Free Oscillations of Simple Systems

• Introduction
• A few definitions
• Equilibrium and concept of potential well
• Motion of a particle in a parabolic potential well (S.H.M.) and its solution
• Dynamical quantities associated with S.H.M.
• Energy of simple harmonic oscillator
• Free oscillations of physical systems

Spring and mass system
Simple pendulum
Helmholtz resonator
Compound pendulum
Oscillations of a LC circuit

• Effective force constant of springs

Springs in parallel
Springs in series

• Linear and transverse oscillations of a mass attached to
  two springs

Linear or longitudinal oscillations
Transverse oscillation

• Oscillations of two masses attached to a spring
• Diatomic molecule and reduced mass

Damped and Forced Oscillations

• Introduction
• Equation of damped harmonic oscillator and its solution

Over damping or heavy damping (r² > ω₀²)
Critical damping (r² = ω₀²)
Under damping or low damping (r² < ω₀²)

• Energy & power dissipation of a damped harmonic oscillator
• Relaxation time and quality factor of damped harmonic oscillator
• Examples of damped harmonic oscillator

LCR circuit
Moving coil galvanometer

• Forced harmonic oscillator or driven harmonic oscillator
• Equation of forced harmonic oscillator and its solution
• Phase of forced oscillator
• Amplitude of forced oscillator
• Amplitude and sharpness of resonance
• Comparison of amplitude and phase at different frequencies
• Velocity of a forced oscillator
• Kinetic energy of a forced oscillator and energy resonance
• Power absorbed and power dissipated by forced oscillator

Power absorbed by forced oscillator
Power dissipated by forced oscillator
Maximum power absorption

• Band width of resonance and quality factor
• Examples of resonance

Number of pendulums connected by a string
Electrical resonance in LCR circuit

• Transient and steady state of oscillations

Free Oscillations of System with Two Degrees of Freedom

• Introduction
• Two coupled pendula or oscillators
• The general method of finding normal modes (or coordinates) and the superposition of normal modes
• Energy exchange (Beats) in coupled oscillations
• Transverse oscillations of a system of many coupled oscillators
• Longitudinal oscillations of many coupled oscillators
• Differentiate equation of a continuous system (stretched string) and normal modes or oscillations of N-particles in continuous approximation and wave equation

Fourier Theorem and its Applications

• Fourier’s theorem
• Exponential form of Fourier series
• Analysis of a rectangular wave or harmonic function by Fourier’s theorem
• Analysis of a square wave by Fourier’s theorem
• Analysis of a saw tooth wave or triangular wave by Fourier’s theorem
• Analysis of a half wave rectifier using Fourier’s theorem
• Analysis of full wave rectifier using Fourier’s theorem
• Fourier integral
• Fourier integral for an odd function
• Fourier integral for an even function
• Complex form of Fourier integral
• Fourier transform
• Applications of Fourier transform

Slit function or top Hat
Gaussian function

• Wave train of finite length, constancy of uncertainty product (line breadth, radiative damping of a charged harmonic oscillator)

Wave equation and Waves in Continuous Media

• Introduction
• Types of wave motion
• Wave equation
• A plane progressive wave
• Classical wave equation (or differential equation of wave motion) wave velocity
• Equation of a plane progressive wave
• Energy, energy density, energy flux and intensity of a plane progressive harmonic wave or energy transmission in wave
• Examples of one-dimensional travelling waves

Speed of transverse waves in a uniform string
Speed of longitudinal waves in a solid bar or rod
Speed of longitudinal waves in gases or fluids (Equation of motion of pressure wave)

• Calculation of velocity of sound in air: Newton’s formula and Laplace correction
• Effect of various factors on velocity of sound

Effect of pressure
Effect of temperature
Effect of molecular weight

• General solution of classical wave equation
• Boundary conditions and normal modes
• Dispersion relation (waves in absorbing medium)
• Dispersion waves
• Phase velocity and group velocity: Relation between group velocity and phase velocity
• Dispersive and non-dispersive medium
• Dynamics of number of oscillators

Wave motion on a one-dimensional monoatomic lattice
Vibrational modes of a finite one-dimensional lattice of identical atoms
The vibrational modes of a diatomic linear lattice

Superposition of Waves

• Principle of superposition of waves
• Limitations of superposition principle
• Phenomena arising from superposition of waves
• Interference
• Interference and conservation of energy
• Sustained interference
• Incoherent sources
• Interference of sound waves
• Experiment for study of interference in sound waves: Quincke’s tube and Tuning fork
• Practical applications of interference of sound
• Interference of light
• Young’s double slit experiment
• Beats
• Distinction between interference and beats
• Beats and amplitude modulation
• Combination tones

Ultrasonic

• Introduction
• Production of ultrasonic waves
• Piezo electric effect and quartz crystal
• Production of ultrasonic waves: Piezo-electric method and Magnetostriction method
• Detection of ultrasonic waves
• Applications of ultrasonic waves

Vibrations in Bounded Systems

• Introduction
• Transverse vibration (or wave) of stretched string
• Modes of vibration in a stretched string
• Laws of vibrations of strings

Law of length
Law of tension
Law of mass
Sonometer

• Longitudinal vibrations of a bar

A fixed bar i.e., a bar clamped (or rigidly fixed) at both ends
A free-free bar i.e., a bar free at both ends
Bar clamped at the middle
A fixed-free bar i.e., a bar clamped at one end and free at the other end

• Transverse vibrations of rods or bars
• Solution of transverse wave equation of a bar

Bar clamped at one end and free at the other end i.e., a clamped-free or fixed-free bar
A bar free at both ends i.e., a free-free bar

• Vibrations of tuning fork
• Vibrations of air columns or gas columns

Open organ pipe
Closed organ pipe
End correction

• Galton’s whistle
• Chladni’s figure
• Vibration of a drum or rotating drum method

Characteristic of Sound and Musical Instruments

• Characteristic of sound
• Characteristics of musical sound

Loudness or intensity
Pitch or frequency
Quality or timbre

• Unit of loudness (bel and decibel)
• Intensity of sound
• Sound pressure level
• Consonance and dissonance
• Musical scale: Diatonic scale and Equally tempered (temperament scale)
• Musical instruments: Violin, Dilruba, Sitar, Guitar, Harmonium and Tabla

Reflection, Refraction and Diffraction of Sound

• Introduction
• Reflection of sound
• Laws of reflection of wave on wave theory
• Reflection of sound from a plane surface
• Reflection of sound from a concave surface
• Some applications of reflection of sound
• Echo
• Echo Sounding
• Phase change on reflection
• Refraction of sound waves
• Examples of refraction of sound in atmosphere

Refraction due to change in temperature
Refraction of sound due to difference of velocity of air

• Silent zone
• Refraction of a plane wavefront at a plane surface
• Diffraction of sound
• Fresnel’s treatment of the wavefront
• Fresnel’s half period zones and rectilinear propagation of light

Construction of half period zones
Radii and area of zones
Resultant distribution at P due to whole wavefront

• Sound ranging
• Characteristic impedance of a string
• Reflection and transmission of waves on a string at a boundary
• Reflection and transmission of wave energy
• Reflection and transmission of energy coefficient
• Impedance matching for transducer
• Acoustic impedance
• Reflection and transmission of sound wave at a boundary between two media
• Intensity of reflected and transmitted sound
• SONAR

Transducer and Acoustics of Buildings

• Introduction of transducer
• Classification of transducer
• Ideal characteristics of a transducer
• Recording and reproduction of sound

Recording of sound
Disc reproduction

• Tape recorder or magnetic sound recording and reproduction
• Recording of sound on cinema films: Variable area method and Variable density method
• Reproduction of sound from cinema films
• Acoustics of buildings
• Reverberation
• Absorption of sound

Absorption coefficient for sound
Sound absorbing materials

• Transmission of sound
• Factors affecting the architectural acoustics

Reverberation time
Focusing of sound waves
Adequate loudness

• Sabine formula
• Optimum reverberation
• Echelon effect