Vibrations and Waves Chapter 14

Vibrations and oscillations  Periodic motions ( )  Periodic motions ( like: uniform circular motion )  usually motions where restoring forces are present ( )  usually motions where restoring forces are present ( restoring forces : forces acting in opposite direction to the distortion )  example 1 : mass hanging on a spring ( )  example 1 : mass hanging on a spring ( restoring force--> elastic )  example 2: simple pendulum ( )  example 2: simple pendulum ( restoring force --> gravitational )  example 3: vibrating membrane or strings ( )  example 3: vibrating membrane or strings ( restoring force--> elastic )

1. Mass hanging on a spring b describing the motion (case with no friction) b new notions -equilibrium position (where there is no net force) -equilibrium position (where there is no net force) -amplitude (maximal deviation from equilibrium, -amplitude (maximal deviation from equilibrium, units: m) units: m) -one cycle (the object reaches its original position and -one cycle (the object reaches its original position and momentum) momentum) -period, T (the length of time for completing one cycle, -period, T (the length of time for completing one cycle, units: s) units: s) -frequency, f (number of cycles per unit time, -frequency, f (number of cycles per unit time, units: 1/s ) units: 1/s ) b period of the system - does NOT depend on amplitude ! - does NOT depend on amplitude ! - stiffness of the spring ( spring constant, k, - stiffness of the spring ( spring constant, k, units: N/m ) units: N/m ) -mass of the object, m (units: kg) -mass of the object, m (units: kg)

2. Pendulum b Describing the motion (case with no friction) b period - does NOT depend on the amount of swing (amplitude) - does NOT depend on the amount of swing (amplitude) - does NOT depend on the mass ! - does NOT depend on the mass ! - depends on its length, l (units: m) - depends on its length, l (units: m) -depends on the strength of gravity, g (units: m/s 2 ) -depends on the strength of gravity, g (units: m/s 2 )

Vibrations and oscillations: key to the clocks! b Early clocks: - motion of the heavens: Sun, Moon, stars - motion of the heavens: Sun, Moon, stars - flow of substance: water, sand - flow of substance: water, sand b pendulum clocks: - Christian Huygens, Christian Huygens, 1656 b spring clocks: - John Harrison, John Harrison, 1756 b today’s instruments: -electronic clocks () -electronic clocks (electronic oscillators) -atomic clocks --> using the frequency of atomic transitions -atomic clocks --> using the frequency of atomic transitions

Resonance b each oscillating system has its distinctive, natural frequency : f n b driving an oscillator with a frequency which is strongly different of this is usually not effective b driving the system with a frequency close to its natural frequency increases strongly the amplitude--> this phenomenon is called RESONANCE b resonance type increases can be achieved also by driving with with frequencies with integer parts of f n b complex systems have all natural frequencies (depends on the stiffness of material, the mass, size and form of the objects) b good effects of resonance: tuning the radio, tuning musical instruments, etc …. b bad effects of resonance: safety of buildings, bridges, airplanes etc.

Waves: Vibrations that Propagate b Simplest waves: perturbations (or pulses) that propagates b example: domino waves (just one pulse propagation possible, no mechanism for restoring the dominoes) b better example: a chain of balls connected by springs (existence of restoring forces) b other examples: tidal waves, earthquake, light, sound b although the waves travel, the individual particles vibrate around their equilibrium position b waves transport energy rather than matter b in real medium a part of the energy dissipates by friction b two basic type of waves: longitudinal and transverse waves

One dimensional waves in a rope b We study simple cases, but the results will be general ones b wave pulse on a rope - the speed can be changed by the tension in the rope - the speed can be changed by the tension in the rope - the speed depends on the linear density of the rope - the speed depends on the linear density of the rope - amplitude of the pulse have no effect on the pulse speed - amplitude of the pulse have no effect on the pulse speed b when a pulse hits the end that is attached to the post, it “bounces” off and heads back (reflection of the pulse) b if the incident pulse is an “up” pulse (crest), the reflected pulse is “down” pulse (trough)

Superposition of waves b we send a crest and when it reflects as a trough we send another crest to meet it b the two waves pass each other as if the other one were not there ! - very strange word if it would not be like this! - very strange word if it would not be like this! b during the time the waves pass through each other the resulting disturbance is a combination (superposition) of the two pulses b the displacement is the algebraic sum of the displacements of the two pulses

Periodic waves b Moving the rope up and down with a steady frequency and amplitude generates periodic waves b properties of periodic waves - frequency, f: : oscillation frequency of any piece in the medium - wavelength, l : the smallest distance for which the wave pattern repeats (distance between two adjacent crest or troughs) units: m - wavelength, l : the smallest distance for which the wave pattern repeats (distance between two adjacent crest or troughs) units: m - speed of the waves, v: traveling speed of a particular crest, units: m/s - speed of the waves, v: traveling speed of a particular crest, units: m/s

Standing waves b When a periodic wave is confined --> new effects b superposition of reflected waves with the original one--> standing waves b oscillating pattern that does not travel b understanding it by using the superposition principle b portions of the rope do not move at all : nodes b positions on the rope that have largest amplitudes: antinodes b different possible standing waves - all have the same speed but different frequency and wave- length - all have the same speed but different frequency and wave- length - fundamental mode and harmonics - fundamental mode and harmonics - longest wavelength: fundamental mode - longest wavelength: fundamental mode

Interference b Scientific term for the superposition of waves b we consider the 2D case, example: surface of liquids b we assume that there are two sources with the same frequency which oscillate in phase (both sources produce crests at the same time) b superposition--> creates interference patterns b bright regions produced by crests b dark regions produced by troughs b regions with large amplitude (where crest meet crest, or trough meets trough) --> antinodes b regions with little or no amplitude (crest meet trough)- -> nodes b the amplitude at a given point depends on the difference in the path lengths from the two sources  if the difference is a an integer multiple of l we have antinode  if the difference is an odd multiple of l/2 we have antinodes b nodes and antinodes form a complex pattern in space b nodal and antinodal lines

Diffraction b waves can spread out behind the barrier b this bending of waves around obstacles is called diffraction b the amount of diffraction depends on the relative sizes of the opening and the wavelength b if the wavelength is much smaller than the opening, very little diffraction is evident (harder to see it with light) b as the wavelength gets closer to the size of the opening, the amount of diffraction gets bigger b diffraction through many obstacles can produce again interference patterns

Home-Work assignments b Vibrations and Oscillations 358/1, 3-18; 361/1-10; 362/ /1, 3-18; 361/1-10; 362/11-12 b Waves 358/20-21; 359/22, 24-33; 360/34-51; 358/20-21; 359/22, 24-33; 360/34-51; 361/55-56, 58-60; 362/ /55-56, 58-60; 362/15-24

Summary b Vibrations and oscillations are described by: period, T (time required for one cycle); frequency, f (oscillations per unit time); f=1/T ; amplitude (maximum distance from the equilibrium) - examples: mass hanging on a spring and pendulum - examples: mass hanging on a spring and pendulum b all systems have a distinctive natural frequency b when a system is excited at a natural frequency --> resonance b waves are vibrations moving through the medium b waves can be transverse or longitudinal one  waves are characterized by: speed v, frequency f,and wavelength l; v=F l b waves pass through each other, when overlap the total displacement is given by the superposition (sum) of the individual waves b when periodic wave is confined resonant patterns -- standing waves - can be produced nodes (portions in the standing wave that do not move), antinodes (moves with the largest amplitude) ; fundamental standing wave and harmonics nodes (portions in the standing wave that do not move), antinodes (moves with the largest amplitude) ; fundamental standing wave and harmonics b two identical periodic wave sources with a constant phase difference produce an interference pattern which contains nodal and antinodal regions b waves do not go straight through openings or around barriers --> they suffer diffraction. The diffraction pattern depends on the relative sizes of the openings and the wavelength