- Distinguish transverse and longitudinal waves and use v = λf = λ/T, knowing that the medium sets the speed and the source sets the frequency.
- Read wave graphs and vibration graphs and find the direction in which a particle moves.
- Apply superposition: decide where two coherent waves strengthen or weaken each other, and describe diffraction and standing waves.
- Explain the properties of sound and the Doppler effect qualitatively.
In a stadium a “Mexican wave” runs around the stands at great speed, yet no spectator leaves their seat: each one just stands up and sits down a little later than their neighbour. That is exactly what a mechanical wave is: a vibration handed on from particle to particle, carrying energy but not matter. In the previous lesson, «Simple harmonic motion», you followed one oscillating body; here every particle of the medium performs that motion, each a little later than the one before. Light waves behave in many of the same ways; their interference and diffraction are treated in «Physical optics: interference and diffraction».
Transverse and longitudinal waves; v = λf
The spreading of a vibration through an elastic medium (机械波). It needs a source that vibrates and a medium (a string, air, water, the ground); there are no mechanical waves in a vacuum. Every particle vibrates about its own equilibrium position with the frequency of the source, and it starts moving in the same direction in which the source started.
| Feature | Transverse wave (横波) | Longitudinal wave (纵波) |
|---|---|---|
| Particles vibrate | at right angles to the direction of travel | along the direction of travel |
| Pattern | crests and troughs (波峰, 波谷) | compressions and rarefactions |
| Examples | a wave on a rope or a guitar string | sound in air and in water; a pushed slinky spring |
| One wavelength λ | crest to next crest | compression to next compression |
- vwave speed, m/s (
波速) - λwavelength: the distance between the two nearest points that vibrate in phase, m (
波长) - T, fperiod and frequency of the vibration — the same as the source's
In one period each particle makes one full vibration and the wave moves on by exactly one wavelength. The medium sets v and the source sets f; when a wave passes into another medium, f stays the same and λ changes in proportion to v.
1) Neighbouring crests on a rope are 0.5 m apart, and the end of the rope is shaken 4 times per second. Find the wave speed. 2) A 500 Hz sound travels first in air (340 m/s) and then in water (1500 m/s). Find the wavelength in each. 3) A boat bobs up and down 12 times a minute on waves whose crests are 5 m apart. How fast do the waves travel?
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2) Air: λ = v/f = 340/500 = 0.68 m; water: λ = 1500/500 = 3 m. The frequency stays 500 Hz, and λ grows in the same ratio as v.
3) f = 12/60 = 0.2 Hz (T = 5 s), so v = λf = 5 · 0.2 = 1 m/s.
At t = 0 the end O of a long rope starts to vibrate with f = 5 Hz, moving upward first; the wave travels at 4 m/s. 1) What is the wavelength? 2) When does point P, 2 m from O, start to move, and in which direction? 3) How many wavelengths fit between O and P at that moment?
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2) The wave front needs t = d/v = 2/4 = 0.5 s to reach P. Like every particle, P starts upward, copying the first motion of the source.
3) 2/0.8 = 2.5 wavelengths (the source has made 0.5 s · 5 Hz = 2.5 vibrations).
Wave graph and vibration graph
Both graphs are sine curves, and CSCA loves to mix them up. A wave graph (波的图像) plots the displacement y of all particles against their position x at one instant: it is a photograph of the rope. A vibration graph (振动图像) plots the displacement of one particle against time t: it is a video of a single point. One repeat of the wave graph is λ, one repeat of the vibration graph is T; combine them and you get v = λ/T.
| Feature | Wave graph (y–x) | Vibration graph (y–t) |
|---|---|---|
| Shows | all particles at one moment | one particle at all moments |
| One repeat along the axis | wavelength λ | period T |
| Direction of a particle's motion | found from the direction of travel (rule below) | read directly: where y goes next |
| A little later | the whole curve slides on by vΔt | the curve simply continues |
- 1Mark the direction of travel
Draw an arrow along the x-axis in the direction the wave travels.
- 2Walk along the arrow
Walk in the direction of travel: if at the particle the curve goes uphill, the particle moves down; if it goes downhill, the particle moves up (
上坡下，下坡上). At a crest or a trough the particle is momentarily at rest. - 3Check by shifting
Redraw the curve moved a little in the direction of travel: the new height above the particle shows where it goes next.
A transverse wave on a rope has a wavelength of 2 m. At t = 0 particle P at x = 1 m is at equilibrium; the nearest crest is at x = 0.5 m and the nearest trough at x = 1.5 m. The vibration graph of P shows that it moves upward at t = 0 and repeats every 0.4 s. Which statement is correct?
A) The wave travels in the +x direction at 5 m/s
B) The wave travels in the −x direction at 5 m/s
C) The wave travels in the +x direction at 0.8 m/s
D) The wave travels in the −x direction at 0.8 m/s
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Direction: from P the curve rises to the crest on the left and falls to the trough on the right. If the wave travelled in +x, walking right from P would go downhill ⇒ P moves up. That matches the vibration graph, so the wave travels in +x: answer A.
B is the trap for anyone who applies the rule while walking against the direction of travel.
Superposition, interference and diffraction
Where two waves meet, the displacement of each particle is the sum of the displacements the waves would cause separately (叠加原理). After meeting, the waves go on as if nothing had happened: two pulses running toward each other on a rope pass through each other unchanged.
Two coherent sources (same frequency, constant phase difference, 相干波源) create a stable pattern (干涉): at some points the waves always strengthen each other (amplitude A₁ + A₂), at others they always weaken each other (amplitude |A₁ − A₂|). What decides it is the path difference Δr = |r₁ − r₂| from the two sources.
- Δrpath difference |r₁ − r₂|, m
- λwavelength, m
- na whole number: 0, 1, 2, …
For sources vibrating in phase. If the sources are in antiphase, the two conditions swap. A whole number of wavelengths means crest meets crest; an odd number of half-wavelengths means crest meets trough.
Two dippers in a ripple tank vibrate in phase and make water waves of wavelength 4 cm. 1) Point P is 30 cm from one dipper and 36 cm from the other. 2) Point Q is 22 cm and 30 cm away. 3) The waves arriving at P and Q have amplitudes of 3 mm and 2 mm. What is the amplitude of the water's vibration at P and at Q?
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2) Δr = 8 cm = 2λ: Q is a maximum.
3) At Q the amplitude is 3 + 2 = 5 mm, at P it is 3 − 2 = 1 mm. With equal amplitudes P would not move at all. The water at Q still goes up and down: its displacement changes between −5 mm and +5 mm; only the amplitude is 5 mm.
In a ripple tank two coherent sources vibrate in phase with the same amplitude A. Point M lies on a line of strengthening (constructive interference). Which statement about M is correct?
A) The water at M is always at a crest
B) The water at M vibrates with amplitude 2A, and its displacement is sometimes zero
C) M is strengthened and weakened in turn
D) The water at M does not move
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A is the classic trap: “strengthened” describes the amplitude, not a permanent crest. C would need a changing phase difference, but coherent sources keep it constant; D describes a point of weakening.
The bending of waves around obstacles and their spreading after passing through a gap (衍射). It is clearly noticeable when the size of the gap or obstacle is smaller than or comparable to the wavelength. Sound (λ from about 1.7 cm to 17 m) bends around doors and walls, so you hear someone around a corner; visible light (λ ≈ 0.4–0.7 μm) does not, so you cannot see them.
Standing waves, sound and the Doppler effect
The result of two identical waves travelling in opposite directions (驻波), for example a wave and its reflection on a string fixed at both ends. Some points never move — the nodes (波节); halfway between them are the antinodes (波腹), which vibrate with the largest amplitude. Neighbouring nodes are λ/2 apart, and a standing wave carries no energy along the string.
- Llength of the string fixed at both ends, m
- nnumber of loops (antinodes); n = 1 is the fundamental
- vspeed of the travelling waves on the string, m/s
Both ends are nodes, so a whole number of half-wavelengths must fit on the string. The lowest frequency (n = 1, λ = 2L) is the fundamental; the others are whole multiples of it.
A string 1.0 m long is fixed at both ends; waves travel on it at 40 m/s. 1) Find the wavelength and frequency of the fundamental. 2) The string vibrates in 3 loops. Find λ, f and the number of nodes. 3) How far apart are neighbouring nodes then?
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2) n = 3: λ = 2L/3 = 2/3 m ≈ 0.67 m, f₃ = 3f₁ = 60 Hz. There are 4 nodes (the two ends and two inside) and 3 antinodes.
3) λ/2 = 1/3 m ≈ 33 cm, which is also L/3.
Sound is a longitudinal mechanical wave. It cannot travel through a vacuum; in air at room temperature it moves at about 340 m/s, in water at about 1500 m/s and in steel at about 5000 m/s. Humans hear roughly 20 Hz–20 000 Hz; lower frequencies are infrasound, higher ones ultrasound (used in medical scans and sonar). Pitch is set by the frequency and loudness by the amplitude. An echo is sound reflected from an obstacle; the distance to it is d = vt/2, because the sound goes there and back.
1) A hiker shouts toward a cliff and hears the echo 1.2 s later (v = 340 m/s). How far away is the cliff? 2) A bat emits ultrasound at 40 kHz. What is its wavelength in air? 3) In the same air the frequency of a sound is doubled. What happens to its wavelength and its speed?
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2) λ = v/f = 340/40 000 = 0.0085 m = 8.5 mm: short enough to reflect from small insects.
3) The medium sets the speed, so v does not change, and λ = v/f is halved.
When a source and an observer move toward each other, the observer receives more waves per second and hears a higher frequency; when they move apart, a lower one (多普勒效应). The source's own frequency does not change, and neither does the speed of sound in the air: in front of a moving source the waves are squeezed (shorter λ), behind it they are stretched.
An ambulance with its siren on (frequency 700 Hz) drives past a pedestrian standing at the roadside at constant speed. What does the pedestrian hear?
A) 700 Hz all the time, only louder while the ambulance is near
B) A frequency above 700 Hz as it approaches and below 700 Hz as it moves away
C) A frequency below 700 Hz as it approaches and above 700 Hz as it moves away
D) A frequency that rises steadily the whole time
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A is tempting because the siren itself really does stay at 700 Hz, but the received frequency does not. D is wrong: at constant speed the pitch stays nearly constant while the ambulance approaches and then drops as it passes.
How CSCA asks about this
The test is taken in English or in Chinese, so learn to recognise the key terms in both languages:
| Term | 中文 | Pinyin |
|---|---|---|
| mechanical wave | 机械波 | jīxièbō |
| transverse wave | 横波 | héngbō |
| longitudinal wave | 纵波 | zòngbō |
| wavelength | 波长 | bōcháng |
| wave speed | 波速 | bōsù |
| crest / trough | 波峰 / 波谷 | bōfēng / bōgǔ |
| wave graph | 波的图像 | bō de túxiàng |
| vibration graph | 振动图像 | zhèndòng túxiàng |
| superposition | 叠加 | diéjiā |
| interference | 干涉 | gānshè |
| coherent sources | 相干波源 | xiānggān bōyuán |
| diffraction | 衍射 | yǎnshè |
| standing wave | 驻波 | zhùbō |
| node / antinode | 波节 / 波腹 | bōjié / bōfù |
| Doppler effect | 多普勒效应 | Duōpǔlè xiàoyìng |
- v = λf with a change of medium: which of f, v and λ changes (f never does).
- Wave graph and direction: given the direction of travel, find the direction of a particle (or the reverse) —
上坡下，下坡上. - Wave graph + vibration graph: λ from the first, T from the second, v = λ/T, and the direction of travel from the particle's motion.
- Timing: when the wave reaches a point (t = d/v), which way the point first moves (like the source), where a particle is after a fraction of T.
- Interference: path difference in half-wavelengths → strengthened or weakened; amplitude A₁ + A₂ or |A₁ − A₂|.
- Standing waves, diffraction, sound, Doppler: nodes λ/2 apart, L = nλ/2; qualitative items — pick the true statement.
Key points
- A mechanical wave is a vibration spreading through a medium: it carries energy, not matter; the particles vibrate in place at the source's frequency.
- v = λf = λ/T; the medium sets v, the source sets f; in a new medium f stays the same.
- The wave graph (y–x) gives λ, the vibration graph (y–t) gives T; direction rule: uphill – down, downhill – up.
- In-phase sources: Δr = nλ — strengthening (A₁ + A₂); Δr = (2n + 1)λ/2 — weakening (|A₁ − A₂|).
- Diffraction is clear when the gap is smaller than or close to λ; in a standing wave neighbouring nodes are λ/2 apart, and for a string fixed at both ends L = nλ/2.
- Sound is longitudinal and cannot cross a vacuum; an approaching source sounds higher, a receding one lower (Doppler).
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