Physics 214 2: Waves in General Travelling Waves Waves in a string Basic definitions Mathematical representation Transport of energy in waves Wave Equation Principle of Superposition Interference Standing waves Propagating vibrations Forcing vibration Material of string is vibrating perpendicularly to direction of propagation TRANSVERSE WAVE If the vibrations were in same direction LONGITUDINAL Each part of vibration produces an oscillating force on string atoms & molecules, which cause neighboring atoms to vibrate wavelength amplitude A frequency number of waves passing a fixed point in one second cps Hz period T= time taken for one wave to pass a fixed point T= 1 T s; m ; A m speed of wave v= y x y = f(x,t) = y m sin(kx- t) the position, x, of points on the wave are functions of time i.e. x = x(t) phase consider points of a fixed amplitude y fixed yx,t y m sinkx t for these points kx t constant as t increases x must increase k dx dt - =0 k v= v= k phase velocity If the wave is propagating to left yx,t y m sinkx t v k Energy Transport If the waves are of small amplitude Hookes Law holds F= -k y k is the force constant of string medium and the waves are made up of propagating simple harmonic vibrations Linear Waves each string element of mass dm has K.E. K 1 2 dm y t 2 = 1 2 dx 2 y m 2 cos 2 kx t where y t = y m cos kx t & is mass per unit length dK dt 1 2 v 2 y m 2 cos 2 kx t Amplitude of standing wave 2Asin kx note that it varies with x The amplitude is zero= positions of NODES sin kx 0 0, ,2 ,, ,n ,, x n 2 The amplitude is a max. = positions of ANTINODES x n 1 2 Standing waves are formed by incident wave + reflected wave Length of the string must be half integer multiples of the wavelengthLength of the string must be half integer multiples of the wavelength The wave with wave length 1 is called the FUNDAMENTAL wave The the other waves are called OVERTONES or HIGHER HARMONICS 2 is called the First Overtone Second Harmonic 3 is called the Second Overtone Third Harmonic Frequency of a HARMONIC FAMILY of standing waves isFrequency of a HARMONIC FAMILY of standing waves is 3 n 3 n HARMONIC SEQUENCEHARMONIC SEQUENCE The overtone level is characterized by the number of NodesThe overtone level is characterized by the number of Nodes standing wave frequencies in string depend on geometry of string length: L inertial property density: elastic property tension: Every object can vibrate in the form of standing waves, whose frequencies form harmonic families and are characteristic of the object and depend on the geometry, inertial and elastic properties of the object i.e. on the geometry and forces (external and internal) experienced by the object. A forcing vibration can make an object vibrate and produce waves in the object These waves have the frequency of the forcing vibration These waves will die out unless they can form standing waves i.e are vibrating at the natural frequencies of the object When this is the case most energy is transferred from the forcing vibration to the object Then the amplitude of the standing waves increases RESONANCE