Vibrations and Waves Chapter 14 Vibrations and oscillations

Vibrations and Waves Chapter 14

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

1. Mass hanging on a spring b b b describing the motion (case with no friction) new notions -equilibrium position (where there is no net force) -amplitude (maximal deviation from equilibrium, units: m) -one cycle (the object reaches its original position and momentum) -period , T (the length of time for completing one cycle, units: s) -frequency, f (number of cycles per unit time, units: 1/s ) period of the system - does NOT depend on amplitude ! - stiffness of the spring ( spring constant, k, units: N/m ) -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 mass ! - depends on its length, l (units: m) -depends on the strength of gravity, g (units: m/s 2)

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

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

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

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

Superposition of waves b b we send a crest and when it reflects as a trough we send another crest to meet it the two waves pass each other as if the other one were not there ! - very strange word if it would not be like this! during the time the waves pass through each other the resulting disturbance is a combination (superposition) of the two pulses 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 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 - speed of the waves, v: traveling speed of a particular crest, units: m/s b

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

Interference b b b b Scientific term for the superposition of waves we consider the 2 D case, example: surface of liquids we assume that there are two sources with the same frequency which oscillate in phase (both sources produce crests at the same time) superposition--> creates interference patterns bright regions produced by crests dark regions produced by troughs regions with large amplitude (where crest meet crest, or trough meets trough) --> antinodes regions with little or no amplitude (crest meet trough)-> nodes 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 and antinodes form a complex pattern in space nodal and antinodal lines

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

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

Summary b b b b 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 all systems have a distinctive natural frequency when a system is excited at a natural frequency --> resonance waves are vibrations moving through the medium waves can be transverse or longitudinal one waves are characterized by: speed v, frequency f , and wavelength l; v=Fl waves pass through each other, when overlap the total displacement is given by the superposition (sum) of the individual waves 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 two identical periodic wave sources with a constant phase difference produce an interference pattern which contains nodal and antinodal regions 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