Lesson 10 of 10 · 15 min

Chapter review

Must-know facts

16 facts

1A wave carries energy and information; the particles of the medium only oscillate about their mean positions.
2Mechanical waves need a medium; electromagnetic waves do not and travel at c = 299,792,458 m s⁻¹ in vacuum.
3Transverse: particle motion ⊥ travel, needs shear, only in solids and strings. Longitudinal: particle motion ∥ travel, in solids, liquids and gases.
4y = a sin(kx − ωt + φ) travels towards +x; a sin(kx + ωt + φ) travels towards −x.
5k = 2π/λ, ω = 2π/T = 2πν.
6v = ω/k = λν = λ/T.
7String: v = √(T/µ), independent of λ and ν. The source fixes ν; the medium fixes v.
8Longitudinal: v = √(B/ρ); thin bar v = √(Y/ρ).
9Newton v = √(P/ρ) ≈ 280 m s⁻¹ in air at STP; Laplace v = √(γP/ρ) = 331.3 m s⁻¹ with γ = 7/5.
10Superposition of two equal waves with phase difference φ: amplitude 2a cos(φ/2).
11Rigid end: reflection with phase change π. Free end: no phase change.
12Standing wave y = 2a sin kx cos ωt: nodes at nλ/2, antinodes at (n + ½)λ/2, spacing λ/2.
13String fixed at both ends and open pipe: ν = nv/2L, all harmonics.
14Closed pipe: ν = (n + ½)v/2L; fundamental v/4L, only odd harmonics.
15Beat frequency = ν₁ − ν₂; 11 Hz and 9 Hz give 2 beats per second.
16Ex 14.3: 0.72 m, 5.0 g wire at 60 N gives 93 m s⁻¹. Ex 14.6: 427 Hz, 5 then 3 beats, so 422 Hz.

Common traps

Where marks are lost

Thinking the particles of the medium travel with the wave.

Only the disturbance and its energy travel. Each particle oscillates about its own mean position.

Reading λ as 1/k or ν as ω from a wave equation.

λ = 2π/k and ν = ω/2π. In y = 0.005 sin(80.0x − 3.0t), λ = 7.85 cm and ν = 0.48 Hz.

Believing a higher-frequency source makes waves on the same string travel faster.

v = √(T/µ) is fixed by the string. A higher ν gives a shorter λ = v/ν at the same speed.

Using Newton's isothermal formula for the speed of sound.

Sound compressions are adiabatic: v = √(γP/ρ). Newton's √(P/ρ) comes out about 15% too low.

Giving a closed pipe the harmonics 2v/4L, 4v/4L and so on.

A closed pipe has a node at the closed end and an antinode at the open end, so only odd multiples of v/4L occur.

Assuming a wave reflected from a rigid wall comes back upright.

At a rigid end the reflected wave is inverted (phase change π). Only at a free end is there no phase change.

Taking the node-to-node distance as λ.

Neighbouring nodes are λ/2 apart; a node and the next antinode are λ/4 apart.

Adding the two frequencies to get the beat frequency.

Beat frequency is the difference ν₁ − ν₂. The sound itself is heard at the average (ν₁ + ν₂)/2.

Formulas

13 to know

Progressive wave

y(x, t) = a sin(kx − ωt + φ)

Travels towards +x; kx + ωt for −x.

Angular wave number

k = 2π/λ

rad m⁻¹.

Angular frequency

ω = 2π/T = 2πν

rad s⁻¹.

Wave speed

v = ω/k = λν = λ/T

Holds for all progressive waves.

Wave on a string

v = √(T/µ)

T tension, µ mass per unit length.

Longitudinal wave

v = √(B/ρ); bar: v = √(Y/ρ)

B = −ΔP/(ΔV/V).

Newton's formula

v = √(P/ρ)

Isothermal; about 280 m s⁻¹ in air at STP.

Laplace's formula

v = √(γP/ρ)

Adiabatic; γ = 7/5 for air gives 331.3 m s⁻¹.

Superposed waves

y = 2a cos(φ/2) sin(kx − ωt + φ/2)

Amplitude 2a cos(φ/2).

Standing wave

y = 2a sin kx cos ωt

Nodes x = nλ/2; antinodes x = (n + ½)λ/2.

String fixed at both ends, open pipe

ν = nv/2L, n = 1, 2, 3, …

All harmonics.

Closed pipe

ν = (n + ½)v/2L, n = 0, 1, 2, …

v/4L, 3v/4L, 5v/4L: odd harmonics.

Beats

ν_beat = ν₁ − ν₂

Heard at (ν₁ + ν₂)/2.

Key terms

17 terms

Wave: A disturbance that travels through a medium, or through space, carrying energy without carrying matter along.
Mechanical wave: A wave that needs a material medium and is governed by Newton's laws, such as sound or a wave on a string.
Transverse wave: A wave in which particles oscillate perpendicular to the direction of travel.
Longitudinal wave: A wave in which particles oscillate along the direction of travel.
Compression and rarefaction: Regions of higher and of lower density than normal in a longitudinal wave.
Progressive wave: A wave whose pattern travels from one part of the medium to another.
Crest and trough: Points of largest positive and largest negative displacement.
Wavelength: The least distance between two points in the same phase.
Angular wave number: k = 2π/λ, the phase change per metre.
Linear mass density: Mass per unit length of a string, µ.
Bulk modulus: B = −ΔP/(ΔV/V), the resistance of a medium to compression.
Principle of superposition: Where waves overlap, each point's displacement is the sum, with signs, of what each wave alone would give it.
Standing wave: The non-travelling pattern formed by two identical waves moving in opposite directions.
Node and antinode: Points of a standing wave with zero amplitude and with the largest amplitude.
Normal mode: A natural frequency at which a bounded system such as a string or air column can oscillate.
Fundamental and harmonics: The lowest normal mode and the modes at whole-number multiples of it.
Beats: Regular rise and fall of loudness when two sounds of nearly equal frequency are heard together.

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