'Waves'

Introduction to Wave Motion

A wave is a disturbance that transfers energy from one point to another without the transfer of matter.

Types of Waves

Mechanical waves: Require a medium for propagation. Examples: sound waves, water waves, waves on a string.

Electromagnetic waves: Do not require a medium. Examples: light, radio waves, X-rays.

Matter waves: Associated with moving particles (de Broglie waves).

Transverse and Longitudinal Waves

Transverse Waves

Particles of the medium vibrate perpendicular to the direction of wave propagation.

• Crests (maxima) and troughs (minima).
• Examples: waves on a string, light waves.

Longitudinal Waves

Particles of the medium vibrate parallel to the direction of wave propagation.

• Compressions (high pressure) and rarefactions (low pressure).
• Examples: sound waves, seismic P-waves.

Comparison Table

Speed of Wave

Speed of Transverse Wave on a String

v = sqrt(T/mu)

Where T = tension in the string, mu = mass per unit length.

Speed of Longitudinal Wave in a Solid

v = sqrt(Y/rho) (Y = Young's modulus, rho = density)

Speed of Sound in a Gas

v = sqrt(gamma P/rho) = sqrt(gamma RT/M)

For air at STP: v approx 332 m/s. Speed increases with temperature: v_t = v_0 sqrt((t+273)/273).

Progressive Wave Equation

For a wave travelling in the positive x-direction: y(x,t) = A sin(omega t - kx + phi)

Parameters

• A = amplitude
• omega = angular frequency (= 2pi/T = 2pi f)
• k = wave number (= 2pi/lambda)
• lambda = wavelength
• T = time period
• f = frequency
• phi = initial phase

Relation Between Parameters

v = f lambda = omega/k

Superposition Principle

When two or more waves overlap, the resultant displacement is the vector sum of the individual displacements.

y = y_1 + y_2 + y_3 + ...

Reflection of Waves

Reflection from Fixed End

The reflected wave undergoes a phase change of pi (180 degrees). Crest reflects as trough.

Reflection from Free End

No phase change upon reflection. Crest reflects as crest.

Transmission at Boundary

Part of the wave transmits and part reflects. The amplitudes depend on the properties of the two media.

Standing Waves (Stationary Waves)

When two identical waves travel in opposite directions, they produce standing waves.

Equation: y(x,t) = 2A sin(kx) cos(omega t)

Nodes and Antinodes

• Nodes: Points of permanent rest (displacement always zero). x = n lambda/2.
• Antinodes: Points of maximum displacement. x = (2n+1) lambda/4.

Distance between successive nodes = lambda/2. Distance between successive antinodes = lambda/2.

Standing Waves in Stretched Strings

Fundamental mode (1st harmonic): f_1 = (1/2L) sqrt(T/mu). nth harmonic: f_n = n f_1 = n/(2L) sqrt(T/mu).

Allowed frequencies: f_n = nv/(2L), where n = 1, 2, 3, ....

Standing Waves in Organ Pipes

Open pipe: f_n = nv/(2L) (all harmonics present). Closed pipe: f_n = nv/(4L) (only odd harmonics: n = 1, 3, 5, ...).

Beats

When two waves of nearly equal frequencies f_1 and f_2 superpose, the intensity varies periodically. This phenomenon is called beats.

Beat frequency: f_beat = |f_1 - f_2|

Applications:

• Tuning musical instruments.
• Detecting presence of ultrasound.
• Determining unknown frequency.

Doppler Effect

The apparent change in frequency of a wave due to relative motion between source and observer.

For Sound Waves

f' = f (v +- v_O)/(v -+ v_S)

Where v = speed of sound, v_O = observer velocity, v_S = source velocity.

• Observer moving towards source: numerator v + v_O (plus).
• Source moving towards observer: denominator v - v_S (minus).
• Signs reverse for moving away.

Applications

• Radar speed detection (using EM waves).
• Red shift in astronomy (distant galaxies).
• Medical ultrasound (blood flow measurement).

Worked Examples

Example 1: A string of length 2 m and mass 5 g has tension 100 N. Find wave speed. Solution: mu = 0.005/2 = 0.0025 kg/m. v = sqrt(100/0.0025) = sqrt(40000) = 200 m/s.

Example 2: Fundamental frequency of a 1 m string under 50 N tension is 100 Hz. Find mass per unit length. Solution: f_1 = (1/2L) sqrt(T/mu) => 100 = 1/2 sqrt(50/mu) => sqrt(50/mu) = 200 => 50/mu = 40000 => mu = 0.00125 kg/m.

Example 3: Two tuning forks produce 4 beats/s. One is 256 Hz. The other is? Answer: f = 256 +/- 4 = 260 Hz or 252 Hz.

Common Mistakes

1. Phase change on reflection: Fixed end gives pi phase change; free end gives none.
2. Open vs closed pipe: Open pipe has all harmonics; closed pipe has only odd harmonics.
3. Doppler effect signs: Carefully determine relative motion before substituting.
4. Particle vs wave velocity: Particle velocity is at a point; wave velocity is the speed of energy transfer.

ISC Exam Focus

• Theory (70%): Wave equation, standing waves, beats, Doppler effect derivations.
• Application (30%): Numerical problems on string harmonics, organ pipes, Doppler effect.
• ISC frequently asks: "Derive expression for standing waves in a stretched string" or "frequencies of organ pipes".
• Doppler effect numericals and beat frequency problems are common.

Self-Test Questions

Q1: Distinguish between transverse and longitudinal waves. Answer: Transverse: particle motion perpendicular to wave direction (e.g., string waves). Longitudinal: particle motion parallel (e.g., sound).

Q2: A wave travels at 340 m/s with frequency 256 Hz. Find wavelength. Answer: lambda = v/f = 340/256 = 1.328 m.

Q3: A 50 cm string has fundamental frequency 200 Hz. Find speed of wave. Answer: v = 2Lf = 2*0.5*200 = 200 m/s.

Q4: An open organ pipe of length 40 cm produces sound. Find fundamental frequency (v = 340 m/s). Answer: f_1 = v/(2L) = 340/(0.8) = 425 Hz.

Q5: A source of 500 Hz moves towards a stationary observer at 30 m/s. Find apparent frequency (v = 340 m/s). Answer: f' = f * v/(v - v_S) = 500 * 340/310 = 548.4 Hz.

Q6: What are beats? Two tuning forks of 256 Hz and 260 Hz are sounded together. Find beat frequency. Answer: Beats are periodic variations in intensity. Beat frequency = |260-256| = 4 Hz.