LECTURE 5 WAVES ON STRINGS & HARMONIC WAVES. Instructor: Kazumi Tolich

Transcription

1 LECTURE 5 WAVES ON STRINGS & HARMONIC WAVES Instructor: Kazumi Tolich

2 Lecture 5 2 Reading chapter Waves on a string Speed of waves on a string Reflections Harmonic waves

3 Speed of waves 3 The speed of a wave is determined by the properties of the material through which it propagates (medium). Speed of waves on a string is given by v = F μ F is the tension in the string. μ is the mass density (mass per unit length) of the string.

4 Demo: 1 4 Tension dependence of wave speed v = & '

5 Quiz: 1 5 A string under tension carries transverse waves traveling at speed v. If the same string is under four times the tension, what is the wave speed?

6 Quiz: 5-1 answer 2v v = & ' v +,- =.& ' = 2 & ' = 2v

7 Quiz: 2 Suppose you pluck the thinnest guitar string, and at the same time, your friend plucks the thickest guitar string which is under the same tension as the thinnest string. Which note travels faster in the air? A. Your note B. Your friend s note C. They travel at the same speed.

8 Quiz: 5-2 answer 8 They travel at the same speed. Speed of waves relative to the medium in which they are traveling depends on the medium the air. Both notes travels in the same air. Speed of waves do not depend on the motion of the source. Otherwise you cannot enjoy music played by different instruments on a stage.

9 Example: 1 (Walker Ch ) 9 A brother and sister try to communicate with a string tied between two tin cans. If the string is L = 7.6 m, has a mass of m = kg, and is pulled taut with a tension of F = 18 N, how much time does it take for a wave to travel from one end of the string to the other?

10 Reflection on a boundary 10 The reflected wave has the same speed, wavelength, and amplitude (and energy) as the incident wave. For a fixed boundary case, the situation can be simulated as an un-terminated string with positive and negative amplitude waves moving in opposite directions and meeting at the boundary. For a free-boundary case, the rope is always perpendicular to the rod. The situation can be simulated as an un-terminated string with two waves of the same amplitude moving in the opposite directions and meeting at the rod.

11 Demo: 2 11 Spring on table Demonstration of reflection of a wave

12 Quiz: 3 12 A pulse moves in a string toward a free end (a ring on a rod) as indicated in the sketch. On reflection, which sketch most nearly represents the pulse?

13 Quiz: 5-3 answer 13 The situation can be simulated as an un-terminated string with two waves of the same amplitude moving in the opposite directions and meeting at the rod.

14 Quiz: 4 14 The pulse shown is moving in the string toward a fixed end at the wall. After reflection at the wall, which figure correctly represents the pulse?

15 Quiz: 5-4 answer 15 The situation can be simulated as an un-terminated string with positive and negative amplitude waves moving in opposite directions and meeting at the boundary.

16 Traveling harmonic waves 16 If a harmonic wave is traveling through a medium, each point of the medium oscillates in simple harmonic motion. For a transverse harmonic wave traveling in the positive x-direction, the wave function is y x, t = A cos 9: ; x λ >? = A cos 9: ; x 9:? t

17 Moving displacement 17 Suppose we wish to move a parabolic function f x = x 9 so that it is centered at greater and greater values of the independent variable x. Then we change x to x 2, x 4, etc. Similarly, if we wish to displace some arbitrary function f x by a distance d in the positive x-direction, we replace f x by f x d. If we wish to make the displacement increase with time, i.e., move the curve with speed v, we make d = vt, so that the function becomes f x vt. Similarly, to displace the function in the negative x direction, we use f x + vt. y x, t = A cos 9: ; x ;? t

18 Quiz: 5 A traveling wave passes a point of observation. At this point, the time between successive crests is 0.2 s. Which of the following statements can be justified? Choose all that apply. A. The wavelength is 5 m. B. The frequency is 5 Hz. C. The velocity of propagation is 5 m/s. D. The wavelength is 0.2 m. E. The frequency is 0.2 Hz. F. The velocity of propagation is 0.2 m/s. G. There is not enough information to justify any of these statements.

19 Quiz: 5-5 answer The frequency is 5 Hz. T = 0.2 s f = E = E = 5 Hz? F.9 H

20 Quiz: 6 20 A wave is traveling with a speed v along the x axis in the positive direction. The upper graph shows the displacement y versus the distance x for a given instant of time. The lower graph shows the displacement y versus the time t for any given point x. From the information in the graphs, what is the wave speed v in m/s?

21 Quiz: 5-6 answer 21 8 m/s The wavelength is 4 m. λ The period is E 9 s. v = ;? T

22 Example: 2 22 The wave function for a harmonic wave on a string is y x, t = 1.00 mm cos 62.8 m QE x s QE t. a) Find the wavelength, period, and frequency of this wave. b) In what direction does this wave travel, and what is the wave s speed?