Oscillations and Waves: SHM, Wave Properties, Sound, Superposition
This domain tests the mathematics of vibration and the physics of wave propagation: recognising simple harmonic motion from its defining relation, applying the period formulas, and handling superposition in strings, pipes and sound. Questions usually combine a calculation (period, frequency, wavelength) with an explanation of a standing-wave or resonance pattern.
Simple harmonic motion
SHM is defined by a = −ω²x: acceleration is proportional to displacement and opposite in direction. With x = A cos(ωt + φ), the maximum speed is ωA at the equilibrium position and the maximum acceleration is ω²A at the extremes. The period is independent of amplitude. Standard results: T = 2π√(m/k) for a mass on a spring (k is the spring constant, not the force), and T = 2π√(l/g) for a simple pendulum at small angles. Total energy ½kA² is shared continuously between kinetic and potential forms.
Wave properties
For any wave v = fλ. On a stretched string v = √(F/μ), where μ is mass per unit length. Sound in air travels at about 340 m/s, essentially independent of frequency; intensity falls with distance as the energy spreads. When waves meet, displacements add: constructive interference needs a path difference of nλ, destructive (2n+1)λ/2.
Standing waves and sound
Nodes are separated by λ/2. A string fixed at both ends, or a pipe open at both ends, has fundamental λ = 2L; a pipe closed at one end has λ = 4L and supports only odd harmonics. Resonance occurs when the driving frequency matches a natural frequency. Two close frequencies produce beats at f_beat = |f₁ − f₂|. The Doppler effect shifts observed frequency upward when source and observer approach.
Worked example
A string of length 0.80 m and linear density 5.0 × 10⁻³ kg/m is stretched by a tension of 50 N. Find its fundamental frequency. Wave speed v = √(F/μ) = √(50 / 5.0 × 10⁻³) = √(1.0 × 10⁴) = 100 m/s. For the fundamental, λ = 2L = 1.60 m, so f = v/λ = 100 / 1.60 = 62.5 Hz.
Traps that cost marks
- Confusing ω (rad/s) with f (Hz); always check the calculator mode when using ωt or phase differences.
- Forgetting that a closed pipe gives only odd harmonics and has λ = 4L for the fundamental.
- Counting nodes and antinodes incorrectly - a string of length L in its nth harmonic has n antinodes.
- Using λ instead of λ/2 as the node spacing, or the reverse.
- Applying the Doppler formula with the wrong sign convention for an approaching versus receding source.