Download presentation

Presentation is loading. Please wait.

Published byAnissa Huddle Modified over 2 years ago

1
Physics 1B03summer-Lecture 9 Test 2 - Today 9:30 am in CNH-104 Class begins at 11am

2
Physics 1B03summer-Lecture 9 Wave Motion Sinusoidal waves Standing Waves

3
Physics 1B03summer-Lecture 9 Sine Waves f(x) = y(x,0) = A sin(kx) For sinusoidal waves, the shape is a sine function, eg., Then, at any time: y (x,t) = f(x – vt) = A sin[k(x – vt)] A -A y x ( A and k are constants)

4
Physics 1B03summer-Lecture 9 Sine wave: A -A y x v (“lambda”) is the wavelength (length of one complete wave); and so (kx) must increase by 2π radians (one complete cycle) when x increases by. So k = 2 , or k = 2π / λ y (x,t) = A sin[kx – t] since kv=ω

5
Physics 1B03summer-Lecture 9 Rewrite: y = A sin [kx – kvt]=A sin [kx – t] ω = 2πf y = A sin [ constant – t] “angular frequency” radians/sec frequency: cycles/sec (=hertz) The displacement of a particle at location x is a sinusoidal function of time – i.e., simple harmonic motion: The “angular frequency” of the particle motion is =kv ; the initial phase is kx (different for different particles). Review: SHM is described by functions of the form y(t) = A cos( t+ ) = A sin( /2 – – t), etc., with

6
Physics 1B03summer-Lecture 9 Example A -A y x Shown is a picture of a wave, y=A sin(kx t), at time t=0. a b c d e i) Which particle moves according to y=A cos( t) ? A B C D E ii) Which particle moves according to y=A sin( t) ? A B C D E iii ) Sketch a graph of y(t) for particle e.

7
Physics 1B03summer-Lecture 9 y(x,t) = A sin (kx ± t – ) angular wave number k = 2π / λ ( radians/metre ) angular frequency ω = 2πf (radians/second) amplitude “phase” The most general form of sine wave is y = Asin(kx ± ωt – ) The wave speed is v = 1 wavelength / 1 period, so v = fλ = ω / k phase constant

8
Physics 1B03summer-Lecture 9 Transverse waves on a string: (proof from Newton’s second law – Pg.625) Electromagnetic wave (light, radio, etc.): v = c 2.998 10 8 m/s (in vacuum) v = c/n (in a material with refractive index n ) (proof from Maxwell’s Equations for E-M fields ) The wave velocity is determined by the properties of the medium: Wave Velocity

9
Physics 1B03summer-Lecture 9 A) it will decrease by 4 B) it will decrease by 2 C) it will decrease by √2 D) it will stay the same E) it will increase by √2 You double the diameter of a string. How will the speed of the waves on the string be affected? Quiz

10
Physics 1B03summer-Lecture 9 Exercise (Wave Equation) What are and k for a 99.7 MHz FM radio wave?

11
Physics 1B03summer-Lecture 9 Particle Velocities Particle displacement, y (x,t) Particle velocity, v y = dy/dt ( x held constant) (Note that v y is not the wave speed v – different directions ! ) Acceleration, This is for the particles (move in y), not wave (moves in x) !

12
Physics 1B03summer-Lecture 9 “Standard” sine wave: maximum displacement, y max = A maximum velocity, v max = A maximum acceleration, a max = 2 A Same as before for SHM !

13
Physics 1B03summer-Lecture 9 Example string: 1 gram/m; 2.5 N tension Oscillator: 50 Hz, amplitude 5 mm y x Find: y (x, t) v y (x, t) and maximum speed a y (x, t) and maximum acceleration v wave

14
Physics 1B03summer-Lecture 9 10 min rest

15
Physics 1B03summer-Lecture 9 1)Identical waves in opposite directions: “standing waves” 2)2 waves at slightly different frequencies: “beats” 3)2 identical waves, but not in phase: “interference” Superposition of Waves

16
Physics 1B03summer-Lecture 9 Practical Setup: Fix the ends, use reflections. node L (“fundamental mode”) We can think of travelling waves reflecting back and forth from the boundaries, and creating a standing wave. The resulting standing wave must have a node at each fixed end. Only certain wavelengths can meet this condition, so only certain particular frequencies of standing wave will be possible. example:

17
Physics 1B03summer-Lecture 9 λ2λ2 Second Harmonic Third Harmonic.... λ3λ3

18
Physics 1B03summer-Lecture 9 In this case (a one-dimensional wave, on a string with both ends fixed) the possible standing-wave frequencies are multiples of the fundamental: f 1, 2f 1, 3f 1, etc. This pattern of frequencies depends on the shape of the medium, and the nature of the boundary (fixed end or free end, etc.).

19
Physics 1B03summer-Lecture 9 Sine Waves In Opposite Directions: y 1 = A o sin(kx – ωt) Total displacement, y(x,t) = y 1 + y 2 y 2 = A o sin(kx + ωt) Trigonometry : Then:

20
Physics 1B03summer-Lecture 9 Example 8mm y 1.2 m f = 150 Hz x a)Write out y(x,t) for the standing wave. b)Write out y 1 (x,t) and y 2 (x,t) for two travelling waves which would produce this standing wave. wave at t=0

21
Physics 1B03summer-Lecture 9 Example When the mass m is doubled, what happens to a) the wavelength, and b) the frequency of the fundamental standing-wave mode? What if a thicker (thus heavier) string were used? m

22
Physics 1B03summer-Lecture 9 Example a) m = 150g, f 1 = 30 Hz. Find μ (mass per unit length) b) Find m needed to give f 2 = 30 Hz c) m = 150g. Find f 1 for a string twice as thick, made of the same material. m 120 cm

23
Physics 1B03summer-Lecture 9Solution

24
Standing sound waves Sound in fluids is a wave composed of longitudinal vibrations of molecules. The speed of sound in a gas depends on the temperature. For air at room temperature, the speed of sound is about 340 m/s. At a solid boundary, the vibration amplitude must be zero (a standing wave node). node antinode Physical picture of particle motions (sound wave in a closed tube) graphical picture

25
Physics 1B03summer-Lecture 9 Standing sound waves in tubes – Boundary Conditions -there is a node at a closed end -less obviously, there is an antinode at an open end (this is only approximately true) node antinode graphical picture

26
Physics 1B03summer-Lecture 9 L Air Columns: column with one closed end, one open end

27
Physics 1B03summer-Lecture 9 Exercise: Sketch the first three standing-wave patterns for a pipe of length L, and find the wavelengths and frequencies if: a)both ends are closed b)both ends are open

Similar presentations

OK

Lecture 11 Chapter 16 Waves I Forced oscillator from last time Slinky example Coiled wire Rope Transverse Waves demonstrator Longitudinal Waves magnetic.

Lecture 11 Chapter 16 Waves I Forced oscillator from last time Slinky example Coiled wire Rope Transverse Waves demonstrator Longitudinal Waves magnetic.

© 2018 SlidePlayer.com Inc.

All rights reserved.

Ads by Google

Ppt on eid festival in pakistan Ppt on amplitude shift keying demodulation Ppt on idea cellular ltd Ppt on schottky diodes Ppt on quality control in industrial management Free download ppt on nitrogen cycle Ppt on health management information system Ppt on causes of road accidents Hrm ppt on recruitment agencies A ppt on waste management