Oscillations and Waves

Oscillations and waves describe periodic motion and energy transfer without matter transport.

Overview

graph TD
    A[Oscillations & Waves] --> B[SHM]
    A --> C[Wave Motion]
    B --> B1[Simple Pendulum]
    B --> B2[Spring System]
    C --> C1[Transverse]
    C --> C2[Longitudinal]
    C --> C3[Standing Waves]

Simple Harmonic Motion (SHM)

Interactive Demo: Spring-Mass Oscillation

Watch how a mass on a spring oscillates. Adjust amplitude and frequency to see how they affect the motion:

Definition

Motion in which acceleration is proportional to displacement and directed towards mean position:

$$\boxed{a = -\omega^2 x}$$

Equations of SHM

$$x = A\sin(\omega t + \phi)$$

or

$$x = A\cos(\omega t + \phi)$$ $$v = A\omega\cos(\omega t + \phi) = \omega\sqrt{A^2 - x^2}$$ $$a = -A\omega^2\sin(\omega t + \phi) = -\omega^2 x$$

Key Relations

$$\boxed{\omega = \frac{2\pi}{T} = 2\pi f}$$ $$\boxed{v_{max} = A\omega}$$

(at mean position)

$$\boxed{a_{max} = A\omega^2}$$

(at extreme position)

Energy in SHM

$$KE = \frac{1}{2}m\omega^2(A^2 - x^2)$$ $$PE = \frac{1}{2}m\omega^2 x^2$$ $$\boxed{E_{total} = \frac{1}{2}m\omega^2 A^2 = \frac{1}{2}kA^2}$$

Properties:

• KE maximum at mean position
• PE maximum at extreme position
• Total energy constant

Simple Pendulum

$$\boxed{T = 2\pi\sqrt{\frac{L}{g}}}$$

For small oscillations (θ < 10°)

Factors Affecting T

• T increases with L
• T decreases with g
• T independent of mass and amplitude

Spring-Mass System

$$\boxed{T = 2\pi\sqrt{\frac{m}{k}}}$$

Combinations of Springs

Series:

$$\frac{1}{k_{eq}} = \frac{1}{k_1} + \frac{1}{k_2}$$

Parallel:

$$k_{eq} = k_1 + k_2$$

Wave Motion

Types of Waves

Type	Particle Motion	Examples
Transverse	Perpendicular to propagation	Light, string waves
Longitudinal	Parallel to propagation	Sound, spring waves

Wave Equation

$$\boxed{y = A\sin(kx - \omega t + \phi)}$$

where:

• $k = \frac{2\pi}{\lambda}$ (wave number)
• $\omega = 2\pi f$ (angular frequency)
• $v = f\lambda = \frac{\omega}{k}$ (wave velocity)

Wave Velocity

On string:

$$v = \sqrt{\frac{T}{\mu}}$$

where T = tension, μ = mass per unit length

Sound in gas:

$$v = \sqrt{\frac{\gamma P}{\rho}} = \sqrt{\frac{\gamma RT}{M}}$$

Superposition of Waves

Principle

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

Interactive Demo: Wave Superposition

See how two waves combine. Adjust frequencies and phase difference to observe interference and beats:

Interference

Constructive: Phase difference = 0, 2π, 4π… (Path difference = 0, λ, 2λ…)

$$A_{max} = A_1 + A_2$$

Destructive: Phase difference = π, 3π… (Path difference = λ/2, 3λ/2…)

$$A_{min} = |A_1 - A_2|$$

Beats

When two waves of slightly different frequencies superpose:

$$\boxed{f_{beat} = |f_1 - f_2|}$$

Standing Waves

Formed by superposition of two waves of same frequency traveling in opposite directions.

Equation

$$y = 2A\sin(kx)\cos(\omega t)$$

Nodes and Antinodes

• Nodes: Points of zero amplitude (destructive interference)
• Antinodes: Points of maximum amplitude (constructive interference)

Distance between consecutive nodes = λ/2

Standing Waves in Strings

Fixed at Both Ends

$$\lambda_n = \frac{2L}{n}$$ $$\boxed{f_n = \frac{n}{2L}\sqrt{\frac{T}{\mu}} = nf_1}$$

n = 1, 2, 3… (harmonics)

Standing Waves in Organ Pipes

Closed Pipe (One end closed)

Only odd harmonics:

$$\boxed{f_n = \frac{(2n-1)v}{4L}}$$

n = 1, 2, 3… (1st, 3rd, 5th harmonics)

Open Pipe (Both ends open)

All harmonics:

$$\boxed{f_n = \frac{nv}{2L}}$$

n = 1, 2, 3… (all harmonics)

Doppler Effect

Apparent change in frequency due to relative motion between source and observer.

General Formula

$$\boxed{f' = f\left(\frac{v \pm v_o}{v \mp v_s}\right)}$$

Sign convention:

• Upper signs: approaching
• Lower signs: receding

Practice Problems

1. A particle executes SHM with amplitude 4 cm and time period 4 s. Find maximum velocity.

2. A spring of constant 400 N/m has a 0.1 kg mass attached. Find time period.

3. A string of length 1 m vibrates in its third harmonic. If tension is 100 N and mass per unit length is 0.01 kg/m, find frequency.

4. A train approaches a platform at 20 m/s blowing a whistle of 500 Hz. Find frequency heard by observer on platform. (v = 340 m/s)

Further Reading

• Properties of Solids and Liquids
• Electromagnetic Waves