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Transverse Wave Motion Definition of Waves

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Transverse Wave Motion Definition of Waves Transverse Wave Motion Definition of Waves • A wave is a disturbance that moves through a medium without giving the medium, as a whole, any permanent displacement. • The general name for these waves is progressive wave. • If the disturbance takes place perpendicular to the direction of propagation of the wave, the wave is called transverse. • If the disturbance is along the direction of propagation of the wave, it is called longitudinal. Characteristics of Waves • At any point, the disturbance is a function of time and at any instant, the disturbance is a function of the position of the point. • In a sound wave, the disturbance is pressure-variation in a medium. • In the transmission of light in a medium or vacuum, the disturbance is the variation of the strengths of the electric and magnetic fields. • In a progressive wave motion, it is the disturbance that moves and not the particles of the medium. Progressive Waves • To demonstrate wave motion, take the loose end of a long rope which is fixed at the other end quickly up and down • Crests and troughs of the waves move down the rope • If the rope is infinity long such waves are called progressive waves Standing Waves • If the rope is fixed at both ends, the progressive waves traveling on it are reflected and combined to form standing waves The first four harmonics of the standing waves allowed between the two fixed ends of a string Transverse vs Longitudinal Waves • Transverse wave: the displacements or oscillations in the medium are transverse to the direction of propagation e.g. electromagnetic (EM) waves , waves on strings • Longitudinal wave: the oscillations are parallel to the direction of wave propagation e.g. sound waves Plane Waves • Take a plane perpendicular to the direction of wave propagation and all oscillators lying within that plane have a common phase • Over such a plane, all parameters describing the wave motion remain constant • The crests and troughs are planes of maximum amplitude of oscillation, which are rad out of phase • Crest = a plane of maximum positive amplitude • Trough = a plane of maximum negative amplitude The Wave Equation • The wave equation of small element of string of linear density and constant tension T 2 2 where 2 T y 1 y c x 2 c 2 t 2 c is the phase or wave velocity. T (x +dx, y +dy) +d T (x , y ) ds 2 dx 2 dy 2 d 2 s dx 2 2 y dx 2 dx 2 x 2 ds 2 y 1 dx x 2 y 2 y very small and thus can be ignored x x 2 ds dx ds dx 2 y T sin( d ) T sin dx t 2 y very small sin tan x 2 y y y T dx x x t 2 2 x dx x y 1 y y 2 2 2 y y x dx x x T dx dx x dx x 2 2 x t 2 y y y 2 2 dx y y 2 x x x 2 2 x dx x x T t Waves in One Dimension • Suppose a wave moves along the x-axis with constant velocity c and without any change of shape (i.e. with no dispersion) and the disturbance takes place parallel to the y-axis, then y (x, t) = f (ct – x) (1) defines a one-dimensional wave along the positive direction of the x-axis (forward wave) x c t Waves in One Dimension • A wave which is the same in all respect but moving in the opposite direction (i.e. along the direction of x decreasing) is given by Eqn. (1) with the sign of v changed: y (x, t) = f (ct + x) (2) • This is known as backward wave. y f 1 (ct x ) y 2 y f 1 (ct x ), f 1 (ct x ) x x 2 y 2 y 2 c f 1 (ct x ), c f 1 (ct x ) t t 2 2 y 1 2 y x 2 c 2 t 2 Waves in One Dimension • Eqns. (1) and (2) satisfy the second-order partial differential equation: 2 y 1 2 y (3) x 2 c 2 t 2 • Eqn. (3) is known as the non-dispersive wave equation. Locus of oscillator displacements a y Displacement Displacement x -a Solution of Wave equation • A solution to the wave equation 2 y a sin( t ) a sin (ct x ) 2 c 2 , where is the oscillation 2 x frequency and • The wave is moving in the positive x direction. The Wave Equation • At position x = 0, wave equation y a sin t • Any oscillator to its right at some position x will be set in motion at some later2 time t. y a sin( t ) a sin (ct x ) • Have a phase lag with respect to the oscillator at x = 0. • The wavelength is the separation in space between any two oscillators with a phase difference 2 rad. The Wave Equation 1 • The period of oscillation c • An observer at any point would2 be passed by y a sin( t ) a sin (ct x ) wavelengths per second. • If the wave is moving to the left the sign is changed. • Wave moving to right • 2 Wave moving to left y a sin( t ) a sin (ct x ) Equivalent Wave Expressions 2 y a sin (ct x ) x y a sin 2 ( t ) x y a sin (t ) c y a sin( t kx ) 2 where k is called wave number. c • Cosine functions are equally valid. i t kx • For both sine and cosine y ae The Wave Equation y a sin( t kx ) y 2 y a cos( t kx ), 2 a sin( t kx ) t t 2 y 2 y ka cos( t kx ), k 2 a sin( t kx ) x x 2 y y y x y c t k x x t x 2 y k 2 2 y 1 2 y x 2 2 t 2 c 2 t 2 Three Velocities in Wave Motion 1. Particle velocity Simple harmonic velocity of the oscillator about its equilibrium position 2. Wave or phase velocity The velocity with which planes of equal phase, crests or troughs, progress through the medium 3. Group velocity A number of waves of different frequencies, wavelengths and velocities may be superposed to form a group. Motion of such a pulse would be described by its group velocity Particle Velocity y y arrows show c the direction t x and magnitude of the particle velocity • Locus of oscillator displacements in a continuous medium as a wave passes over them travelling in the positive x-direction • The wavelength is defined as the distance between any two oscillators having a phase difference of 2 rad Wave or Phase velocity x • The wave or phase velocity is c t • It is the rate at which disturbance moves across the oscillators. • The oscillator or particle velocity is a simple harmonic velocity y t y a sin( t kx ) y a cos( t kx ) t Wave or Phase Velocity Wave or Phase Velocity = the rate at which disturbance moves across the oscillators x Wave or Phase Velocity = t Oscillator or Particle Velocity is a simple harmonic velocity y Oscillator or Particle Velocity = t Characteristic Impedance of a String (the string as a forced oscillator) • Any medium through which waves propagate will present an impedance to those waves • If the medium is lossless, and possesses no resistive or dissipation mechanism, for a string the impedance is determined by inertia and elasticity • The presence of a loss mechanism will introduce a complex term into the impedance Characteristic Impedance of a String (the string as a forced oscillator) • The transverse impedance is define as: transverse force F Z transverse velocity v • Characteristic impedance of the string: T Z c since T c 2 c Characteristic Impedance of a String (the string as a forced oscillator) it The string as a forced oscillator with a vertical force F0e driving it at one end i t y For small : F 0 e T sin T tan T x Characteristic Impedance of a String (the string as a forced oscillator) displacement of the progressive waves may be represented exponentially by: y A e i ( t kx ) amplitude A may be complex At the end of the string, where x = 0 i t y i ( t k 0 ) F 0 e T ikT A e x x 0 F 0 F 0 c F 0 c i ( t kx ) A y e ikT i T i T Characteristic Impedance of a String (the string as a forced oscillator) c transverse velocity: i ( t kx ) v y F 0 e T velocity amplitude: v F 0 / Z transverse impedance: Characteristic Impedance of the string T Z c since T c 2 Since the velocity c isc determined by the inertia and the elasticity, the impedance is also governed by these properties Reflection and Transmission • Suppose a string consists of two sections smoothly joined at a point x = 0 with a tension T • Waves on a string of impedance Z1= 1c1 reflected and transmitted at the boundary x = 0 where the string changes to impedance Z2= 2c2 Z2 = 2c2 Z1 = 1c1 Reflection and Transmission i ( t k 1 x ) Incident wave: y i A1 e i ( t k 1 x ) Reflected wave: y r B 1 e i ( t k 2 x ) Transmitted wave: y t A 2 e find the reflection and transmission amplitude coefficients i.e. the relative values of B1 and A2 with respect to A1 i ( t k 1 x ) y i A1 e i ( t k 1 x ) y r B 1 e findi (the t kreflectionx ) and y A e 2 t transmission2 amplitude coefficients i.e.
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