To Learn the Basic Properties of Traveling Waves. Slide 20-2 Chapter 20 Preview

Chapter 20 Traveling Waves

Chapter Goal: To learn the basic properties of traveling waves. Slide 20-2 Chapter 20 Preview

Slide 20-3 Chapter 20 Preview

Slide 20-5

• result from periodic disturbance • same period (frequency) as source 1 f  • Longitudinal or Transverse Waves  • Characterized by – amplitude (how far do the “bits” move from their equilibrium positions? Amplitude of MEDIUM) – period or frequency (how long does it take for each “bit” to go through one cycle?) – wavelength (over what distance does the cycle repeat in a freeze frame?) – wave speed (how fast is the energy transferred?) vf  v Wavelength and Frequency are Inversely related: f   The shorter the wavelength, the higher the frequency. The longer the wavelength, the lower the frequency.

3Hz

5Hz Spherical Waves Wave speed: Depends on Properties of the Medium: Temperature, Density, Elasticity, Tension, Relative Motion vf  Transverse Wave • A traveling wave or pulse that causes the elements of the disturbed medium to move perpendicular to the direction of propagation is called a transverse wave Longitudinal Wave A traveling wave or pulse that causes the elements of the disturbed medium to move parallel to the direction of propagation is called a longitudinal wave:

Pulse

Tuning Fork

Guitar String Types of Waves

Sound String Wave PULSE:

• traveling disturbance • transfers energy and momentum • no bulk motion of the medium • comes in two flavors • LONGitudinal • TRANSverse Traveling Pulse

• For a pulse traveling to the right – y (x, t) = f (x – vt) • For a pulse traveling to the left – y (x, t) = f (x + vt) • The function y is also called the wave function: y (x, t) • The wave function represents the y coordinate of any element located at position x at any time t – The y coordinate is the transverse position • If t is fixed then the wave function is called the waveform – It defines a curve representing the actual geometric shape of the pulse at that time Traveling Pulse 2 y(,) x t  Wave Form (xt 3 )2 1 Space Snap Shots 2 @t 0 s , y ( x ,0) (x )2  1

2 @t 1 s , y ( x ,1) (x  3)2 1

2 @t 2 s , y ( x ,2) (x  6)2 1 Time Plot 2 y(,) x t  One position (xt 3 )2 1 2 Changing in time @x 5, y (5, t ) (5 3t )2 1 HW to be Turned in

• Then use excel to generate plots - y vs t for x=0, x=5m, x=10 m - y vs x for t = 0, t = 1s, t = 3s - Try a 3-D plot for fun! Traveling Waves The media moves in SHM. The wave travels at constant speed. The wave has the same frequency as the ‘shaking’ source! Traveling Waves

• The wave represented by the y( x , t ) A sin( kx t ) curve shown is a sinusoidal wave 2 y( x , t ) A sin  x vt • It is the same curve as sin q  plotted against q • This is the simplest example of a periodic continuous wave – It can be used to build more complex waves • Each element moves up and down in simple harmonic motion • Distinguish between the motion of the wave and the motion of the particles of the medium Wave Functions are Solutions to the Wave Equation 22yy1 2  y( x , t ) A sin  x vt x2 v 2 t 2 

2 2  k  2 f vf    T Tk

Derive these: y( x , t ) A sin( kx t )

x xt y( x , t ) A sin 2 f ( t ) y( x , t ) A sin 2  v  T Time Plot y( x , t ) A sin( kx t ) x is fixed

Snap shot in Space. This is an image of one piece of a string and how it moves as the waves goes by in time. The one piece oscillates in SHM. COMPARE: Motion Equations for Simple Harmonic x is fixed

x( t )  A cos ( t ) dx vA  sin(  t   ) dt dx2 aA  2 cos(  t   ) dt 2

Notice: ax 2 Snap Shot y( x , t ) A sin( kx t )

Snap shot in TIME. Time is fixed. This is an image of the entire string or the medium’s displacement from equilibrium at one instant. Can represent either transverse or longitudinal waves!! Wave Speed is Constant! Medium Accelerates!! vf 

String y( x , t ) A sin( kx t ) y max = A v( x , t )  A cos( kx  t ) vy, max =  A 2 a( x , t )  A2 sin( kx  t ) ay, max =  A Speed of wave depends on properties of the MEDIUM vf 

Speed of particle in the y( x , t ) A sin( kx t ) Medium depends on v( x , t )  A cos( kx  t ) SOURCE: SHM a( x , t )  A2 sin( kx  t ) Wave Speed vf 

This gives the relationship between the wavelength and frequency for constant wave speed. The frequency depends on the source and the speed depends on the properties of the medium. The speed of sound is independent of the frequency. When traveling from one medium to another, if the speed changes, the wavelength changes but the frequency (energy) remains the same. QuickCheck 14.3 Phase Constant: Initial Conditions

This is the position graph of a mass oscillating on a horizontal spring. What is

the phase constant 0?

A.  /2 rad. B. 0 rad. C.  /2 rad. D.  rad. E. None of these.

Slide 14-42 QuickCheck 14.3 Phase Constant: Initial Conditions

This is the position graph of a mass oscillating on a horizontal spring. What is

the phase constant 0?

A.  /2 rad. B. 0 rad. C.  /2 rad. D.  rad. Initial conditions: E. None of these. x = –A

vx = 0 Slide 14-43 QuickCheck 14.2 Phase Constant: Initial Conditions

This is the position graph of a mass oscillating on a horizontal spring. What is

the phase constant 0?

A.  /2 rad. B. 0 rad. C.  /2 rad. D.  rad. E. None of these.

Slide 14-40 QuickCheck 14.2 Phase Constant: Initial Conditions

This is the position graph of a mass oscillating on a horizontal spring. What is

the phase constant 0?

A.  /2 rad. B. 0 rad. C.  /2 rad. D.  rad. E. None of these. Initial conditions: x = 0 vx > 0 Slide 14-41 QuickCheck 14.4 Phase Constant: Initial Conditions The figure shows four oscillators at t = 0. For which

is the phase constant 0   / 4?

Slide 14-44 QuickCheck 14.4 Phase Constant: Initial Conditions The figure shows four oscillators at t = 0. For which

is the phase constant 0   / 4?

Initial conditions: x = 0.71A

vx > 0

Slide 14-45 Simple Harmonic Motion x( t ) A cos ( t ) The oscillator is the SOURCE & SHM v( t ) - A sin(  t  ) frequency is the wave frequency. a( t ) -2 A cos(  t  )

Wave Motion vfwave   Speed of wave depends on properties of the MEDIUM

Particle Motion Speed of particle in the Medium depends on SOURCE: SHM y( x , t ) A sin( kx t ) SHM and Wave motion are out of v( x , t )  A cos( kx  t ) phase, hence different functions. a( x , t )  A2 sin( kx  t ) SHM is fixed in x coordinate. Wave Motion on a String

. Shown is a snapshot graph of a wave on a string with vectors showing the velocity of the string at various points. . As the wave moves along x, the velocity of a particle on the string is in the y-direction.

2 2  k  2 f vf    T Tk

Variations: y( x , t ) A sin( kx  t  0 )

x xt y( x , t ) A sin 2 f ( t ) y( x , t ) A sin 2  v  T QuickCheck 20.6

The period of this wave is

A. 1 s. B. 2 s. C. 4 s. D. Not enough information to tell.

Slide 20-49 QuickCheck 20.6

The period of this wave is

A. 1 s. A sinusoidal wave moves B. 2 s. forward one wavelength C. 4 s. (2 m) in one period. D. Not enough information to tell.

Slide 20-50 QuickCheck 20.1

These two wave pulses travel along the same stretched string, one after the other. Which is true?

A. vA > vB

B. vB > vA

C. vA = vB D. Not enough information to tell.

Slide 20-25 QuickCheck 20.1

These two wave pulses travel along the same stretched string, one after the other. Which is true?

A. vA > vB

B. vB > vA Wave speed depends on the properties of the medium, C. vA = vB not on the amplitude of the wave. D. Not enough information to tell.

Slide 20-26 Wave 1

Ocean waves with a crest-to-crest distance of 10.0 m can be described by the wave function y(x, t) = (0.800 m) sin[0.628(x – vt)] where v = 1.20 m/s. (a) Sketch y(x, t) at t = 0. (b) Sketch y(x, t) at t = 2.00 s. Space Snap Shots in Time

Note how the entire wave form has shifted 2.40 m in the positive x direction in this time interval: x= vt =(1.2m/s)(2s)=2.4m!!! Do workbook

Waves can be travelling either to the right or left.

y( x , t ) A sin( kx  t  0 ) To the right: both x and t increase and the y position or depth of the wave decreases.

y( x , t ) A sin( kx  t  0 )

To the left: t increases but x decreases

y( x , t ) A sin( kx  t  0 ) Wave Motion on a String

A wave on a string is traveling to the right. At this instant, the motion of the piece of string marked with a dot is

A. Up. B. Down. C. Right. D. Left. E. Zero. Instantaneously at rest.

A wave on a string is traveling to the right. At this instant, the motion of the piece of string marked with a dot is

A. Up. B. Down. C. Right. D. Left. E. Zero. Instantaneously at rest.

y( x , t ) A sin( kx  t  0 )

A positive value of phi shifts the curve in the negative direction of the x axis; A negative value shifts the curve in the positive direction.

(a) Zero phase shift. (b) Positive phase shift.

Do workbook page 2 Find the function from the plots

y( x , t ) A sin( kx  t  0 ) PhasePhase and Phase and Difference Phase Difference

. The quantity (kx  t + 0) is called the phase of the wave, denoted . . The phase difference  between two points on a wave depends on only the ratio of their separation x to the wavelength :  x  2

v 22  t2  f  t 2   t ( v  t ) k x  x    Phase Difference y = (15.0 cm) cos(0.157x – 50.3t). At a certain instant, let point A be at the origin and point B be the first point along the x axis where the wave is 60.0 out of phase with point A. What is the coordinate of point B? v 2  t2  f  t 2   t ( v  t ) k x  2  x  k  x   x   2

/3 /3 x    6.67 cm x  40 cm  6.67 cm k 0.157 22 Most waves are really spherically symmetric saves (waves on a string are not) Waves in Two and Three Dimensions

. Consider circular ripples spreading on a pond. . The lines that locate the crests are called wave fronts. . If you observe circular wave fronts very, very far from the source, they appear to be straight lines.

© 2013 Pearson Education, Inc. Slide 20-60 Waves in Two and Three Dimensions . Loudspeakers and lightbulbs emit spherical waves. . That is, the crests of the wave form a series of concentric spherical shells. . Far from the source this is a plane wave.

A spherical wave travels outward from a point source. What is the phase difference between the two points on the wave marked with dots?

A. /4 radians. B.  /2 radians. C.  radians. D. 7 /2 radians. E. 7 radians.

Slide 20-63 QuickCheck 20.8Phase Difference

A spherical wave travels outward from a point source. What is the phase difference between the two points on the wave marked with dots?

A. /4 radians. B.  /2 radians. C.  radians. D. 7 /2 radians. E. 7 radians.

Slide 20-64

2 Waves Phase Difference: Ch 21 2 Phase Difference at P:   r   0