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? 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