Review: Waves - I

Waves Particle: a tiny concentration of matter, can transmit energy. Wave: broad distribution of energy, filling the space through which it travels. Quantum Mechanics: Wave Particle

Types of Waves Types of waves: Mechanical Waves, Electromagnetic Waves, Matter Waves, Electron, Neutron, People, etc …… Transverse Waves: Displacement of medium  Wave travel direction Longitudinal Waves: Displacement of medium || Wave travel direction

Parameters of a Periodic Wave : Wavelength, length of one complete wave form T: Period, time taken for one wavelength of wave to pass a fixed point v: Wave speed, with which the wave moves f: Frequency, number of periods per second  = vT v =  T = f

Wave Function of Sinusoidal Waves y(x,t) = y m sin(kx-  t) y m : amplitude kx-  t : phase k: wave number When ∆x=, 2  is added to the phase  : angular frequency When ∆t=T, 2  is added to the phase

Wave Speed How fast does the wave form travel?

Wave Speed How fast does the wave form travel? Pick a fixed displacement  a fixed phase kx-  t = constant y(x,t) = y m sin(kx-  t) v>0 y(x,t) = y m sin(kx+  t) v<0 Transverse Waves (String):

Principle of Superposition Overlapping waves add to produce a resultant wave y ’ (x,t) = y 1 (x,t) + y 2 (x,t) Overlapping waves do not alter the travel of each other

Interference n=0,1,2,... Constructive: Destructive:  n    n  1 2      

Phasor Addition PHASOR: a vector with the amplitude y m of the wave and rotates around origin with  of the wave When the interfering waves have the same  PHASOR ADDITION INTERFERENCE Can deal with waves with different amplitudes

Standing Waves Two sinusoidal waves with same AMPLITUDE and WAVELENGTH traveling in OPPOSITE DIRECTIONS interfere to produce a standing wave The wave does not travel Amplitude depends on position

NODES: points of zero amplitude ANTINODES: points of maximum (2y m ) amplitude

Standing Waves in a String The BOUNDARY CONDITIONS determines how the wave is reflected. Fixed End: y = 0, a node at the end Free End: an antinode at the end The reflected wave has an opposite sign The reflected wave has the same sign

Case: Both Ends Fixed k can only take these values OR where RESONANT FREQUENCIES:

(a) k = 60 cm -1, T=0.2 s, z m =3.0 mm z(y,t)=z m sin(ky-  t)  = 2  /T = 2  /0.2 s =10  s -1 z(y, t)=(3.0mm)sin[(60 cm -1 )y -(10  s -1 )t] (b) Speed u z,min =  z m = 94 mm/s HRW 11E (5 th ed.). (a) Write an expression describing a sinusoidal transverse wave traveling on a cord in the  y direction with an angular wave number of 60 cm -1, a period of 0.20 s, and an amplitude of 3.0 mm. Take the transverse direction to be the z direction. (b) What is the maximum transverse speed of a point on the cord?

f = 500Hz, v=350 mm/s (a) Phase (b) HRW 16P (5 th ed.). A sinusoidal wave of frequency 500 Hz has a velocity of 350 m/s. (a) How far apart are two points that differ in phase by  /3 rad? (b) What is the phase difference between two displacements at a certain point at times 1.00 ms apart? y(x,t) = y m sin(kx-  t)

For HRW 36E (5 th ed.). Two identical traveling waves, moving in the same direction, are out of phase by  /2 rad. What is the amplitude of the resultant wave in terms of the common amplitude y m of the two combining waves?

(a) (b) The angle  is either 68˚ or 112˚. Choose 112˚, since  >90˚. HRW 41E (5 th ed.). Two sinusoidal waves of the same wavelength travel in the same direction along a stretched string with amplitudes of 4.0 and 7.0 mm and phase constant of 0 and 0.8  rad, respectively. What are (a) the amplitude and (b) the phase constant of the resultant wave? y m1 =4.0 mm y m2 =7.0 mm  ymym h  