Test 2 - Today

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

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

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

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

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

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

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

8. Wave Velocity The wave velocity is determined by the properties of the medium: Transverse waves on a string: (proof from Newton’s second law – Pg.625) Electromagnetic wave (light, radio, etc.): v = c 2.998108 m/s(in vacuum) v = c/n(in a material with refractive index n) (proof from Maxwell’s Equations for E-M fields ) Physics 1B03summer-Lecture 9

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

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

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

12. “Standard” sine wave: maximum displacement, ymax = A maximum velocity, vmax = w A maximum acceleration, amax = w 2 A Same as before for SHM ! Physics 1B03summer-Lecture 9

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

14. 10 min rest Physics 1B03summer-Lecture 9

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

16. Practical Setup: Fix the ends, use reflections. 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: L • (“fundamental mode”) • node • node Physics 1B03summer-Lecture 9

17. λ3 λ2 • Second Harmonic • Third Harmonic . . . . Physics 1B03summer-Lecture 9

18. In this case (a one-dimensional wave, on a string with both ends fixed) the possible standing-wave frequencies are multiples of the fundamental: f1, 2f1, 3f1, etc. This pattern of frequencies depends on the shape of the medium, and the nature of the boundary (fixed end or free end, etc.). Physics 1B03summer-Lecture 9

19. Sine Waves In Opposite Directions: y2 = Aosin(kx + ωt) y1 = Aosin(kx – ωt) Total displacement, y(x,t) = y1 + y2 Trigonometry : • Then: Physics 1B03summer-Lecture 9

20. Example wave at t=0 • y • 8mm • x • 1.2 m • f = 150 Hz • Write out y(x,t) for the standing wave. • Write out y1(x,t) and y2(x,t) for two travelling waves which would produce this standing wave. Physics 1B03summer-Lecture 9

21. Example m • Whenthe 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? Physics 1B03summer-Lecture 9

22. Example 120 cm m • m = 150g, f1 = 30 Hz. Find μ (mass per unit length) • Find m needed to give f2 = 30 Hz • m = 150g. Find f1 for a string twice as thick, made of the same material. Physics 1B03summer-Lecture 9

23. Solution Physics 1B03summer-Lecture 9

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). Physical picture of particle motions (sound wave in a closed tube) node antinode antinode node node graphical picture Physics 1B03summer-Lecture 9

25. 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 antinode graphical picture Physics 1B03summer-Lecture 9

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

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