PHYS 218sec. 517-520

ReviewChap. 15

Mechanical Waves

What you have to know

• Defining waves and their properties

• Wave functions and wave dynamics

• Energy or power of waves

• Principle of superposition

Types of mechanical waves

Mechanical wave: a disturbance that travels through some material (called medium).

Transverse wave

The displacements of the medium are perpendicular (transverse) to the direction of travel of the waveE.g: wave on a string. In this case the wave travels in the x-direction while the medium (particle of the string) is moving in the y-direction

Direction of the travel of the wiggle

The point on the string moves vertically.The string itself does NOT move with the wiggle.

Longitudinal wave

Types of mechanical waves (2)

Direction of the travel of the wave

The displacements of the medium are back and forth along (longitudinal) the direction of travel of the waveE.g: when the medium is a liquid or gas in a tube

As the wave passes, each particle of the fluid moves back and forth parallel to the motion of the wave.

But the fluid particles do NOT move away from their original (equilibrium) positions.

In addition, there are waves which have both longitudinal and transverse components.

Wave speed

The speed of the wave propagating through the medium.This is determined by the mechanical properties of the medium.This is NOT the speed of the particle in the medium.

Periodic waves

When each particle in the medium is in periodic motion

Sinusoidal waves

Periodic waves when the particles are in Simple Harmonic Motion.Sine or Cosine wavesAny periodic waves can be represented as a combination of sinusoidal waves.

Wavelength (): the length of one complete wave pattern

wave speed and wavelength: v fl=

Wave function

The function that describes the position (i.e. y-direction) of any particle (i.e., at any x value) at any time (i.e. at any t).

wave function: ( , ) depends on and y x t y x t

Wave function contains all information on the wave.

( )

( )

Sinusoidal wave moving in -direction

( , ) cos 2 cos

Sinusoidal wave moving in -direction

( , ) cos 2 cos

x

x ty x t A A kx t

T

x

x ty x t A A kx t

T

p wl

p wl

+

æ ö÷ç= - = -÷ç ÷çè ø

-

æ ö÷ç= + = +÷ç ÷çè ø

2wave number k

pl

=

Wave function

To read the wave speed from the wave function, set the phase be constant

const

Then take the time derivative, which gives

0

2Since , we have

2

kx t

dxkdt

dx fv v f

dt k

w

w

w pl

p l

- =

- =

= = = =

Particle velocity and acceleration

22 2

2

2

2 2

( , ) cos( )

then

( , )( , ) sin( )

( , )( , )( , ) cos( ) ( , )

1 ( , )( , )

y

yy

y x t A kx t

y x tv x t A kx t

tv x ty x t

a x t A kx t y x tt t

y x ty x t

t

w

w w

w w w

w

= -

¶= = -

¶¶¶

= = =- - =-¶ ¶

¶Þ =-

¶

22 2

2

2 2 2 22 2

2 2 2 2 2

2 2

2 2 2

On the other hand,

( , )cos( ) ( , )

Then we have

( , ) 1 ( , ) ( , )( , )

Since / ,

( , ) 1 ( , )

y x tk A kx t k y x t

x

y x t y x t k y x tk y x t k

x t t

v k

y x t y x t

x v t

w

w w

w

¶=- - =-

¶

æ ö¶ ¶ ¶÷ç ÷=- =- - =ç ÷ç ÷ç¶ ¶ ¶è ø

=

¶ ¶=

¶ ¶

Wave equation

Wave equation

2The wave equation contains , so it holds for the wave

propagating in the negative direction.

You can see that ( , ) cos( ) satisfies the wave equation above.

v

x

y x t A kx tw= +

Wave speed

Wave speed is determined by the properties of the medium. It does not depend on the shape of the wave.

x x x+D

1F

2F2 1

2 1 2 1

In direction,

Since there is no motion in direction

(note that the string itself doesn't move to direction)

0

x x x

x x x x x

x

F F F

x

x

F F F F F F

= -

= - = Þ = =

å

å

( )

1 2

2

2 1 2

In direction,

slope and we set

Therefore,

yx x

x

y y yx x x

y

F yF F F

F x

y y yF F F F F F x

x x x+D

æ ö¶ ÷ç= = = =÷ç ÷çè ø¶

æ öæ ö æ ö¶ ¶ ¶ ÷ç÷ ÷ç ç ÷= - = - = D÷ ÷ çç ç ÷÷ ÷ç ç ç ÷çè ø è ø¶ ¶ ¶è øå

( )

( ) ( )2 2 2 2

2 2 2 2

Since the mass of the segment is

, where is the mass line density, and ,

By comparing with the wave equation, we get

m x F ma

y y y yx F x

t x x F t

Fv

m m

mm

m

= D =

æ ö¶ ¶ ¶ ¶÷ç ÷D = D Þ =ç ÷ç ÷ç¶ ¶ ¶ ¶è ø

=

In general

restoring force

inertiav=

Ex 15.3 Wave speed

length of the rope: 80 m, mass of the rope: 2 kgL m= =

20 kgM =

creates a wave

Speed of the transverse wave

tenstion and for this case,

inertia

2 kgmass density 0.025 kg/m

80 mtension: mass is attached 20 9.8 N 196 N

196 N88.5 m/s

0.025 kg/m

Fv v

m

LM T Mg

Fv

m

m

m

= =

= = =

Þ = = ´ =

\ = = =

If the frequency is 2.0 Hz, how many cycles are there in the rope’s length?

88.5 m/sWe first should know the wavelength of the wave 44.3 m

2 /s

the number of wave cycles in the rope 1.81 cycles

v

f

LN

l

l

= = =

Þ = =

Energy in wave motion

Waves transport energy

power

( . )

( , ) ( , ) ( , )

since slope: the force exerted by the left side of the segment

y y

cf P Fv

y yP x t F x t v x t F

x tF F

=

¶ ¶= =-

¶ ¶=- ´

Remember that the particles are moving vertically

2 2 2 2 2

2 2

0

From

( , ) cos( ),

sin( ), sin( )

( , ) sin ( ) sin ( )

Thus, the average power is

1 1( , )

2

T

av

y x t A kx t

y ykA kx t A kx t

x t

P x t Fk A kx t F A kx t

P P x t F AT

w

w w w

w w m w w

m w

= -

¶ ¶=- - = -

¶ ¶

Þ = - = -

= =ò

Wave intensity

Intensity (I): average power per unit area

1r

2r 22

1 1 2 2

22 2 1 2

1 1 2 2 22 1

, 4 the surface area4

Energy is conserved, so the power is constant.

Therefore, the relation between at and at is

4 4

PI r

r

I r I r

I rP r I r I

I r

pp

p p

= =

= = Þ =

Ex 15.3At a distance of 15 m the intensity of a sound wave is 0.25 W/m2.

At what distance from the siren is the intensity 0.01 W/m2?

( )

21 2

22 1

21

2 1 22

Since ,

0.25 W/m15 m 75 m

0.01 W/m

I r

I r

Ir r

I

=

= = =

Unit of power = WUnit of intensity = W/m2

Principle of superposition

When two waves meet, the net wave function is the algebraic sum of the two wave functions

1 2( , ) ( , ) ( , )y x t y x t y x t= +

When two waves meet, they overlap. This overlapping is called interference.

If the two waves are in phase, their amplitudes are added and the amplitude of the net wave is larger : constructive interference

If the two waves are out of phase, their amplitudes are subtracted and the amplitude of the net wave is smaller : destructive interference

Reflection of a wave

Fixed end Free end

180 phase difference°

Incident wave

Reflected wave

Standing wave

When a sinusoidal wave is reflected by a fixed end.

Incident wave

Reflected wave

Interference of the two waves

This wave does not move.

Therefore,STANDING WAVE

node

antinode

( )

1

2

1 2

incident wave: ( , ) cos( )

reflected wave: ( , ) cos( )

By interference, ( , ) ( , ) ( , ) 2 sin sin

At ( 0,1,2....) , 0 always: positions of nodes2

y x t A kx t

y x t A kx t

y x t y x t y x t A kx t

n nx n y

k

w

w

w

p l

=- +

= -

= + =

= = = =

Destructive interferenceConstructive interference

Standing wave

Normal modes of a string

If the string has length L

For standing waves, both ends of the string should be positions for nodes of a standing wave

Therefore, ( 1, 2,3....)2

L n nl

= =

1

1

In other words, the wavelength of a standing wave should be

2 (There are many possible values of . We label it as )

2

: called the fundamental frequency

n n

nn

L

nv v

f n nfL

f

l l l

l

=

Þ = = =

1, : fundamental frequency2

L fl

= 2 1, ( 2 ) : second harmonicL f fl= =