Wave Applications — SONAR, Ultrasound, Music, and Seismology — Summary

Musical instruments (strings): Frequency determined by $f_n$ = $\frac{nv}{2L}$ ; tightening string raises tension T → higher v → higher f (higher pitch). Multiple harmonics create musical timbre.
Tuning forks and beats: Two forks of slightly different frequencies produce beats (| $f_{1}$ − $f_{2}$ | per second). Used to tune instruments — zero beats means identical frequencies.
Open organ pipes: Support all harmonics; richer sound. Examples: flute (open), most brass instruments.
Closed organ pipes: Only odd harmonics; simpler tone. Examples: some flutes and reed instruments when one end is stopped.
SONAR (underwater): Emits ultrasonic pulse; measures echo return time t → depth d = vt/2 (v ≈ 1500 m/s in seawater). Used for ocean floor mapping and submarine detection.
Medical ultrasound (1–20 MHz): Impedance mismatch at tissue boundaries causes partial reflection. Echo timing constructs 2D cross-sectional images. Doppler ultrasound: blood flow velocity from frequency shift.
Seismic P-waves: Longitudinal; travel through all media (solids, liquids, gases); speed ≈ 8 km/s. Arrive first at seismographs.
Seismic S-waves: Transverse; travel only through solids; speed ≈ 4 km/s. Cannot pass through Earth's liquid outer core — this observation revealed the core's liquid state.
Epicentre location: Time difference between P and S wave arrivals at multiple stations triangulates earthquake epicentre.
Noise-cancelling headphones: Generate anti-phase wave (phase shift π) of equal amplitude. Superposition: $y_{cancel}$ = A sin(kx−ωt) + A sin(kx−ωt+π) = 0. Destructive interference silences ambient noise.
Echolocation $\frac{bats}{dolphins}$ : Emit ultrasonic pulses; Doppler shift of returning echo reveals speed of prey.
Earthquake engineering: Buildings designed with natural frequencies away from typical seismic frequencies to avoid resonance amplification.
Doppler radar: Speed cameras and weather radar use Doppler shift of reflected radio waves to measure velocity.