Senior 5 practice book · Unit 3 of 11

Propagation of Mechanical Waves

50 questions that complete the Senior 5 quiz for this unit: 20 core and 30 challenge. Easy and hard questions are mixed. Try each question, then tap “Show answer”.

Common misconceptions

A wave of higher frequency travels faster.The speed of a wave is set by the medium (tension and mass per unit length, or stiffness and density). A higher frequency simply gives a shorter wavelength, since v = fλ.
Loud sounds travel faster than quiet sounds.Loudness depends on amplitude, and amplitude does not change the wave speed: a whisper and a shout in the same air arrive at the same speed.
Where two waves cancel each other, their energy is destroyed.Energy is conserved. At places of destructive interference there is less energy and at places of constructive interference there is more; the energy is redistributed.
Sound travels fastest in air because air is the lightest medium.Sound usually travels fastest in solids and slowest in gases, because the stiffness of solids is very much larger, which outweighs their larger density.
A pipe closed at one end has a node at its open end too.Air at the open end is free to move, so there is an antinode (just beyond the end); only the closed end, where the air cannot move, is a displacement node.

What this unit covers

Topics marked new are not tested much in the quiz, so this book gives them extra questions.

Mechanical waves: transverse and longitudinal, compressions and rarefactions
Wave quantities: amplitude, wavelength, period, frequency and v = fλ
Progressive wave equation y = A sin(ωt − kx), direction of travel and particle motion
Phase difference between two points on a wave, Δφ = 2πΔx/λ
Principle of superposition and interference of waves from two coherent sources
Formation of stationary waves: nodes, antinodes and their spacing
Stationary waves on strings fixed at both ends: harmonics f_n = nv/2L
Speed of transverse waves on a stretched string, v = √(T/μ)
Stationary waves in air columns: closed and open pipes, resonance tube and end correction
Reflection, refraction and diffraction of mechanical waves (ripple tank, phase change on reflection)
Energy and intensity of waves: I ∝ A², inverse-square law
Speed of sound in solids, liquids and gases and the effect of temperature

Go to the questions

Questions (1–50)

Easier and harder questions are mixed, just like in a real exam. The 30 harder ones are marked ★ Challenge.

1True or false
At the closed end of a pipe there is always a displacement node.
Show answer
Answer: True
The air molecules at a closed end cannot move along the pipe, so their displacement is always zero: a node.
Common mistakeLearners sometimes mix displacement and pressure: the closed end is a displacement NODE but a pressure antinode.
2True or false · ★ Challenge
Sound travels faster in the air on a hot afternoon than on a cold morning.
Show answer
Answer: True
In a gas the speed of sound increases with temperature (v ∝ √T in kelvin), because the molecules move faster.
Common mistakeLearners often think warm air is 'thinner' so sound must be slower; the faster-moving molecules pass the disturbance on more quickly.
3True or false · ★ Challenge
When two waves interfere destructively at a point, the energy they carried is destroyed.
Show answer
Answer: False
Energy is conserved: less arrives at the minima and more at the maxima, so it is only redistributed.
Common mistakeLearners often think cancellation removes energy; the total energy in the pattern equals the energy sent out by the sources.
4True or false
A 'Mexican wave' moving round the crowd in Amahoro Stadium is like a transverse wave, even though nobody moves around the stadium.
Show answer
Answer: True
Each person only stands up and sits down (moves at right angles to the direction of travel); the pattern travels round, which is how a transverse wave works.
Common mistakeLearners often think something must travel round with the wave; only the disturbance (and energy) travels, not the people.
5True or false · ★ Challenge
Water waves passing through a gap in a harbour wall spread out most when the gap is about the same size as their wavelength.
Show answer
Answer: True
Diffraction is greatest when the gap width is similar to the wavelength; a gap much wider than λ lets the waves pass almost straight through.
Common mistakeLearners often think a wider gap gives more spreading because more wave gets through; the spreading depends on the gap compared with λ.
6True or false
In y = A sin(ωt − kx), the particles of the medium move with a speed that keeps changing, while the wave itself moves at a constant speed ω/k.
Show answer
Answer: True
Each particle moves in SHM, so its speed varies between 0 and Aω; the wave pattern moves steadily at v = ω/k.
Common mistakeLearners often think the particles move at the wave speed; the two speeds are different quantities.
7True or false · ★ Challenge
In a stationary wave on a string, there are instants in each cycle when the whole string is straight, even at the antinodes.
Show answer
Answer: True
All points pass through their rest positions at the same moment (twice per cycle), so the string is momentarily flat; the energy is then all kinetic.
Common mistakeLearners often think antinodes are always displaced; an antinode is a point of maximum AMPLITUDE, but its displacement still passes through zero.
8Multiple choice · ★ Challenge
Which of these mechanical waves can be either transverse or longitudinal?
1. ASound waves travelling in air
2. BWaves travelling through solid rock
3. CSound waves travelling in sea water
4. DWaves in the gas inside a balloon
Show answer
Answer: B. Waves travelling through solid rock
Solids resist both squashing and sideways shearing, so they carry both kinds. Gases and liquids (in their bulk) do not resist shearing, so they carry only longitudinal waves.
Common mistakeChoosing sea water confuses sound inside the water (longitudinal only) with ripples on its SURFACE, which are a different kind of wave.
9Fill in the blank
In a longitudinal wave, regions where the particles are pushed closer together than normal are called ______.
Show answer
Answer: compressions
Compressions are regions of high pressure and density; between them are rarefactions, where the particles are spread out.
Common mistakeWriting 'crests' uses a word for transverse waves; longitudinal waves have compressions and rarefactions.
10Multiple choice · ★ Challenge
Which set of frequencies can a pipe closed at one end produce, if its fundamental is f?
1. Af, 2f, 3f, 4f …
2. B2f, 4f, 6f …
3. Cf, 2f, 4f, 8f …
4. Df, 3f, 5f, 7f …
Show answer
Answer: D. f, 3f, 5f, 7f …
A closed pipe must have a node at the closed end and an antinode at the open end, so its length holds 1, 3, 5 … quarter wavelengths: only odd harmonics.
Common mistakeChoosing f, 2f, 3f … gives the series for a string or an OPEN pipe; a closed pipe cannot have an even harmonic.
11Multiple choice
In which order does sound travel, from fastest to slowest?
1. AAir, water, steel
2. BWater, steel, air
3. CAir, steel, water
4. DSteel, water, air
Show answer
Answer: D. Steel, water, air
Typical speeds: steel about 5000 m/s, water about 1500 m/s, air about 340 m/s. Stiff solids carry sound fastest.
Common mistakeChoosing 'air first' assumes a light medium lets sound move more easily; stiffness matters more than density.
12Multiple choice · ★ Challenge
In a Kigali office corridor you can hear a person talking round a corner but you cannot see them. What is the best explanation?
1. ASound travels faster than light, so it reaches you first round the corner
2. BSound reflects off walls but light is always absorbed by walls
3. CLight waves are transverse, and transverse waves can never be diffracted
4. DSound wavelengths are similar to the width of a doorway, so sound diffracts a lot; light wavelengths are far smaller
Show answer
Answer: D. Sound wavelengths are similar to the width of a doorway, so sound diffracts a lot; light wavelengths are far smaller
Speech has wavelengths of about 0.3 m to 3 m, comparable to doorways, so it spreads round corners; light (about 5 × 10⁻⁷ m) hardly diffracts at such openings.
Common mistakeChoosing 'sound is faster than light' is simply false (340 m/s against 3 × 10⁸ m/s); the difference is in the wavelength, not the speed.
13Multiple choice · ★ Challenge
A guitarist touches a string lightly at its exact midpoint while plucking it, which forces a node there. Which harmonics can still sound?
1. AOnly the odd harmonics (1st, 3rd, 5th …)
2. BOnly the even harmonics (2nd, 4th, 6th …)
3. COnly the fundamental
4. DAll harmonics, louder than before
Show answer
Answer: B. Only the even harmonics (2nd, 4th, 6th …)
The fundamental and the other odd harmonics have an antinode at the midpoint, so they are stopped; even harmonics already have a node there and keep sounding.
Common mistakeChoosing 'only the fundamental' is the opposite of the truth: the fundamental needs an antinode in the middle, which the finger prevents.
14Fill in the blank
Wave intensity is the power carried per unit area at right angles to the wave. Its SI unit is ______.
Show answer
Answer: W/m² (W m⁻²)
Intensity = power ÷ area, so its unit is watts per square metre.
Common mistakeWriting 'W' or 'J' forgets that intensity is power spread over an area.
15Fill in the blank · ★ Challenge
A bat sends out ultrasound of frequency 40 kHz in air, where sound travels at 340 m/s. The wavelength is ______ mm.
Show answer
Answer: 8.5
λ = v/f = 340 ÷ 40 000 = 0.0085 m = 8.5 mm.
Common mistakeWriting 8.5 × 10⁻³ mm or 0.0085 mm forgets the conversion: 0.0085 m is 8.5 mm.
16Multiple choice · ★ Challenge
A learner doubles the frequency at which she shakes one end of a stretched rope, keeping the tension the same. What happens to the waves?
1. AThe speed doubles and the wavelength stays the same
2. BThe speed and the wavelength both double
3. CThe speed halves and the wavelength doubles
4. DThe speed stays the same and the wavelength halves
Show answer
Answer: D. The speed stays the same and the wavelength halves
The speed depends only on the rope's tension and mass per unit length, so it is unchanged; λ = v/f halves.
Common mistakeChoosing 'the speed doubles' assumes shaking faster pushes the wave faster; the medium sets the speed, the source sets the frequency.
17Fill in the blank
Points on a progressive wave exactly one wavelength apart have a phase difference of ______ rad, so they move in phase.
Show answer
Answer: 2π
Δφ = 2πΔx/λ = 2π × λ/λ = 2π rad, one whole cycle.
Common mistakeWriting π rad gives the phase difference for points half a wavelength apart, which move in antiphase.
18Fill in the blank · ★ Challenge
Two loudspeakers are driven in phase at 850 Hz (speed of sound 340 m/s). The smallest non-zero path difference that gives a loud sound is ______ m.
Show answer
Answer: 0.40
λ = v/f = 340 ÷ 850 = 0.40 m; constructive interference needs a path difference of a whole number of wavelengths, the smallest non-zero being 1λ = 0.40 m.
Common mistakeWriting 0.20 m gives λ/2, which is the condition for a QUIET sound, not a loud one.
19Multiple choice
The amplitude of a water wave is tripled while its frequency stays the same. By what factor does the energy it carries each second change?
1. A9
2. B3
3. C6
4. D√3
Show answer
Answer: A. 9
The energy carried (and the intensity) is proportional to the square of the amplitude: 3² = 9.
Common mistakeChoosing 3 assumes energy is proportional to amplitude; like an oscillator's energy, wave energy depends on A².
20Multiple choice · ★ Challenge
The speed of sound in air is 331 m/s at 0 °C, and v ∝ √T with T in kelvin. At what temperature would sound travel at twice this speed?
1. A819 °C
2. B273 °C
3. C546 °C
4. D1092 °C
Show answer
Answer: A. 819 °C
Doubling v needs T to be 2² = 4 times larger: 4 × 273 K = 1092 K, which is 1092 − 273 = 819 °C.
Common mistakeChoosing 273 °C (546 K) only doubles the kelvin temperature; because v ∝ √T, the temperature must be multiplied by four.
21Multiple choice
Straight water waves in a ripple tank pass from deep water into shallow water. What happens?
1. AThe speed, wavelength and frequency all decrease
2. BThe speed and wavelength decrease; the frequency stays the same
3. CThe speed decreases and the wavelength increases
4. DThe frequency decreases; the speed and wavelength stay the same
Show answer
Answer: B. The speed and wavelength decrease; the frequency stays the same
Waves travel more slowly in shallow water. The frequency is fixed by the vibrator, so λ = v/f must decrease too.
Common mistakeChoosing 'frequency decreases' forgets that the number of waves arriving each second must equal the number leaving; the source sets the frequency.
22Multiple choice · ★ Challenge
A small siren sends sound equally in all directions. The intensity 2.0 m away is 0.080 W/m². What is the intensity 8.0 m away?
1. A0.020 W/m²
2. B0.010 W/m²
3. C0.0050 W/m²
4. D0.32 W/m²
Show answer
Answer: C. 0.0050 W/m²
I ∝ 1/r²: the distance is 4 times larger, so the intensity is 4² = 16 times smaller: 0.080 ÷ 16 = 0.0050 W/m².
Common mistakeChoosing 0.020 W/m² divides by 4 instead of 4²; the same power spreads over a sphere whose area grows as r².
23Fill in the blank · ★ Challenge
A wire has a mass per unit length of 5.0 × 10⁻³ kg/m. To make transverse waves travel along it at 200 m/s, the tension must be ______ N.
Show answer
Answer: 200
v = √(T/μ) gives T = μv² = 5.0 × 10⁻³ × 200² = 200 N.
Common mistakeWriting 1.0 N uses T = μv and forgets to square the speed.
24Multiple choice
A pulse of height 3.0 cm (upwards) and a pulse of depth 2.0 cm (downwards) travel towards each other along a rope. What is the displacement of the rope where they exactly overlap?
1. A5.0 cm upwards
2. B1.0 cm upwards
3. C1.0 cm downwards
4. DZero
Show answer
Answer: B. 1.0 cm upwards
By the principle of superposition, displacements add with their signs: +3.0 + (−2.0) = +1.0 cm.
Common mistakeChoosing 5.0 cm adds the sizes without signs; a downward displacement must be counted as negative.
25Multiple choice · ★ Challenge
The string of an umuduri (musical bow) is 0.90 m long between its fixed ends, and waves travel along it at 180 m/s. Which of these frequencies can the string NOT vibrate at?
1. A100 Hz
2. B200 Hz
3. C300 Hz
4. D150 Hz
Show answer
Answer: D. 150 Hz
f_n = nv/2L = n × 180 ÷ 1.8 = 100n Hz: 100, 200, 300 … Hz. 150 Hz is not a whole-number multiple of 100 Hz.
Common mistakeChoosing 100 Hz rejects the fundamental itself; the allowed frequencies are all whole-number multiples of f₁, starting with f₁.
26Multiple choice · ★ Challenge
Two dippers vibrate in phase in a ripple tank and make waves of wavelength 2.0 cm. Point P is 12.0 cm from one dipper and 17.0 cm from the other. What is seen at P?
1. ALarge ripples, because the path difference is a whole number of centimetres
2. BLarge ripples, because the path difference is 2.5 wavelengths
3. CCalm water, because the path difference is 2.5 wavelengths
4. DCalm water, because P is closer to one dipper
Show answer
Answer: C. Calm water, because the path difference is 2.5 wavelengths
Path difference = 17.0 − 12.0 = 5.0 cm = 2.5λ, an odd number of half wavelengths, so the waves arrive in antiphase: destructive interference.
Common mistakeChoosing 'whole number of centimetres' compares the path difference with 1 cm instead of with the wavelength; only whole numbers of WAVELENGTHS give maxima.
27Multiple choice
A pipe 0.85 m long is closed at one end. Taking the speed of sound as 340 m/s and ignoring the end correction, what is its fundamental frequency?
1. A100 Hz
2. B200 Hz
3. C400 Hz
4. D50 Hz
Show answer
Answer: A. 100 Hz
For a closed pipe the fundamental has a node at the closed end and an antinode at the open end: L = λ/4, so λ = 3.4 m and f = 340 ÷ 3.4 = 100 Hz.
Common mistakeChoosing 200 Hz uses L = λ/2, which is the rule for a pipe open at BOTH ends.
28Fill in the blank · ★ Challenge
A tube closed at one end first resonates with a 320 Hz tuning fork when the air column is 0.250 m long. Taking the speed of sound as 340 m/s, the end correction is about ______ cm.
Show answer
Answer: 1.6
λ = 340 ÷ 320 = 1.0625 m, so λ/4 = 0.266 m. λ/4 = l + e gives e = 0.266 − 0.250 = 0.016 m = 1.6 cm.
Common mistakeWriting −1.6 cm or forgetting the end correction assumes the antinode is exactly at the top of the tube; it is slightly above it, so λ/4 is LONGER than l.
29Short answer
What are coherent sources? Explain why two loudspeakers playing two different radio stations do not give a steady pattern of loud and quiet places.
Show answer
Model answer: Coherent sources have the same frequency and a constant phase difference. Two radios playing different stations give sounds of many different, changing frequencies, so the phase difference at any point keeps changing; loud and quiet places move about too quickly to notice.
Common mistakeLearners often think any two sources of sound interfere to give fixed loud and quiet spots; a steady pattern needs a constant phase difference.
30Multiple choice · ★ Challenge
A string vibrates at 250 Hz in a stationary wave. The distance from the first node to the fifth node is 0.60 m. What is the speed of the waves on the string?
1. A150 m/s
2. B75 m/s
3. C38 m/s
4. D60 m/s
Show answer
Answer: B. 75 m/s
From the 1st to the 5th node there are 4 node-to-node gaps, each λ/2: 4 × λ/2 = 0.60 m, so λ = 0.30 m and v = fλ = 250 × 0.30 = 75 m/s.
Common mistakeChoosing 60 m/s counts 5 gaps instead of 4; between the first and the fifth node there are only four half-wavelengths.
31Multiple choice · ★ Challenge
Two points 0.15 m apart on a wave of frequency 200 Hz have a phase difference of π/3 rad. What is the wave speed?
1. A30 m/s
2. B90 m/s
3. C360 m/s
4. D180 m/s
Show answer
Answer: D. 180 m/s
Δφ = 2πΔx/λ gives λ = 2π × 0.15 ÷ (π/3) = 0.90 m; v = fλ = 200 × 0.90 = 180 m/s.
Common mistakeChoosing 30 m/s multiplies f by the separation 0.15 m, as if the points were a whole wavelength apart.
32Short answer
Explain why sound travels much faster in a steel rod than in air, although steel is far denser.
Show answer
Model answer: The speed depends on how stiff the medium is compared with its density (v = √(stiffness/density)). The particles in steel are held by very strong bonds, so a disturbance is passed on very quickly. Steel's stiffness is so much greater than that of air that it outweighs its larger density.
Common mistakeLearners often say 'particles are closer, so sound jumps across faster'; the key is the strength of the forces between particles (stiffness).
33Multiple choice · ★ Challenge
Every particle of a medium carrying the wave y = 0.02 sin(100t − 5x) (SI units) oscillates in SHM. What is the largest acceleration that any particle has?
1. A2.0 m/s²
2. B200 m/s²
3. C0.50 m/s²
4. D20 m/s²
Show answer
Answer: B. 200 m/s²
Each particle moves in SHM with ω = 100 rad/s and A = 0.02 m, so a_max = ω²A = 100² × 0.02 = 200 m/s².
Common mistakeChoosing 20 m/s² mixes up the particle with the wave: ω/k = 20 m/s is the wave speed, not an acceleration.
34Short answer
Describe how you would use a long slinky spring on a bench to show (a) a transverse wave and (b) a longitudinal wave. Name the parts of the longitudinal wave.
Show answer
Model answer: (a) Move one end quickly from side to side, at right angles to the slinky: the coils move sideways while the wave travels along. (b) Push and pull one end along the length of the slinky: the coils move back and forth along the direction of travel. Regions where the coils are squeezed together are compressions; regions where they are spread out are rarefactions.
Common mistakeLearners often say the coils travel along with the wave; each coil only moves to and fro about its own position.
35Multiple choice · ★ Challenge
A worker strikes a long steel railway line. A learner 680 m away, with an ear near the rail, hears two bangs. Taking 5100 m/s in steel and 340 m/s in air, what is the time between the two bangs?
1. A2.00 s
2. B0.13 s
3. C1.87 s
4. D2.13 s
Show answer
Answer: C. 1.87 s
Time in air = 680 ÷ 340 = 2.00 s; time in steel = 680 ÷ 5100 = 0.13 s; the gap is 2.00 − 0.13 = 1.87 s.
Common mistakeChoosing 2.00 s gives only the time through air; the first bang arrives through the steel, so its travel time must be subtracted.
36Short answer
A pulse is sent along a rope towards a wall. Describe the reflected pulse when the end of the rope is (a) tied tightly to the wall, (b) attached to a light ring that slides freely on a vertical pole.
Show answer
Model answer: (a) At a fixed end the reflected pulse is inverted (an upward pulse returns downward): there is a phase change of π. (b) At a free end the pulse is reflected the same way up, with no phase change. In both cases it returns with the same speed and almost the same size.
Common mistakeLearners often think the reflected pulse is always the same way up, as for a mirror image; a fixed end pulls the rope the opposite way and inverts it.
37Multiple choice · ★ Challenge
Two strings are under the same tension. String B has four times the mass per unit length of string A. How does the speed of waves on B compare with that on A?
1. AHalf as fast
2. BA quarter as fast
3. CTwice as fast
4. DThe same speed
Show answer
Answer: A. Half as fast
v = √(T/μ): μ is 4 times larger, so v is divided by √4 = 2.
Common mistakeChoosing 'a quarter' forgets the square root in v = √(T/μ).
38Multiple choice · ★ Challenge
A transverse wave on a washing line obeys y = 0.03 sin(20πt + 4πx), x and y being in metres and t in seconds. State which way the wave moves and how fast.
1. AIn the +x direction at 5.0 m/s
2. BIn the −x direction at 0.20 m/s
3. CIn the −x direction at 5.0 m/s
4. DIn the +x direction at 10 m/s
Show answer
Answer: C. In the −x direction at 5.0 m/s
v = ω/k = 20π ÷ 4π = 5.0 m/s. The + sign between ωt and kx means the wave moves towards negative x.
Common mistakeChoosing '+x' ignores the sign: y = A sin(ωt − kx) moves in +x, but y = A sin(ωt + kx) moves in −x.
39Multiple choice
Ripples from a stone dropped in a pond reach the bank 6.0 m away after 4.0 s. The crests are 0.30 m apart. What is the frequency of the ripples?
1. A5.0 Hz
2. B0.45 Hz
3. C20 Hz
4. D0.20 Hz
Show answer
Answer: A. 5.0 Hz
v = 6.0 ÷ 4.0 = 1.5 m/s; f = v/λ = 1.5 ÷ 0.30 = 5.0 Hz.
Common mistakeChoosing 0.45 Hz multiplies v by λ; from v = fλ the frequency is v DIVIDED by λ.
40Multiple choice · ★ Challenge
In a ripple tank, crests are 1.5 cm apart over the deep part and 1.0 cm apart over a sheet of glass that makes the water shallow. The ripples cross the deep part at 12 cm/s. How fast do they move over the glass?
1. A18 cm/s
2. B8.0 cm/s
3. C12 cm/s
4. D9.0 cm/s
Show answer
Answer: B. 8.0 cm/s
The frequency is the same in both parts, so v is proportional to λ: v = 12 × (1.0 ÷ 1.5) = 8.0 cm/s.
Common mistakeChoosing 18 cm/s uses the ratio of wavelengths upside down; a shorter wavelength at the same frequency means a SLOWER wave.
41Short answer
On a guitar the lower notes come from the thicker strings, even though all strings are under similar tension and have the same length. Explain why.
Show answer
Model answer: A thicker string has a larger mass per unit length μ. With similar tension, the wave speed v = √(T/μ) is smaller, and since f = v/2L for the same length, its frequency (pitch) is lower.
Common mistakeLearners often say thick strings are 'tighter'; at similar tension it is the larger mass per unit length that slows the waves and lowers the note.
42Short answer · ★ Challenge
A wave travels to the right along a rope. At one instant, point P is on the front slope of a crest (the crest is just to the left of P, about to reach it). Is P moving up or down at that instant? Explain.
Show answer
Model answer: P is moving up. The wave moves right, so a moment later the shape has shifted to the right and the crest arrives at P; for that to happen P must be rising.
Common mistakeLearners often say 'down' because P is on a downward slope; the direction of motion depends on what part of the wave arrives next, not on the slope alone.
43Short answer · ★ Challenge
Explain the difference between a displacement–distance graph and a displacement–time graph for a wave, and say which wave quantity can be read directly from each.
Show answer
Model answer: A displacement–distance graph is a 'snapshot' of the whole wave at one instant; the distance between successive crests gives the wavelength λ. A displacement–time graph shows how ONE point moves as time passes; the time between successive crests gives the period T (and f = 1/T). Both show the amplitude.
Common mistakeLearners often read the 'wavelength' from a displacement–time graph; the repeat distance on that graph is a time, the period.
44Multiple choice
A wave of amplitude 2.0 cm and frequency 5.0 Hz passes along a rope. How far does one point of the rope travel during one period?
1. A4.0 cm
2. BOne wavelength
3. C8.0 cm
4. DZero, because it ends where it started
Show answer
Answer: C. 8.0 cm
In one period the point moves from the centre up to +A, down to −A and back to the centre: a total distance of 4A = 8.0 cm.
Common mistakeChoosing 'one wavelength' gives the distance moved by the WAVE in one period; the particle only moves up and down.
45Short answer · ★ Challenge
An open guitar string 0.65 m long has a fundamental of 110 Hz. A finger presses it against a fret so that only 0.52 m vibrates. Find the new fundamental frequency.
Show answer
Model answer: With the same tension and string, v is unchanged and f = v/2L ∝ 1/L. f′ = 110 × 0.65 ÷ 0.52 = 137.5 Hz ≈ 138 Hz.
Common mistakeLearners often think a shorter string gives a lower note; halving the vibrating length doubles the frequency, so shortening it raises the pitch.
46Short answer · ★ Challenge
A small loudspeaker gives out 0.50 W of sound power equally in all directions. Calculate the sound intensity 5.0 m away.
Show answer
Model answer: The power spreads over a sphere of area 4πr² = 4π × 5.0² = 314 m². I = P/4πr² = 0.50 ÷ 314 = 1.6 × 10⁻³ W/m².
Common mistakeLearners often use the area of a circle (πr²) instead of a sphere (4πr²); the sound spreads in three dimensions.
47Multiple choice
Two points on a progressive wave are exactly 3 wavelengths apart. How do their motions compare?
1. AThey move in phase
2. BThey move in antiphase (half a cycle apart)
3. CThey are a quarter of a cycle apart
4. DThey are 3 cycles apart, so one moves three times faster
Show answer
Answer: A. They move in phase
Δφ = 2π × 3 = 6π rad, which is a whole number of cycles, so the two points are always at the same stage: in phase.
Common mistakeChoosing 'antiphase' comes from thinking an odd number means opposite; it is an odd number of HALF wavelengths that gives antiphase.
48Short answer · ★ Challenge
A bamboo flute can be treated as a pipe 0.50 m long, open at both ends (speed of sound 340 m/s). (a) Find its fundamental frequency and the next two harmonics. (b) What would the fundamental become if one end were blocked?
Show answer
Model answer: (a) Open pipe: L = λ/2, so λ = 1.00 m and f₁ = 340 ÷ 1.00 = 340 Hz; the next harmonics are 2f₁ = 680 Hz and 3f₁ = 1020 Hz. (b) Closed at one end: L = λ/4, λ = 2.00 m, f = 340 ÷ 2.00 = 170 Hz (an octave lower).
Common mistakeLearners often think closing one end raises the note; the closed pipe's fundamental wavelength is twice as long, so its frequency is halved.
49Short answer
Compare a stationary wave with a progressive wave under three headings: amplitude of the particles, phase of the particles, and energy transfer.
Show answer
Model answer: Amplitude: in a progressive wave all particles have the same amplitude; in a stationary wave it varies from zero at nodes to maximum at antinodes. Phase: in a progressive wave the phase changes steadily along the wave; in a stationary wave all particles between two adjacent nodes are in phase, and those in neighbouring loops are in antiphase. Energy: a progressive wave transfers energy along the medium; a stationary wave stores it without net transfer.
Common mistakeLearners often say every particle in a stationary wave has the same amplitude as in a progressive wave; the amplitude depends on position.
50Short answer · ★ Challenge
A wave is described by y = 0.04 sin(10πt − 2πx) in SI units. Show that the points at x = 0 and x = 0.50 m always move in opposite directions.
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Model answer: k = 2π m⁻¹, so λ = 2π/k = 1.0 m. Phase difference Δφ = kΔx = 2π × 0.50 = π rad. Points half a wavelength apart are in antiphase: when one is at a crest the other is at a trough, so they always move opposite ways.
Common mistakeLearners often compare x-values directly with 1 m instead of with λ; the phase difference depends on Δx compared with the wavelength.