OSCILLATIONS AND WAVE MOTION

Oscillation/Periodic/Vibrational is a type of motion in which an object moves along the same path in a regular manner.

Wave pulse: Isolated disturbance travelling in undisturbed medium.

Continuous wave: Regular periodic disturbance to a medium

Progressive/Travelling wave: Movement of disturbance from a source transferring energy but not material to surrounding medium.

Transverse wave: Displacement of particles of medium perpendicular to direction of wave motion

EM waves:

-       Transverse waves

-       Travel through vacuum

-       Two sinusoidal fields (E and B) in phase, perpendicular

-       Undergo reflection, refraction, interference, diffraction, polarization

(Gamma – X – UV – Visible – Infra-red – Micro – Radio) –> increasing

Longitudinal wave: Displacement of particles of medium parallel to direction of wave motion. (series of compressions and rarefactions)

For hammer hitting a rod, transmitted wave pulse in rod is longitudinal: Hammer causes the atoms of the rod to vibrate in the same direction of the horizontal motion of the hammer.

Wavefront: Line joining points of a wave in phase

Wavelength: Distance between to points in phase

Ray: Direction of wave motion

Wave speed: Distance wave profile moves per unit time (speed of oscillating particles within wave)

Intensity: Rate of flow of energy per unit cross-sectional area perpendicular to direction of wave motion, Wm-2.

Graph of distance x (from source) moved by particles against time:

Direction of motion always towards equilibrium

For longitudinal wave, x-axis intercepts are alternating points of compression and rarefaction.

Graph of displacement d of particles against distance x from the source at a particular moment in time:

Can be used to determine wavelength.

To determine direction of motion/velocity of particles, draw 2nd wave slightly displaced to the right along the wave motion.

For both graphs, acceleration always directed towards equilibrium point. Hence by comparing direction of velocity and acceleration, can determine if a wave a speeding up or slowing down at that point.

To identify regions of compression and rarefaction for longitudinal wave:

  1. C, R must lie on x-intercept
  2. Determine point where particles to the right and left are moving towards equilibrium à compression
  3. Determine point where particles to the left and right are moving away à rarefaction

Beat frequency:

When two waves of nearly the same frequency interfere, the intensity of the resulting wave varies with a frequency called the beat frequency.

Rate at which volume is heard to be oscillating from high to low volume, equal to difference in frequency of two notes which interfere to produce beats (sounds from two sources e.g. two tuning forks, produce sounds with slightly different frequencies that interfere to produce detectable beats)

Hence 2Hz for beat frequency = When 2 complete cycles of high and low volumes heard every second

For car B moving away with frequency fB and car A moving towards observer with frequency fA, beat frequency heard = fA – fB

For situation involving two tuning forks: To decrease beat frequency/ difference in frequency, add a weight e.g. gum to the mass of the higher frequency tuning fork. The weight increases the mass of the prong and hence decreases its vibrational frequency, lowering its frequency.

Simple Harmonic Motion

A type of oscillation (transverse wave) in which acceleration/net force acting is directly proportional to displacement from equilibrium position and acceleration/net force acting is always directed towards this position.

Displacement: Distance from rest position to stated direction, a vector.

Amplitude: Maximum magnitude of displacement from equilibrium position, scalar.

Frequency: No. of complete oscillations per unit time.

Period: Time taken for one complete oscillation.

Angular frequency

SHM for object starting at equilibrium position/ Graph of x, v, a against time t:

SHM for objecting starting at amplitude/ Graph of x, v, a against time t::

Graph of velocity against displacement x of object:

Velocity is maximum when object at equilibrium position,

Velocity is minimum/zero when object at amplitude.

Graph of acceleration against displacement x of object:

Acceleration is maximum when object at amplitude.

Acceleration is minimum/zero when object at equilibrium position.

Phase difference: Phase angle between two oscillations which have the same frequency.

Graph of energy against displacement x of SHM motion:

INSANELY IMPORTANT TO KNOW!!!!!!!!!!!

Hence

Thus for mass moving on a horizontal spring on a frictionless surface with amplitude A and time period T and energy E…

If amplitude is doubled to 2A, time period stays constant, however energy is 4E.

Resonance (SHM WAVES):

Damped oscillation: An SHM that is subjected to frictional or other dissipative forces.

Light-damping: Minimal dissipative force, definite oscillations, amplitude decays exponentially with time

Heavy-damping: No oscillations occur, amplitude decays very slowly over time

Critical damping: Suitable dissipative conditions, returns to equilibrium in shortest possible time

Application: Shock absorbers in cars use slightly under critical damping. Light-damping à Too bouncy. Over-damping à Car cannot respond fast enough to further bumps in the road.

Natural frequency: Frequency of an oscillating system with no external force

Forced oscillation: Oscillating system in which periodic force is applied

Driving frequency: Frequency of a periodic force

Resonance: Forced oscillation where driving frequency equals the natural frequency, resulting in maximum energy transfer from the periodic force to the system, causing it to oscillate with maximum amplitude.

Microwave oven: Frequency of microwaves corresponds to natural frequency of water molecules in food. Resonance occurs, molecules in food absorb energy and heat up.

Radio station: Natural frequency of electrical receiving circuit in radio matches frequency of radio waves emitted by station. Resonance occurs and required frequency is isolated and amplified.

Wind instruments: Resonance occurs between vibration of air columns and vibrating reeds, amplify the note.

Bridge collapse: When frequency of external driving force (cars passing over, wind) matches natural frequency of the mechanical bridge structure, resonance occurs and resulting oscillation may cause structure to rupture.

Aircraft: Designers must ensure that natural frequency of wing vibration does not match angular frequency of the engines at cruising speed.

Motion sickness: When subsonic frequencies match natural frequency of internal organs in human body. Giddiness, blurred vision.

Barton’s pendulum: Used to demonstrate forced oscillation and resonance. Driver pendulum released, after some time pendulums oscillate with very nearly the same frequency but at different amplitudes.

Reflection and Transmission:

When waves pass through a medium, they are reflected, transmitted, or absorbed.

Obstacle/fixed/rigid end: Inverted (shape) reflection. Wall exerts an equal and opposite force to resist movement; a short impulse. From less dense to denser, hence phase change occurs for reflected pulse. (No transmission if boundary is infinitely dense; total reflection)

Free end: Upright (shape) reflection. From denser to less dense, hence reflected pulse no phase change.

(For both, inverted sequence of reflected pulse, and transmitted pulse has no phase change.)

Transmission: DIFFRACTION OF WAVES:

Bending/Spreading of waves passing an obstacle or through an aperture

Observable only when

IMPORTANT:

Sound wave has long wavelength hence can diffract easily (hence sounds can be heard from different rooms). However light has very short wavelength hence can only be diffracted through very small openings (card pinhole, fine silk handkerchief).

Ripple tank: Used to demonstrate diffraction of waves.

Principle of superposition: When two or more waves of same kind exist simultaneously in a medium, resultant displacement of the waves is the vector sum of the displacement due to each wave acting independently

Interference: When two or more coherent sources superpose one another producing a resultant wave/ When two waves meet at a point the amplitude of the resultant wave is the vector sum of the two waves acting independently.

An interference pattern consisting of clear lines of constructive and destructive interference is seen.

Coherent sources:

-       Same frequency

-       Constant phase difference

For observable interference pattern:

-       Coherent sources (hence saying same frequency and constant phase difference only counts as one point)

-       Roughly the same amplitude/intensity (for the sources)

-       Same plane of polarization

-       Waves from source meet at a point

Constructive interference: Waves arrive in phase at the same point, and the resultant displacement due to the vector sum of the two waves acting separately gives a maximum displacement.

Path difference of the waves from the two sources is zero/ differ by

Destructive interference: Waves arrive out of phase at the point, path difference of the waves from the two sources is

Explaining why as source moves relative to another stationary source, sound detected from both decreases and increases in intensity: Sound from two sources undergo interference. As source moves, path difference of sound from the two sources changes. When the pd is an integral number of wavelengths, CI occurs and sound increases in intensity. When pd is an odd integral number of half wavelengths, DI occurs and intensity decreases.

Standing wave (SOUND/WATER WAVES):

How is standing wave formed? As tube is vibrated, wave travels along tube and is reflected at B. On reflection, the wave becomes inverted. The reflected wave interferes with the second incoming wave from A. Maximum displacement thus occurs at the midpoint between A and B. Since there are always nodes at A and B, the resulting pattern is produced. The nodes do not appear to change with time, wave appears to not be moving, hence a seeming ‘standing’ wave.

Formed by superposition of two waves of same type, amplitude and frequency, but travelling in opposite directions. Know how to state conditions necessary for formation of standing wave (max 2).

Travelling and reflected waves interfere, resulting in seeming standing still of waves

Nodes: Points of DI, no displacement

Antinodes: Points of CI, maximum displacement

Resonant frequencies/Fundamental and harmonics: Frequencies of standing waves

Vibrating air columns:

-       For tube open at both ends:  and all harmonics present

-       For tube open at one end:  and only odd harmonics present

Open at both ends: Closed at one end: Closed at both ends:

(n+1) harmonic = n overtone

Stationary/Standing Travelling
Amplitude Amplitude varies with time All points have same amplitude
Energy Energy not transmitted along the string Energy transmitted
Wave pattern Does not move Moves
Frequency All points have same frequency All points have same frequency
Phase Same phase between all points/nodes All particles have different phases
Wavelength Double the distance between two nodes Distance between two points in phase

Maximum kinetic energy of each segment of the string is proportional to the square of the amplitude of the segment.

For tube closed at one end with fine powder sprinkled along its length and source of sound placed at open end of tube, powder will be seen to form equally spaced heaps in the tube. A standing wave has been set up in the tube causing heaps to form at the nodes (points of DI, no displacement) and the powder to be pushed away at the antinodes (points of CI, max displacement).

Increasing the temperature for the same frequency of the sound increases the separation of the heaps in the tube. The wavelengths of the sound waves have increased for the same frequency, indicating that the speed of sound increases when the temperature rises.

Doppler effect (SOUND WAVES):

The change in observed frequency of a wave due to relative motion between a source of sound and the observer (and the medium)

(wavelength remains the same, detected wave speed different)

Hence for moving observer: Apparent frequency heard by the observer will be lower, as the observer moves away from the source, the relative velocity of the source and the observer is smaller.

When drawing convergence of the waves, remember that the wavefronts cannot overlap.

 

Next: Chapter 7: Diffraction and Polarization

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