Waves

A wave is a disturbance that transfers energy from one point to another without the transfer of matter. Waves are fundamental to music, optics, communication, and the structure of matter.

Types of Waves

Mechanical waves require a medium for propagation (sound, waves on a string, water waves).
Electromagnetic waves do not require a medium (light, radio, X-rays).

Transverse waves: Particle oscillation is perpendicular to wave propagation (e.g., waves on a string, EM waves).
Longitudinal waves: Particle oscillation is parallel to wave propagation (e.g., sound in air).

Wave Parameters

Wavelength (lambda): Distance between two successive crests (or compressions), unit: m.
Frequency (f): Number of complete oscillations per second, unit: Hz.
Time period (T): T = 1/f
Wave speed (v): v = f lambda = lambda / T
Angular frequency (omega): omega = 2 pi f
Wave number (k): k = 2 pi / lambda
Wave equation: y(x, t) = A sin(kx - omega t) (for a wave travelling in +x direction)

Speed of Waves

Speed of transverse waves on a string:
v = √(T/mu) where T = tension, mu = linear mass density (kg/m)

Speed of longitudinal waves in a medium:
v = √(B/rho) (in a fluid, B = bulk modulus, rho = density)
v = √(Y/rho) (in a solid rod, Y = Young's modulus)

Speed of sound in air (Newton's formula): v = √(P/rho) — gives 280 m/s (incorrect)
Laplace correction: Sound propagates adiabatically: v = √(gamma P/rho) ~ 331 m/s at 0C ✓

Speed of sound increases with temperature: v proportional to √(T).

Principle of Superposition

When two or more waves overlap, the resultant displacement is the algebraic sum of individual displacements:
y = y₁ + y₂

Reflection of Waves

At a rigid boundary (fixed end): Wave reflects with phase reversal (crest becomes trough).
At a free end: Wave reflects without phase change.

Standing Waves

When two identical waves travel in opposite directions, they form standing (stationary) waves.
y = 2A sin(kx) cos(omega t)

Positions of zero displacement: nodes (at x = 0, lambda/2, lambda, ...)
Positions of maximum displacement: antinodes (at x = lambda/4, 3lambda/4, ...)

Strings fixed at both ends (e.g., guitar string):
Fundamental (1st harmonic): f₁ = v / (2L) = (1/(2L)) √(T/mu)
nth harmonic: f_n = n f₁ (n = 1, 2, 3, ...)

Open organ pipe (open at both ends): Frequencies = n v / (2L), n = 1, 2, 3, ...
Closed organ pipe (closed at one end): Frequencies = (2n-1) v / (4L), n = 1, 2, 3, ... (odd harmonics only)

Beats

When two waves of slightly different frequencies f₁ and f₂ superpose:
Beat frequency = |f₁ - f₂|
Beats are periodic variations in loudness heard when two close frequencies are played together.

Doppler Effect

The apparent change in frequency due to relative motion between source and observer:
f' = f (v + v_o) / (v - v_s)

where v = speed of sound, v_o = observer speed (+ve toward source), v_s = source speed (+ve toward observer).

---

Example 1

A wave has frequency 440 Hz and wavelength 75 cm. Find wave speed.
v = f lambda = 440 x 0.75 = 330 m/s

Example 2

A string of length 60 cm, mass 3 g, is under tension 100 N. Find speed of transverse waves.
mu = (3 x 10^-3) / 0.6 = 5 x 10^-3 kg/m
v = √(T/mu) = √(100 / 5 x 10^-3) = √(20000) = 141.4 m/s

Example 3

Find fundamental frequency of a string (L = 0.5 m, mu = 5 x 10^-3 kg/m, T = 80 N).
v = √(T/mu) = √(80/5x10^-3) = √(16000) = 126.5 m/s
f₁ = v/(2L) = 126.5/(2 x 0.5) = 126.5 Hz

Example 4

An ambulance siren (f = 1000 Hz) approaches a stationary observer at 20 m/s. Speed of sound = 340 m/s. Find apparent frequency.
f' = f(v)/(v - v_s) = 1000 x 340/(340 - 20) = 340000/320 = 1062.5 Hz

Example 5

Two tuning forks of 256 Hz and 260 Hz are sounded together. Find beat frequency.
Beat frequency = |260 - 256| = 4 Hz. The sound waxes and wanes 4 times per second.

Example 6

Find the fundamental frequency of a closed organ pipe of length 0.5 m. Speed of sound = 340 m/s.
f₁ = v/(4L) = 340/(4 x 0.5) = 340/2 = 170 Hz

Example 7

Show that an open pipe of length L and a closed pipe of length L/2 have the same fundamental frequency.
Open: f_open = v/(2L). Closed: f_closed = v/(4 x L/2) = v/(2L). They are equal.

---

Common mistakes

Closed organ pipes produce only odd harmonics (1st, 3rd, 5th...). Open pipes produce all harmonics.
In the Doppler formula, take velocities as positive when toward each other and negative when away.
Beats arise from two close frequencies — do not confuse beats with standing waves.
Sound is a longitudinal wave; light is a transverse electromagnetic wave.

Summary

Waves transfer energy without transferring matter. Wave speed on a string depends on tension and mass density. Standing waves arise from superposition of opposing waves. Beats occur between close frequencies. The Doppler effect shifts perceived frequency with relative motion.