Lecture 19 Waves So far we ve essentially treated light in terms of rays But in the 19th century is was realized that light really exhibited wave like behavior To understand this part of the nature of light we need to explore some of the characteristics of waves The basics what is a wave It s a transfer of energy from one place to another without transferring matter Common examples are waves in water some of the energy from a pebble thrown into a pond is transmitted to the shore sound waves energy from my voice is transmitted to your ears In both cases matter wiggles as the wave passes by but then returns to its normal position Types of Waves The way that matter wiggles is different for waves on a string and sound waves for waves on a string the molecules in the string wiggle in a direction perpendicular to the direction of the wave any wave that behaves this way is called a transverse wave for sound waves the air molecules wiggle back and forth in the same direction the wave is traveling any wave that behaves this way is called a longitudinal wave Sinusoidal Waves Waves typically oscillate back and forth periodically We can represent this behavior with a sine wave the distance by which the wiggling matter is displaced from its equilibrium position is a function of both position and time y x t Asin kx t Amplitude Wave number Angular frequency Wavelength Frequency and Period The wave is periodic in both space and time First consider a wave frozen at an instant in time t 0 Wave repeats when Wavelength sin k x sin kx sin kx 2 k 2 2 k We can similarly consider a single position x 0 As the wave goes by the matter at that position wiggles up and down and back to where it started in one period T sin t T sin t sin t 2 T 2 2 T The frequency is the inverse of the period 1 f T 2 Speed and Direction Since we defined a wave as something that s carrying energy from place to place the wave must have a speed and direction For sinusoidal waves we can find this by considering the motion of a point at the peak of the wave Want y Asin kx t A That means kx t t x k dx v dt k Note that wave is moving in the x direction 2 f v f 2 Note also that k Superposition and Interference What happens if two waves are traveling in the same medium i e two waves on the same string or two sources of sound in the same room The medium responds to the sum of all the waves y x t y1 x t y2 x t yN x t This is the principle of superposition Consider two waves with the same amplitude and frequency but different values at x t 0 this value is known as the phase of the wave y1 x t Asin kx t y2 x t Asin kx t The nature of the resulting wave depends on y x t A sin kx t sin kx t A sin kx t sin kx t cos cos kx t sin A sin kx t 1 cos cos kx t sin Special cases 1 0 y x t A sin kx t 1 1 cos kx t 0 2Asin kx t Twice the amplitude of the original wave 2 180o y x t A sin kx t 1 1 cos kx t 0 0 No wave at all Interference in Two Dimensions If waves are traveling in more than one dimension across the surface of water for example even more interesting patterns of interference can result Assume we have two identical wave sources lines below represent wave crests When a crest overlaps a trough get small amplitude destructive interference When two crests or troughs overlap get a large amplitude constructive interference We can tell whether a given point in space will have a maximum or minimum of the amplitude by finding the distance from that point to each source d1 and d2 If the distances differ by an integer number of wavelengths d1 d2 n n 0 1 2 then crests and troughs will line up to produce large amplitudes On the other hand if 1 d1 d2 n n 0 1 2 2 then the crest of one wave will coincide with the trough of the other and the two waves will cancel Non sinusiodal waves So far we ve only considered waves that can be represented with a sine function but there are many examples of waves with different shapes why do we care so much about sine waves then Due to Fourier s Theorem which tells us that any periodic function can be be formed from the sum of sine waves y t An sin nt Bn cos nt n The fundamental frequency 1is the frequency corresponding to the period of the function T 2 1 f1 1 T The other frequencies are multiples of the fundamental frequency n n 1 The coefficients An and Bn represent the amplitude of each sine wave in the sum you ll learn how to calculate them in a more advanced course