Sound in Music

Musical instruments exploit standing waves. A sitar string (both ends fixed) vibrates at $f_n$ = $\frac{nv}{2L}$. Harmonics above the fundamental create timbre. Tightening the string increases tension T → increases v → increases f (higher pitch).

SONAR (Sound Navigation And Ranging)

Submarines emit an ultrasonic pulse; the echo return time t gives the depth: d = v·t/2 (factor of 2 because sound travels to object and back). Speed of sound in seawater ≈ 1500 m/s. Used to map ocean floors and detect submarines.

Medical Ultrasound

Frequencies 1–20 MHz (above human hearing) penetrate soft tissue. Impedance mismatch at tissue boundaries causes partial reflection; echoes are timed to construct cross-sectional images. Doppler ultrasound measures blood flow velocity using the Doppler shift.

Seismic Waves

Earthquakes generate P-waves (longitudinal, travel through all media, faster ~8 km/s) and S-waves (transverse, travel only through solids, slower ~4 km/s). The P–S arrival time difference locates the epicentre. S-waves cannot pass through Earth's liquid outer core, revealing its structure.

Noise-Cancelling Headphones

Generate a wave with equal amplitude but opposite phase (π phase shift) to incoming sound. By superposition, the two waves cancel: $y_{cancel}$ = A sin(kx − ωt) + A sin(kx − ωt + π) = 0.