WAVES

1. WAVES • wave = disturbance that propagates • “disturbance” e.g., displacement of medium element from its equilibrium position; • propagation can be in medium or in space (disturbance of a “field”); • mechanical waves: • when matter is disturbed, energy emanates from the disturbance, is propagated by interaction between neighboring particles; this propagation of energy is called wave motion; • a traveling mechanical wave is a self-sustaining disturbance of a medium that propagates from one region to another, carrying energy and momentum. • examples: • waves on strings, • surface waves on liquids, • sound waves in a gas (e.g. in air), • compression waves in solids and liquids; • it is the disturbance that advances, not thematerial medium • transverse wave displacements perpendicular to direction of propagation; • longitudinal wave sustaining medium displaced parallel to direction of propagation (e.g. sound waves, some seismic waves, compression waves in a bell);

2. periodic wave motion • periodic wave motion: • particles oscillate back and forth, same cycle of displacement repeated again and again; (we only discuss periodic waves) • terms describing waves: • crest of the wave = position of maximum displacement (“highest point of the wave”) • amplitude = amount of maximum displacement (height of crest above undisturbed position) • wave velocity v = velocity of propagation of wave crest • wavelength = distance between successive same-side crests • frequency f = number of same-side crests passing by a fixed point per second • period T = time for one complete wave oscillation: period = 1/frequency • unit of frequency: 1 hertz = 1Hz = 1/second • wave velocity (speed of waves) depends on properties of the carrying medium; in general: speed of mechanical waves in solids greater than in liquids, and greater in liquids than in gases. • relation between speed, wavelength and frequency: v = f  , i.e. speed = frequency times wavelength

3. Energy in a wave • intensity of a wave is a measure of how much power is transported to a point by the wave; • intensity = energy flow per unit time, per unit area = power per unit area, (where area = area perpendicular to propagation direction) • energy flow carried by wave: is proportional to the square of the amplitude and the square of the frequency; • “inverse square law of wave intensity”: the intensity of a wave is inversely proportional to the square of the distance from the source of the wave I = P/(4R2) (source = object emitting the wave) (I = intensity, P = total power emitted by source, R = distance from source) (strictly speaking, only for point-like or spherically symmetric sources, or if size of the source much smaller than R)

4. Superposition of waves, interference • Superposition principle: • two or more waves moving through the same region of space will superimpose and produce a well-defined combined effect; the resultant of two or more waves of the same kind overlapping is the algebraic sum of the individual contributions at each point, i.e. the (signed) displacements (elongations) add. • Huygens' principle • every point on a wavefront can be considered as a source, emitting a wave; the superposition of all these waves results in the observed wave. • consequences: interference, diffraction • interference: superposition of two waves of same frequency can lead to reinforcement (constructive interference) or partial or complete cancellation (destructive interference; • constructive interference: two waves “in phase”, (i.e. crests of two waves coincide in time) reinforce each other, resultant amplitude bigger than that of individual waves; • destructive interference: two waves “completely out of phase” (i.e. out of phase by 1/2 period, so that crests of one wave coincide with troughs of the other)  cancellation; complete cancellation (extinction) if both waves have same amplitude.

5. Interference, cont’d • phase differences can be caused by: • differences in pathlength; given a pathlength difference, the phase difference depends on the wavelength; • travel time difference due to difference in speed in different media; • reflection; • examples: • colors of thin films (oil on water, soap bubbles) • dead spots in auditorium • diffraction grating: • many narrow parallel slits spaced closely together; • every slit forms source for wave; • differences in pathlength from different slits to some point in space  phase difference  wavelength dependent interference pattern; • can be used to measure wavelength; • interferometers: • Michelson - Morley (used to measure “ether wind”) • Fabry - Perot

6. SOUND • Sound waves propagate in any medium that can respond elastically and thereby transmit vibrational energy. • sound waves in gases and liquids are longitudinal (alternating compression and rarefaction); in solids, both longitudinal and transversal; • speed of sound is independent of frequency; • speed of sound in air  340m/s at 20o C; increases with temperature;  1500m/s in water; • three frequency ranges of sound waves: • below 20 Hz: infrasonic • 20 Hz to 20 kHz: audible, i.e. sound proper • above 20 kHz: ultrasonic, “ultrasound” • pitch is given by frequency e.g. “standard a” corresponds to 440 Hz • intervals between tones given by ratio of frequencies (e.g. doubling of frequency - one octave) • male voice range 80 Hz to 240 Hz for speech, up to 700 Hz for song; • female voice range 140 Hz to 500 Hz for speech, up to 1100 Hz for song.