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Review: Waves - I

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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

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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

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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

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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

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Wave Speed How fast does the wave form travel?

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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):

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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

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Interference n=0,1,2,... Constructive: Destructive: n n 1 2

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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

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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

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

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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

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

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(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?

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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)

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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?

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(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

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