1

Sinusoidal WavesSinusoidal Waves The waves produced in SHM are sinusoidal, i.e., they can be described by a sine or cosine function with appropriate amplitude, frequency, and phase. The figure at the right shows a sinusoidal wave moving along the x axis.

The history plot and snapshot plot of sinusoidal waves shown show an interesting feature:Ø In the history plot (D vs. t), the distance between adjacent sinusoid maxima is the period T.Ø In the snapshot (D vs. x), the distance between adjacent sinusoid maxima is the wavelength .

2

The Fundamental Relationshipof Sinusoidal Waves

The Fundamental Relationshipof Sinusoidal Waves

Consider a sinusoidal wave moving along the x axis with velocity v. In the time interval of one period T, the crest of the traveling sinusoidal wave moves forward by one wavelength .

distance

time 2v f

T

Note that while this relation works for periodic waves, a non-periodic pulse will have a definite velocity (speed of its peak) but has neither a frequency nor a wavelength.

3

Clicker Question 2Clicker Question 2

What is the frequency of this traveling wave?

(a) 0.10 Hz;(b) 0.20 Hz;(c) 2.0 Hz;(d) 5.0 Hz;(e) 10.0 Hz.

4

The Mathematicsof Sinusoidal Waves

The Mathematicsof Sinusoidal Waves

0( , 0) sin 2x

D x t A

0

0 0

( ) sin 2

sin 2 2 sin 2 ( )

xD x A

x xA A D x

0 0( , ) sin 2 sin 2x vt x t

D x t A AT

5

Angular Frequency and Wave Number k

Angular Frequency and Wave Number k

0( , ) sin 2x t

D x t AT

22 angular frequency (rad/s)f

T

2 wave number (rad/m)k

2/ or

2v f k vk

k

0( , ) sinD x t A kx t

0Note that: (0,0) sinD A

6

Example:Analyzing a Sinusoidal Wave

Example:Analyzing a Sinusoidal Wave

A sinusoidal wave with amplitude A = 1.0 m and frequency f = 100 Hz travels at v =200 m/s in the +x direction. At t=0, the point at x=1.0 m is on the crest of the wave.

1. Find A, v, l, k, f, w, T, and f0 for this wave.2. Write the wave equation.3. Draw a snapshot graph at t-0.

1.0 m; 200 m/s; 100 Hz;A v f / 0.02 m;v f

2 / rad/m 3.14 rad/m;k

2 200 rad/s 628 rad/s;f

1/ 0.10 s;T f

0(1.0 m, 0) sin( (1.0 m) )D A A k

0 0(1.0 m)2 2 2

k

( , ) (1.0 m)sin[( rad/m) (200 rad/s) ] / 2]D x t x t

7

Velocity of Waves on a StringVelocity of Waves on a String( , ) cos( ), so sin( ) at 0

dyy x t A kx t k kx t

dx

( , ) sin( )y

dyv x t A kx t

dt

2 2( , ) cos( ) @ crestya x t A kx t A

net( ) ( )y y yF ma x a

net/ 2

2

( ) 2 sin 2 tan 2

2 sin2

y s s sx

s s

dyF T T T

dx

k xT kA k AT x

2 2Therefore, ( ) sA x k AT x

2 2( / ) / or /s sk v T v T

8

Example:Generating a Sinusoidal Wave

Example:Generating a Sinusoidal Wave

A very long string with = 2.0 g/m is stretched along the x axis with a tension of Ts = 5.0 N. At position x=0 it is tied to a 100 Hz simple harmonic oscillator that vibrates perpendicular to the string with an amplitude of 2.0 mm. The oscillator is at its maximum positive displacement at t=0.

1. Write the displacement equation of waves on the string.2. At t = 5.0 ms, what is the string’s displacement at a point

2.7 m from the oscillator? Peak at (0,0), so ( , ) cos( )D D x t A kx t

2.0 mm 0.002 m; 2 (100 Hz) 200 rad/sA

/ (5.0 N) /(0.002 kg/m) 50 m/sv T

/ (200 rad/s) /(50 m/s) 4 rad/mk v

( , ) (0.002 m)cos[(4 rad/m) (200 rad/s) ]D x t x t

(2.7 m,0.005 s) 0.00162 mD

9

Waves in 2D and 3DWaves in 2D and 3D So far we have only considered one dimensional waves, like those on a vibrating string. However, waves may move in two dimensions on a surface (ripples on a pond), or in three dimensions in a volume (sound waves from a firecracker). A small source in a large volume emits spherical waves. However, if the observer is very far away from the source, the curvature of the wave front is small and the wave becomes a plane wave. Circular or spherical waves can be described by changing D(x,t) to D(r,t), where r is the distance from the source:

0

00

( , ) ( )sin( )

sin( )

D r t A r kr t

Akr t

r

10

Phase and Phase DifferencePhase and Phase Difference The quantity is called the phase of the wave. The wave fronts we have seen in the previous figures are surfaces of constant phase because each point on such a surface has the same displacement, and therefore the same phase. The displacement can be written as D(x,t) = Asin.

The figure shows a snapshot of a traveling wave. The phase difference between points x1 and x2 is:

2 1 2 0 1 0

2 1

( ) ( )

( ) 2

kx t kx t

xk x x k x

2

x t

T

Phase difference over 2equals space separation over .

tkx

11

Example: The Phase Difference between points of a Sound WaveExample: The Phase Difference

between points of a Sound Wave

A 100 Hz sound wave travels at 343 m/s.(a) What is the phase difference between two points 60

cm apart in the direction the wave is traveling?(b) How far apart are two points with phase difference

900?-1/ (343 m/s) /(100 s ) 3.43 mv f

(0.60 m)=2 2 0.350 rad 63.0

(3.43 m)

x

/ 2 1

2 4

x

(3.43 m)

0.858 m4 4

x

12

Longitudinal WavesLongitudinal Waves

Longitudinal waves (e.g., sound) are produced in a compressible medium by longitudinal motion of each particle of the medium, participating in the wave motion by moving in a horizontal path as the wave propagates. This produces moving regions of compression and rarefaction in the medium. Note that although the wave moves to the right, the individual particles return to their original positions.

sound (wave speed of sound); =1.402 for airv RT

13

Sound WavesSound Waves We usually think of sound waves as traveling through air, but actually sound can travel through any gas liquid, or solid. The figure shows sound as traveling regions of compression and rarefaction, traveling out from a loudspeaker as a longitudinal wave.

Sound waves in gases and liquids are always longitudinal, but sound in solids can be both longitudinal compression waves and transverse “shear” waves, which usually travel at differing speeds in the medium.

We hear sound in the range of 20 Hz to 20 kHz, but sound waves at higher and lower frequencies are common.

14

Example: Sound WavelengthsExample: Sound Wavelengths

What are the wavelengths of sound waves at the limits of human hearing and at the midrange frequency of 500 Hz?

-120 Hz / (343 m/s) /(20 s ) 17.2 mf v f

-1500 Hz / (343 m/s) /(500 s ) 0.690 mf v f

-120 kHz / (343 m/s) /(20,000 s ) 0.0172 mf v f

15

16

17

Electromagnetic WavesElectromagnetic Waves8

light 299,792,458 m/s 3.00 10 m/s (electromagnetic wave speed in vacuum)v c

814

-9

(3.00 10 m/s)600 nm; 5.00 10 Hz

(6.00 10 m)

cf

18

Example:Traveling at the Speed of Light

Example:Traveling at the Speed of Light

A satellite exploring Jupiter transmits data to the Earth as a radio wave with a frequency of 200 MHz. What is the wavelength of the electromagnetic wave? How long does it take for the signal to travel 800 million km from Jupiter to Earth?

8

8

(3.00 10 m/s)1.5 m

(2.00 10 Hz)

c

f

11

8

(8.0 10 m)2,700 s 45 min

(3.00 10 m/s)

xt

c

19

Index of RefractionIndex of Refraction

speed of light in vacuum

speed of light in material

cn

v

vacmat

mat mat vac

v c c

f nf nf n

Typically, light slows down when it passes through a transparent material like water or glass. The slow-down effect is characterized by the index of refraction of the material:

20

Example:Light Traveling through Glass

Example:Light Traveling through Glass

Orange light with wavelength 600 nm is incident on a 1 mm thick microscope slide.

(a) What is the speed of light in the glass?(b) How many wavelengths of light are inside the slide?

glass vac1.50; 600 nmn

88

glassglass

(3.00 10 m/s)2.00 10 m/s

(1.50)

cv

n

-7vacglass

glass

(600 nm)400 nm 4.00 10 m

(1.50)n

-3

-7glass

(1.00 10 m)2,500

(4.00 10 m)

dN

21

Clicker Question 2Clicker Question 2

Which inequality describes the three indices of refraction?

• n1 > n2 > n3;• n1 > n2 > n3;• n2 > n1 > n3;• n1 > n3 > n2;• n3 > n1 > n2;

22

Power and IntensityPower and Intensity

Intensity: I = P/a (units – W/m2)

23

Example:Intensity of a Laser Beam

Example:Intensity of a Laser Beam

A red helium-neon laser emits 1.0 mW of light power in a laser beam that is 1.0 mm in diameter. What is the intensity I of the laser beam?

-32

2 -3 2

(1.0 10 W)1,270 W/m

(0.5 10 m)

P PI

a r

24

Inverse Square LawInverse Square Law

sphere

2 (intensity for spherical waves)

4

PIa

P

r

2 21 1 2

2 22 2 1

/ 4

/ 4

I P r r

I P r r

1 22

E kA

2 ( is a constant)I CA C

The intensity of a wave is proportional to the square of its amplitude.

25

Chapter 20 - Summary (1)Chapter 20 - Summary (1)

26

Chapter 20 - Summary (2)Chapter 20 - Summary (2)

27

Chapter 20 - Summary (3)Chapter 20 - Summary (3)

End of Lecture 4End of Lecture 4