃənəri weɪv/ | 驻波 | zhù bō |
| node/nəʊd/ | 波节 | bō jié |
| antinode/ˌæntɪˈnəʊd/ | 波腹 | bō fù |
| progressive wave/prəˈɡresɪv weɪv/ | 行波 | xíng bō |
Cancelling sound with sound
- Noise-cancelling headphones play a wave that cancels the noise around you.
- Two waves can add up — or wipe each other out.
- This adding-up of waves is the idea behind this whole topic.
用声音抵消声音
- 降噪耳机播放一个 抵消 你周围噪声的波。
- 两个波可以相加——或者互相抵消。
- 波的这种相加,是整个这个主题背后的核心思想。
The principle of superposition 叠加
- Where waves overlap, the displacement is the sum of the separate displacements.
- Afterwards the waves carry on, unchanged.
叠加原理
- 在波重叠的地方,位移是各自位移的 和。
- 之后,波继续传播,不发生改变。
Stationary waves · 驻波
y = y₁ + y₂
Two waves superpose: where they reinforce you get antinodes, where they cancel, nodes. · 两个波 叠加:它们加强处是波腹,抵消处是波节。
When two waves overlap at a point, the resultant displacement is: · 当两个波在某点重叠时,合位移是:
That is the principle of superposition — displacements add (as vectors) at each point. · 这就是叠加原理——位移在每一点(作为矢量)相加。
Match each term to the definition the examiner marks. · 把每个术语与评分认可的定义配对。
Displacements add. Amplitudes and intensities do not, which is why two equal waves can give zero at one point and four times the intensity at another. · 相加的是位移。振幅和强度不相加,这就是两列相同的波能在一处给出零、在另一处给出四倍强度的原因。
Constructive 相长 and destructive 相消
- In phase 同相 (crest meets crest) → amplitudes add → constructive.
- Out of phase (crest meets trough) → amplitudes cancel → destructive.
- Since $I \propto A^{2}$, two equal waves in phase give 4× the intensity of one.
Two waves arriving in phase add to give double the amplitude (constructive)
相长与相消
- 同相(波峰遇波峰)→ 振幅相加 → 相长(constructive)。
- 反相(波峰遇波谷)→ 振幅抵消 → 相消(destructive)。
- 由于 $I \propto A^{2}$,两个相等的波同相时,强度是单个波的 4 倍。
两个同相到达的波相加,给出加倍的振幅(相长)
Standing waves & harmonics · 驻波与谐波
A string fixed at both ends only resonates at its harmonics. Drag n to see the nodes, antinodes and how the wavelength changes. · 两端固定的弦只在它的谐波上共振。拖动 n,看波节、波腹以及波长如何变化。
Match what happens when the two waves meet. · 把两个波相遇时发生的情况配对。
In phase → a bigger wave; exactly out of phase → they cancel. · 同相 → 更大的波;恰好反相 → 它们抵消。
Two equal waves meeting in phase give four times the intensity of one wave alone. · 两个相等的波同相相遇,给出单个波四倍的强度。
The amplitude doubles to $2A$, and $I \propto A^{2}$, so $I \propto (2A)^{2} = 4A^{2}$. · 振幅加倍到 $2A$,而 $I \propto A^{2}$,所以 $I \propto (2A)^{2} = 4A^{2}$。
Stationary waves 驻波
- Two identical waves travelling in opposite directions overlap to make a stationary wave.
- It happens when a wave reflects back on itself — on a string, or in an air column.
驻波
- 两个相同的波沿 相反方向 传播,重叠形成 驻波(stationary wave)。
- 当波反射回它自身时就会发生——在弦上,或在空气柱中。
Nodes 波节 and antinodes 波腹
- A node never moves (the waves always cancel); an antinode has the biggest swing.
- Neighbouring nodes are $\dfrac{\lambda}{2}$ apart; a node and the next antinode are $\dfrac{\lambda}{4}$ apart.
Two waves arriving exactly out of phase cancel to zero (destructive)
波节与波腹
- 波节(node) 永不移动(波总是抵消);波腹(antinode) 摆动最大。
- 相邻波节相距 $\dfrac{\lambda}{2}$;一个波节和下一个波腹相距 $\dfrac{\lambda}{4}$。
两个恰好反相到达的波抵消为零(相消)
A point on a stationary wave that is always at zero displacement is called a ____. · 驻波上始终处于零位移的点叫做 ____。
At a node the two waves always cancel. Halfway between nodes is an antinode (biggest swing). · 在波节处两个波总是抵消。两个波节正中间是波腹(摆动最大)。
Neighbouring nodes on a stationary wave are $0.30\ \text{m}$ apart. What is the wavelength? · 驻波上相邻的波节相距 $0.30\ \text{m}$。波长是多少?
Neighbouring nodes are $\dfrac{\lambda}{2}$ apart, so $\lambda = 2 \times 0.30 = 0.60\ \text{m}$. · 相邻波节相距 $\dfrac{\lambda}{2}$,所以 $\lambda = 2 \times 0.30 = 0.60\ \text{m}$。
What is needed to form a stationary wave? Select all · 所有 that apply. · 形成驻波需要什么?选出所有适用的。
Equal amplitudes give the cleanest nodes, but the marked requirements are same frequency, same speed, opposite directions. · 振幅相等能给出最干净的波节,但评分要求的是频率相同、速率相同、方向相反。
Stationary vs progressive
- A stationary wave does not carry energy along, and its pattern stays put.
- A progressive wave 行波 moves along and carries energy with it.
驻波与行波
- 驻波 不沿长度方向传递能量,它的图案保持不动。
- 行波 沿传播方向移动,并带着能量一起走。
A stationary wave transfers energy along its length. · 驻波沿它的长度方向传递能量。
No — the pattern stays put and no energy travels along it. A progressive wave is the one that carries energy. · 不——图案保持不动,没有能量沿它传播。带着能量传播的是行波。
On a stationary wave, adjacent nodes are 0.30 m apart. What is the wavelength, in metres? · 驻波上相邻波节相距 0.30 m。波长是多少米?
Adjacent nodes are HALF a wavelength apart, so the wavelength is 0.60 m. Taking the node spacing as a full wavelength halves every answer that follows. · 相邻波节相距半个波长,所以波长是 0.60 m。把波节间距当成一整个波长,后面每个答案都会小一半。
Put the comparison of a stationary and a progressive wave in order, stationary first. · 把驻波与行波的对比按顺序排列,先驻波。
Those three contrasts are what a compare question is marked on, and the phase one is the one most often left out. · 对比题正是按这三条来评分的,而相位那一条最常被漏掉。
Pipes and strings
- A closed pipe end is a node; an open end is an antinode.
- Closed-pipe fundamental: $L = \dfrac{\lambda}{4}$. Open both ends: $L = \dfrac{\lambda}{2}$. Measure node spacing → $\lambda$, then $v = f\lambda$.
管与弦
- 闭合的管端是一个 波节;开口端是一个 波腹。
- 闭管基音:$L = \dfrac{\lambda}{4}$。两端开口:$L = \dfrac{\lambda}{2}$。测量波节间距 → $\lambda$,再用 $v = f\lambda$。
In a pipe closed at one end, the fundamental fits a length of: · 在一端封闭的管中,基音对应的长度是:
A node at the closed end and an antinode at the open end is a quarter of a wave, so $L = \dfrac{\lambda}{4}$. · 闭端是波节、开口端是波腹,正好是四分之一个波,所以 $L = \dfrac{\lambda}{4}$。
Marks that slip away
- The principle of superposition adds displacements, never amplitudes or intensities. The resultant intensity then follows from the resultant amplitude squared.
- Adjacent nodes are half a wavelength apart, and a node to the next antinode is a quarter.
- A stationary wave needs two progressive waves of the same frequency and speed travelling in opposite directions. State all of it.
- On a stationary wave the amplitude varies with position and every point between two nodes is in phase. On a progressive wave the amplitude is the same everywhere and the phase varies steadily.
- Nodes are points of permanently zero displacement, not points where the wave is momentarily flat.
容易丢掉的分
- 叠加原理相加的是位移,绝不是振幅或强度。合成强度再由合成振幅的平方得出。
- 相邻波节相距半个波长,而波节到相邻波腹是四分之一个波长。
- 驻波需要两列频率和速率相同、方向相反传播的行波。要全部说出来。
- 驻波上振幅随位置变化,而两个波节之间的每一点都同相。行波上振幅处处相同,而相位稳定变化。
- 波节是位移永远为零的点,不是波瞬间变平的点。
You've got it
- superposition: overlapping displacements add (constructive) or cancel (destructive)
- a stationary wave has fixed nodes and antinodes, $\dfrac{\lambda}{2}$ apart
- it carries no energy along — unlike a progressive wave
你掌握了
- 叠加:重叠的位移相加(相长)或抵消(相消)
- 驻波 有固定的 波节 和 波腹,相距 $\dfrac{\lambda}{2}$
- 它不沿传播方向传递 能量——这与行波不同