Transverse and longitudinal waves
A ripple spreads on a pond, but a floating leaf only bobs up and down; a shout reaches you across a field. Link to earlier classes : sound and waves of class 8 and 9.
A wave carries energy and information from one place to another without carrying the matter along. In a transverse wave the particles vibrate at right angles to the direction of travel; in a longitudinal wave they vibrate along it.
Rule 1: Kinds of waves — Transverse waves (waves on a string, light) travel as crests and troughs. Longitudinal waves (sound) travel as compressions and rarefactions. Transverse mechanical waves can travel only in solids (and on liquid surfaces); longitudinal waves travel in solids, liquids and gases.
- Easy : A wave on a rope shaken up and down is transverse.
- Medium : Sound in air is longitudinal.
- Exam : The distance between two successive crests (or two successive compressions) is the wavelength λ.
Transverse : particles ⟂ direction; longitudinal : particles ∥ direction; waves carry energy, not matter.
Believing that the water or air itself moves along with the wave.
Memory clue : "Transverse = Tilted (right angle); Longitudinal = Long-way (along)."
Spot the difference
| Wave | Particle motion | Example |
|---|---|---|
| Transverse | perpendicular | rope wave |
| Longitudinal | parallel | sound |
Displacement relation for a progressive wave
A ripple moving outwards is a pattern that travels while each particle only oscillates. Link to earlier classes : SHM of chapter 13 is the motion of every particle in the wave.
A progressive (travelling) wave moving along +x is described by a function of x and t. The wave number k = 2π ÷ λ tells how fast the phase changes with distance.
Rule 1: Displacement of a progressive wave — y(x, t) = A sin(k x − ω t + φ) with k = 2π ÷ λ, ω = 2π ÷ T and v = ω ÷ k = ν λ. A minus sign between kx and ωt means a wave moving in +x; a plus sign means a wave moving in −x.
In the figure : The sine curve repeats every wavelength λ and its height is the amplitude A.
- Easy : λ = 2 m and ν = 50 Hz ⇒ v = 100 m s⁻¹.
- Medium : For y = 0.05 sin(4π x − 200π t) : k = 4π ⇒ λ = 0.5 m; ω = 200π ⇒ ν = 100 Hz; v = ω ÷ k = 50 m s⁻¹.
- Exam : In the same wave two points 0.125 m apart have a phase difference k Δx = 4π × 0.125 = π ÷ 2.
v = νλ = ω ÷ k; phase difference = 2π Δx ÷ λ.
Mixing the speed of the wave v with the speed of a particle (which changes in time).
Memory clue : "kx minus ωt : the wave moves forward."
Spot the difference
| Quantity | Symbol | Formula |
|---|---|---|
| Wavelength | λ | v ÷ ν |
| Wave number | k | 2π ÷ λ |
| Angular frequency | ω | 2πν |
The speed of a travelling wave
A tightly stretched guitar string carries a pulse faster than a loose one; thunder is heard a few seconds after lightning. Link to earlier classes : speed = distance ÷ time.
The speed of a mechanical wave depends on the properties of the medium : its elastic (restoring) property and its inertia (density).
Rule 1: Speed on a stretched string — v = √(T ÷ μ) (T = tension in the string, μ = mass per unit length).
- Easy : T = 100 N and μ = 0.01 kg m⁻¹ ⇒ v = √(100 ÷ 0.01) = 100 m s⁻¹.
- Medium : If the tension becomes 4 times, the speed doubles.
- Exam : A thicker (heavier) string of the same tension carries waves more slowly.
Rule 2: Speed of sound — v = √(B ÷ ρ) for a fluid; for a gas (Newton–Laplace) v = √(γ P ÷ ρ). In air v ∝ √T (T in kelvin).
- Easy : For air at 0 °C : v = √(1.4 × 1.013 × 10⁵ ÷ 1.29) = 332 m s⁻¹.
- Medium : At 20 °C the speed is 332 × √(293 ÷ 273) = 344 m s⁻¹.
- Exam : Sound travels faster in solids than in gases because the elastic constant is much larger.
String : v = √(T ÷ μ); gas : v = √(γP ÷ ρ); the speed of sound in air rises with temperature but does not change with pressure at a fixed temperature.
Thinking that sound is faster in air at higher pressure at a constant temperature (it is not).
Memory clue : "Tight and Thin is fast : v = root of T over mu."
Spot the difference
| Medium | Approximate speed of sound |
|---|---|
| Air | 340 m s⁻¹ |
| Water | 1500 m s⁻¹ |
| Steel | 5000 m s⁻¹ |
Superposition and reflection of waves
Two ripples crossing on a pond pass through each other, and an echo returns from a wall. Link to earlier classes : reflection of sound of class 8.
The principle of superposition : when two or more waves meet, the resultant displacement is the sum of the individual displacements; afterwards each wave moves on unchanged.
Rule 1: Superposition — y = y₁ + y₂. For two waves of equal amplitude A and phase difference φ : A_R = 2A cos(φ ÷ 2). The intensity is proportional to the square of the amplitude.
- Easy : Two pulses of 3 cm and 4 cm in the same phase give 7 cm; in opposite phase they give 1 cm.
- Medium : Equal waves in phase (φ = 0) give 2A (constructive); with φ = π they give 0 (destructive).
- Exam : For A = 2 mm each and φ = 120°, A_R = 2 × 2 × cos 60° = 2 mm.
Rule 2: Reflection — At a fixed (rigid) end the reflected pulse is inverted (a phase change of π). At a free end it returns without inversion.
- Easy : A pulse sent along a rope tied to a wall comes back upside down.
- Medium : When the rope end is free to slide on a pole, the pulse comes back the right way up.
- Exam : The reflected wave has the same speed and wavelength; only the phase can change.
Add the displacements; fixed end inverts, free end does not.
Adding the intensities instead of the amplitudes.
Memory clue : "Fixed flips, Free follows."
Spot the difference
| Interference | Phase difference | Resultant |
|---|---|---|
| Constructive | 0 | 2A |
| Destructive | π | 0 |
Standing waves on a string
A plucked guitar string appears to vibrate in fixed loops. Link to earlier classes : superposition of the previous section.
A standing (stationary) wave is formed when two identical waves travel in opposite directions. It has fixed points of no motion, the nodes, and points of the largest motion, the antinodes, at equal distances of λ ÷ 2 and λ ÷ 4 from each other.
Rule 1: Normal modes of a string fixed at both ends — ν_n = n v ÷ (2L) where n = 1, 2, 3, … (L = length, v = wave speed). n = 1 is the fundamental (first harmonic); all harmonics are possible.
- Easy : L = 0.5 m and v = 100 m s⁻¹ ⇒ ν₁ = 100 Hz, ν₂ = 200 Hz, ν₃ = 300 Hz.
- Medium : A wire of length 1.0 m vibrates with the fundamental frequency of 220 Hz ⇒ v = 2Lν = 440 m s⁻¹.
- Exam : If the tension is increased 4 times, v doubles and every frequency doubles : the pitch rises by an octave.
ν_n = n v ÷ 2L; nodes are λ ÷ 2 apart; both ends of a fixed string are nodes.
Taking the distance between a node and the next node as λ.
Memory clue : "Fixed ends : one, two, three … all the harmonics."
Spot the difference
| Wave | Energy moves? | Nodes? |
|---|---|---|
| Travelling | yes | no |
| Standing | no | yes |
Vibrations of air columns : open and closed pipes
Blowing across a bottle or playing a flute makes a note. Link to earlier classes : standing waves of the previous section, now with air in a tube.
In a pipe open at both ends there is an antinode at each end; in a pipe closed at one end there is a node at the closed end and an antinode at the open end.
Rule 1: Normal modes of pipes — Open pipe : ν_n = n v ÷ (2L), n = 1, 2, 3 … (all harmonics). Closed pipe : ν = (2n − 1) v ÷ (4L), n = 1, 2, 3 … (only odd harmonics).
- Easy : v = 340 m s⁻¹ and L = 0.5 m : an open pipe has ν₁ = 340 Hz; a closed pipe has ν₁ = 170 Hz.
- Medium : The closed pipe above gives 170, 510 and 850 Hz (the first three allowed modes).
- Exam : To get a fundamental of 256 Hz from a closed pipe, L = v ÷ (4ν) = 340 ÷ (4 × 256) = 0.33 m.
Open pipe : all harmonics, ν₁ = v ÷ 2L; closed pipe : odd harmonics only, ν₁ = v ÷ 4L.
Using v ÷ 2L for a closed pipe.
Memory clue : "Open = Every harmonic; Closed = Only odd."
Spot the difference
| Pipe | Fundamental | Harmonics |
|---|---|---|
| Open | v ÷ 2L | all |
| Closed | v ÷ 4L | odd only |
Beats and the Doppler effect
The pitch of an ambulance siren drops as it passes you, and two slightly different guitar strings give a wobbling sound. Link to earlier classes : frequency and pitch of sound.
Beats are the periodic rise and fall of loudness when two sounds of slightly different frequencies are heard together. The Doppler effect is the change in the observed frequency when the source and the observer are in relative motion.
Rule 1: Beats — beat frequency = |ν₁ − ν₂| (beats per second).
- Easy : Two forks of 256 Hz and 260 Hz give 4 beats per second.
- Medium : Loading a fork lowers its frequency; if the beat rate then rises, the other fork had a lower frequency.
- Exam : Beats are used to tune musical instruments.
Rule 2: Doppler effect — ν′ = ν (v ± v_o) ÷ (v ∓ v_s) (v = speed of sound, v_o = speed of observer, v_s = speed of source). Use the upper signs when they approach each other.
- Easy : A source of 500 Hz approaches a still observer at 34 m s⁻¹ (v = 340 m s⁻¹) ⇒ ν′ = 500 × 340 ÷ 306 = 556 Hz.
- Medium : The same source moving away gives ν′ = 500 × 340 ÷ 374 = 455 Hz.
- Exam : An observer walking at 20 m s⁻¹ towards a still source of 400 Hz hears ν′ = 400 × 360 ÷ 340 = 424 Hz.
Approach raises the frequency; recession lowers it; beats = |ν₁ − ν₂|.
Thinking that the source itself changes its frequency.
Memory clue : "Closer, Climb in pitch; Away, Abate."
Spot the difference
| Motion | Observed frequency |
|---|---|
| source approaching | higher |
| source receding | lower |