Chapter 10: Oscillations, Waves, Sound, and Light
Simple harmonic motion
A system performs SHM when its acceleration is proportional to displacement and directed towards equilibrium: . The solution is , with period . For a mass–spring system, ; for a simple pendulum (small angles), . The total energy is , exchanged between kinetic and potential forms.
Examples: a 0.5 kg mass on a spring with has s. A pendulum 1 m long has s.
Damping and resonance. Damping dissipates energy. A system driven at its natural frequency has maximum amplitude (resonance), which is desirable in radio tuning and musical instruments and dangerous in bridges and buildings.
Wave properties
A wave carries energy without transporting matter. In transverse waves the oscillation is perpendicular to the direction of travel (light, waves on strings); in longitudinal waves it is parallel (sound). For any wave,
Superposition: where waves overlap, displacements add. Interference is constructive when the path difference is and destructive when it is . Standing waves form from two identical waves travelling in opposite directions; they have nodes and antinodes.
| System | Allowed frequencies |
|---|---|
| String fixed at both ends, or pipe open at both ends | , |
| Pipe closed at one end | , (odd harmonics only) |
Example: a closed pipe of length 0.5 m with has fundamental frequency Hz.
Sound
Intensity is power per unit area (). The sound intensity level is decibels, with . Doubling the intensity adds about 3 dB. Beats occur when two close frequencies superpose; the beat frequency is . The Doppler effect: for a source and observer moving along the line joining them, , with the upper signs when moving towards each other. Example: a source of 500 Hz moves towards a stationary observer at ; with , Hz.
Geometrical optics
Refraction: (Snell’s law), where is the refractive index. Total internal reflection occurs when light passes from a denser to a rarer medium at an angle greater than the critical angle , where . It is the principle of optical fibres.
For thin lenses and mirrors (real-is-positive convention):
Example: an object 30 cm from a converging lens of cm: , so cm (real image) and (diminished). Optical instruments (magnifying glass, microscope, telescope) combine lenses.
Wave optics
Young’s double slit: bright fringes are separated by , where is the slit separation and the distance to the screen. Example: nm, m, mm gives mm. A diffraction grating gives maxima where . Polarization shows that light is a transverse wave. The electromagnetic spectrum extends from radio waves to gamma rays, all with speed in vacuum.
Common mistakes
Using with the wrong (speed changes with medium, frequency does not).
Mixing sign conventions for lenses; fix one convention and use it consistently.
Forgetting that open and closed pipes have different harmonic series.
Practice questions
A spring–mass system oscillates with s for kg. Find . [≈12.3 N m⁻¹]
A sound has intensity . Find its level. [60 dB]
The critical angle for glass () to air. [≈41.8°]