Flashcards · IB Diploma Programme Physics · Topic 3

C. Wave behaviour flashcards

41 cards on Topic 3, subtopics C.1 to C.5. Pick a subtopic or study them all.

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C.1 Simple harmonic motion

Conditions for simple harmonic motion, and its defining equation: The net force acts towards equilibrium and is proportional to displacement (F = −kx). a = −ω²x.
Period, frequency and angular frequency: T = 1/f = 2π/ω, with ω = 2πf in rad s⁻¹.
Periods of a mass–spring system and a simple pendulum: T = 2π√(m/k) T = 2π√(l/g) (small angles) Neither depends on the amplitude.
A 0.50 kg mass on a spring with k = 200 N m⁻¹. Period and ω?: T = 2π√(0.50/200) = 0.314 s; ω = 2π/T = 20 rad s⁻¹
Where in SHM are the speed and the acceleration greatest?: Speed is greatest at equilibrium, where acceleration is zero. Acceleration is greatest at the turning points, where speed is zero.
Energy changes during one oscillation (no damping): Kinetic energy is maximum at equilibrium and zero at the turning points; potential energy does the opposite; the total stays constant.
Displacement and velocity as functions of time in SHM: x = x₀ sin(ωt + φ) v = ωx₀ cos(ωt + φ) (HL only)
Speed at displacement x, and total energy, in SHM: v = ±ω√(x₀² − x²) E_T = ½mω²x₀² (HL only)
ω = 10 rad s⁻¹, amplitude 0.030 m. Speed at x = 0.020 m?: v = 10 × √(0.030² − 0.020²) = 0.224 m s⁻¹ (HL only)

C.2 Wave model

Travelling wave: A disturbance that transfers energy through a medium or space without the medium as a whole being shifted.
Transverse vs longitudinal waves: Transverse: particles oscillate at right angles to the direction of travel. Longitudinal: particles oscillate along it, forming compressions and rarefactions.
Wavelength, and the wave equation: Wavelength: the distance between neighbouring points in phase. v = fλ = λ/T.
Sound vs electromagnetic waves: Sound: longitudinal, needs a medium, speed depends on the medium. EM: transverse (perpendicular electric and magnetic fields), needs no medium, all travel at c in a vacuum.
A wave has period 0.125 s and wavelength 0.90 m. Speed and frequency?: v = 0.90 ÷ 0.125 = 7.2 m s⁻¹; f = 1/T = 8.0 Hz
Electromagnetic spectrum, long wavelength to short: Radio, microwave, infrared, visible (about 4 to 7 × 10⁻⁷ m), ultraviolet, X-ray, gamma. Frequency increases in that order.
On a longitudinal wave’s displacement graph, where is a compression?: At a point of zero displacement where the particles on either side are displaced towards it (a rarefaction: displaced away from it).

C.3 Wave phenomena

Wavefront and ray: Wavefront: a line or surface joining points at the same phase. Ray: the direction of travel and energy transfer, perpendicular to the wavefronts.
Snell’s law and refractive index: n₁ sin θ₁ = n₂ sin θ₂, i.e. n₁/n₂ = sin θ₂/sin θ₁ = v₂/v₁. n = c/v.
What changes at a boundary when a wave refracts?: Speed and wavelength change (and usually direction). Frequency stays the same.
Critical angle from n = 1.45 into n = 1.20?: sin C = 1.20 ÷ 1.45 = 0.8276, so C = 55.9°. Total internal reflection if the angle of incidence exceeds C.
When is diffraction significant?: When the gap or obstacle is about the same size as the wavelength or smaller. Speed, wavelength and frequency do not change.
Principle of superposition: Where waves meet, the resultant displacement is the sum of the individual displacements, signs included.
Conditions for constructive and destructive interference (coherent sources in phase): Constructive: path difference = nλ. Destructive: path difference = (n + ½)λ. Coherent: same frequency, constant phase difference.
Young’s double slit: λ = 589 nm, d = 0.45 mm, D = 1.35 m. Fringe separation?: s = λD/d = 589 × 10⁻⁹ × 1.35 ÷ 0.45 × 10⁻³ = 1.77 × 10⁻³ m
Single-slit diffraction: angle of the first minimum: θ = λ/b (radians), b the slit width. A narrower slit gives a wider, dimmer central maximum. (HL only)
Diffraction grating equation, with an example: nλ = d sin θ. 700 lines per mm gives d = 1.43 × 10⁻⁶ m; a second order at 40.0° gives λ = d sin θ / 2 = 459 nm. (HL only)

C.4 Standing waves and resonance

Standing wave, node and antinode: The superposition of two matching waves travelling in opposite directions; no net energy is carried along. Node: permanently zero amplitude. Antinode: maximum amplitude.
Distances in a standing wave: Node to adjacent node: λ/2. Node to adjacent antinode: λ/4.
Allowed frequencies for a string fixed at both ends and a pipe open at both ends: f = nv/2L, all harmonics n = 1, 2, 3, …
Allowed frequencies for a pipe closed at one end: f = nv/4L, odd harmonics only (n = 1, 3, 5, …).
A string of length 0.80 m has wave speed 240 m s⁻¹. First harmonic?: f₁ = 240 ÷ (2 × 0.80) = 150 Hz
Resonance: A large-amplitude response when the driving frequency is at or very near a natural frequency of the system.
Light, critical and heavy damping: Light: several oscillations of decreasing amplitude. Critical: fastest return to equilibrium with no overshoot. Heavy: no oscillation, but a slower return than critical.
Effect of increasing damping on a resonance curve: Lower peak amplitude, a broader peak, and the peak moves slightly below the natural frequency.

C.5 Doppler effect

Doppler effect: The change in observed frequency (and wavelength) when the source and observer are in relative motion.
Moving source vs moving observer: Moving source: wavefronts bunch up in front (shorter λ, higher f) and spread out behind. Moving observer: λ is unchanged, but wavefronts are met more or less often.
Doppler shift for light (v much less than c): Δf/f = Δλ/λ ≈ v/c. A positive Δλ (redshift) means recession; blueshift means approach.
Doppler formula for sound from a moving source: f′ = f v / (v ± us): minus when the source moves towards the observer, plus when it moves away. (HL only)
A 300 Hz source recedes at 17 m s⁻¹ (v = 340 m s⁻¹). Observed frequency?: f′ = 300 × 340 ÷ (340 + 17) = 286 Hz (HL only)
Doppler formula for sound with a moving observer: f′ = f (v ± uo) / v: plus when moving towards the source, minus when moving away. (HL only)
Two applications of the Doppler effect: Doppler ultrasound: the shifted echo from moving blood gives its speed and direction. Radar speed checks: microwaves reflected from a moving vehicle are shifted.