P1X: Optics, Waves and Lasers Lectures, 2005-06. 1

Lecture 2: Introduction to wave Lecture 2: Introduction to wave theory (II)theory (II)

Phase velocity: is the same as speed of wave:

constkxtkxtAtxy )sin(),(

fk

vkvdt

dxkkxt

dt

d 0

Phase velocity: the velocity (speed) at which we would have to move to keep up with a point of constant phase on the wave.

Right moving wave:

Derivative with respect to t is zero:

kv

fv

Transverse speed of wave:

kxtdt

dkxtAkxtA

dt

d

dt

dyv y )cos()sin(

)cos( kxtAv y

Mathematical description of waves (Y&F 15.3):

P1X: Optics, Waves and Lasers Lectures, 2005-06. 2

)sin(),( kxtAtxy

Transverse speed maximum:

Transverse speed minimum:

,2,,0)cos( kxtAkxtAv y

,2

3,

20)cos(

kxtkxtAv yy

3/4/2/4x

- A

A

vy

3/4/2/4x

- Aw

Aw

)cos( kxtAv y

Offset by 90o

P1X: Optics, Waves and Lasers Lectures, 2005-06. 3

Example: 15-2 from Y&F (page 556)Find the maximum transverse speed of the example shown in lecture 1. What is the velocity at t=0, at the end of the clothes-line and at 3.0 m from the end.

)cos( kxtAvy 1max, 94.04075.0 msAvy

mxt

kxtAtxy

0.65.02sin075.0sin),(

194.0)0cos(0,0.3 msAAtxvy

At x=t=0, velocity is maximum transverse speed = +0.94 ms-1

At x=3.0 m and t=0:

P1X: Optics, Waves and Lasers Lectures, 2005-06. 4

Definition:• Simple Harmonic Motion (SHM) is motion in which a particle is acted on by a force proportional to its displacement from a fixed (equilibrium) position and is in the opposite direction to the displacement:

Simple Harmonic Motion (Y&F 13.1-2, 13.4-5):

xxm

k

dt

xdakxF 2

2

2

Examples:• Mass vibrating on a spring.• Simple pendulum (only when displacement is small).

P1X: Optics, Waves and Lasers Lectures, 2005-06. 5

Simple pendulum:• Vertical:

• Horizontal:

If is small then when

and

therefore:

and:

,0cos ymaWT

xmaT sin

L

xsin Lx

1cos

mgT

mgW

xxL

g

m

Tax

2sin

x

mT

W

L

The same as the restoring force of a spring but with:L

mgk

P1X: Optics, Waves and Lasers Lectures, 2005-06. 6

Solution:• What function satisfies ?

•Try

xdt

xda 2

2

2

xtAdt

xd

tAdt

dx

tAx

222

2

)sin(

)cos(

)sin(

x

T3T/4T/2T/4 t

- A

A

with the angular frequency (rad/s):

• For the case of the spring:

•For the case of the pendulum:

m

k

L

g

P1X: Optics, Waves and Lasers Lectures, 2005-06. 7

Definitions: a) Amplitude A is maximum displacement (m).

b) Frequency f: number of oscillations per second.(Units: 1 Hertz = 1 cycle/s = 1 s-1)

c) Period T: time (s) between oscillations

d) Phase constant (): gives position of oscillation at t=0.

2

f

21

f

T

)sin()0( Atx

P1X: Optics, Waves and Lasers Lectures, 2005-06. 8

Example: 13-2 from Y&FA spring is mounted horizontally. A force of 6.0 N causes a displacement of 0.030 m. If we attach an object of 0.50 kg to the end and pull it a distance of 0.020 m and watch it oscillate in SHM, find (a) the force constant of the spring, (b) the angular frequency, frequency and period of oscillation.

(a) At x = 0.030 m, F=-6.0 N

(b) m=0.50 kg, k=200 N/m:

The frequency:

The period:

mNm

N

x

Fk /200

030.0

0.6

sradkg

skg

m

k/20

50.0

/200 2

Hzscyclecyclerad

sradf 2.3/2.3

/2

/20

2

ssf

T 31.02.3

111

P1X: Optics, Waves and Lasers Lectures, 2005-06. 9

Example: 13-8 from Y&FFind the frequency and period of a simple pendulum that is 1.0 m long (assume g=9.80 m/s). The angular frequency:

The frequency:

The period:

sradm

sm

L

g/13.3

0.1

/80.9

Hzscyclecyclerad

sradf 4983.0/4983.0

/2

/13.3

2

ssf

T 007.24983.0

111

P1X: Optics, Waves and Lasers Lectures, 2005-06. 10

Example: Vertical SHMVertical oscillations from a spring hanging vertically.1) At rest: Spring is stretched by l such that: 2) x above equilibrium:

3) x below equilibrium:

Same SHM as in vertical case, oscillations with angular frequency:

m

k

mglk

kxmgxlkFnet )()(

kxmgxlkFnet )()( Fnet

Equilibrium is at stretched position l instead of x=0

P1X: Optics, Waves and Lasers Lectures, 2005-06. 11

Example: 13-6 from Y&FShock absorbers of an old car with mass 1000 kg are worn out. When a person weighing 100 kg climbs into the car, it sinks by 2.8 cm. When the car is in motion and hits a bump it oscillates. What is the frequency and period of oscillation?

The spring constant:

The angular frequency:

The frequency:

The period:

sradkg

kgs

m

k/64.5

1100

105.3 24

Hzscyclecyclerad

sradf 898.0/898.0

/2

/64.5

2

ssf

T 11.1898.0

111

241

105.3028.0

8.9100

kgs

m

mskg

x

mg

x

Fk

P1X: Optics, Waves and Lasers Lectures, 2005-06. 12

Simple Harmonic Motion initiates sinusoidal waves and sets the boundary conditions for wave motion

• For example, a string attached to a vertical spring

kxF

mg

• A radio transmitting antenna causes electromagnetic waves by oscillating molecules

v

v

)sin(),()sin()0,( kxtAxtytAxty