Mechanical waves and sound
Mechanical-wave questions test the wave equation, superposition, standing waves, interference and diffraction, and the Doppler effect. The PAU often links a formula to a physical boundary condition, such as a string fixed at both ends or a pipe open at one end.
Core rules
- Travelling wave: y = A sin(ωt - kx), with ω = 2πf, k = 2π/λ and v = λf = ω/k. Wave speed is set by the medium; frequency is set by the source.
- String speed: v = sqrt(T/μ), where T is tension and μ is mass per unit length.
- Superposition: displacements add. Constructive interference occurs when path difference is mλ; destructive when it is (m + ½)λ.
- Standing waves: nodes and antinodes are separated by λ/2. A string fixed at both ends has f_n = nv/(2L). A pipe open at both ends has the same set; a pipe closed at one end has f_n = (2n - 1)v/(4L).
- Sound intensity follows I ∝ 1/r² for a point source. Diffraction is significant when the obstacle or slit size is comparable to λ.
- Doppler: f' = f(v ± v_o)/(v ∓ v_s), with signs chosen so approach raises frequency and recession lowers it.
Worked example
A string of length L = 1.0 m has μ = 5.0×10⁻³ kg/m and tension T = 80 N. Then v = sqrt(80/5.0×10⁻³) = sqrt(1.6×10⁴) = 126.5 m/s. The fundamental is f₁ = v/(2L) = 63.2 Hz; the next harmonic is 126.5 Hz. The wavelength of the fundamental is 2L = 2.0 m. At the midpoint of the string the fundamental has an antinode, while the ends are nodes.
Traps
Do not confuse wave speed with particle speed. A standing wave does not travel; it stores energy between nodes. For a closed pipe, only odd harmonics occur. In Doppler problems, identify whether the source or observer moves and check that approach increases f'. Do not use λ = 2L for every harmonic; for a string f_n = nv/(2L). Diffraction is not just bending at an edge; it depends on λ relative to aperture size. Keep degrees and radians consistent in phase calculations.