Lesson 10 of 10 · 15 min
Chapter review
Must-know facts
16 facts
- 01A wave carries energy and information; the particles of the medium only oscillate about their mean positions.
- 02Mechanical waves need a medium; electromagnetic waves do not and travel at c = 299,792,458 m s⁻¹ in vacuum.
- 03Transverse: particle motion ⊥ travel, needs shear, only in solids and strings. Longitudinal: particle motion ∥ travel, in solids, liquids and gases.
- 04y = a sin(kx − ωt + φ) travels towards +x; a sin(kx + ωt + φ) travels towards −x.
- 05k = 2π/λ, ω = 2π/T = 2πν.
- 06v = ω/k = λν = λ/T.
- 07String: v = √(T/µ), independent of λ and ν. The source fixes ν; the medium fixes v.
- 08Longitudinal: v = √(B/ρ); thin bar v = √(Y/ρ).
- 09Newton 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
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.