1Animations from: Wikipedia and http://zonalandeducation.com/mstm/physics/waves/partsOfAWave/waveParts.htm#pictureOfAWave

WAVES

Antonio J. Barbero, Mariano Hernández, Alfonso Calera, Pablo Muñiz, José A. de Toro and Peter Normile

Dpt. of Applied Physics. UCLM

2

A wave is a periodic disturbance in space and time, able to propagate energy. The wave equation describes mathematically how the disturbance proceeds across the space and over time.

Transverse waves: The oscillations occur perpendicularly to the direction of energy transfer. Exemple: a wave in a tense string. Here the varying magnitude is the distance from the equilibrium horizontal position.

Longitudinal waves: Those in which the direction of vibration is the same as their direction of propagation. So the movement of the particles of the medium is either in the same or in the opposite direction to the motion of the wave. Exemple: sound waves, what changes in this case is the pressure of the medium (air, water or whatever it be).

Vibration

PropagationVibrationPropagation

A kind of transverse waves can propagate in the vacuum (electromagnetic waves). However, longitudinal waves can only propagate in a material medium.

3

INTRODUCTORY MATH OF WAVES

tvxfy Wave equation

Sign +

Waveform traveling to the right

Waveform traveling to the left

Sign -

Space Time

Phase velocity

0,0 0,5 1,0 1,5 2,0 2,5 3,0-0,05

0,00

0,05

0,10

0,15

X

Y

0,0 0,5 1,0 1,5 2,0 2,5 3,0-0,05

0,00

0,05

0,10

0,15

X

Y tvxfy

tvxfy

Waveform f

Waveform f

The wave equation describes a traveling wave if the group (x vt) is present. This is a necessary condition. (The term traveling wave is used to emphasize that we refer here to waves propagating in the medium, not to standing waves that we will consider later)

4

Harmonic wave moving to the right

tvxAy 2

sin

y

x

Wave equation

tvxAy 2

cos

or

HARMONIC WAVES

We can choose any of them by adding an initial phase 0 into the argument of the function…

A wave is said to be harmonic when its waveform f is either a sine or a cosine function ?

…what physically means that we choose the initial time upon our convenience

One more stuff:

Whenever a harmonic wave propagates through a medium, every point in the medium describes a harmonic motion

0xx

For exemple: If the wave reaches a maximum for t = 0 and we choose as a reference the cosine waveform, we have that 0 = 0 and the wave equation becomes simply

2/2

sin

tvxAy

00

2cos

tvxtyx

y

2/0

0

2cos

tvxAy

That describes exactly the same wave

tvxAy 2

cos

What do we have to do to write the same waveform by using the sine form?

Answer:

Remember: cos2/sin cos2/cos sin2/sin

Wave profile for t = 0

y depends only upon the time

0xx is a distance

5Time dependence for x = x0

t

y

Wave profile for t = t0

y

x

HARMONIC WAVES / 2

0

2cos

tvxAy

Harmonic wave equation (choosing cosine form)

Phase velocity

Space Time

Remember: cosine is periodic. Periodic function is that which verifies

See that harmonic waves have double periodicity

Ttftf

Period

0

2cos

tvxAy

Phase

Amplitude

Initial phase

Displacement

1tt

10 , txy

2tt

20 , txy

T

T

space

time

Trough

Crest

A

-A

01, txy

1xx

02 , txy

2xx

Same phase points

Wavelength

Period

Snapshot graph History graph

6

(s) t2

2(m) x

HARMONIC WAVES / 3

Harmonic wave equation (choosing cosine form)

Displacement: current value of the magnitude y, depending upon space and time. Its maximum value is the amplitude A.

Wavelength : distance between two consecutive points whose difference of phase is 2.

Wavenumber k: is the number of waves contained into a turn (2 radians). Sometimes it is called angular or circular wavenumber.

m 3/2 1-m 3

3/2

22

k

Its units (I.S.) are rad/m, but often they are referred as m-1.

1st wave 2nd wave 3rd wave

Period T: time elapsed till the phase of the harmonic wave increases 2 radians.

Frequency f: is the inverse of the period, so the frequency tells us the number of oscillations per unit of time. Its units (I.S.) are s-1 (1 s-1 = 1 Hz).

Angular requency : is the number of oscillations in a phase interval of 2 radians.

2

k

fT

22

Tf

1

Phase velocity is given by the quotientkT

v

0

2cos

tvxAy

Phase velocity

Space Time

Amplitude

Initial phase

Displacement

0

2cos

tvxAy

Phase

In terms of wavenuber and angular frequency the harmonic wave equation can be written as txkAy cos rad/s 8

2

T

Hz 41

Tf

s 4/T

7

Wave equation 24

4

tvxy

where x, y are in meter, t in seconds, v = 0.50 m/s

Let us to plot y for different values of time

-8 -6 -4 -2 0 2 4 6 80,0

0,2

0,4

0,6

0,8

1,0

-8 -6 -4 -2 0 2 4 6 80,0

0,2

0,4

0,6

0,8

1,0

-8 -6 -4 -2 0 2 4 6 80,0

0,2

0,4

0,6

0,8

1,0

x (m)

y (m) t = 0

t = 5t = 10

SOME EXAMPLES

Example 1: traveling pulse

Each of those profiles indicates the shape of the pulse for the given time.

This pulse moves to the right (positive direction of X axis) with a velocity of 0.50 m/s

8

Wave equation 221

2sen

tx

txy

where x, y are in meter, t in seconds

Plotting for different values of time

Exemple 2: traveling pulse

-4 -3 -2 -1 0 1 2 3 4-0,5

-0,4

-0,3

-0,2

-0,1

0,0

0,1

0,2

0,3

0,4

0,5

x (m)

y (m)

-4 -3 -2 -1 0 1 2 3 4-0,5

-0,4

-0,3

-0,2

-0,1

0,0

0,1

0,2

0,3

0,4

0,5

-4 -3 -2 -1 0 1 2 3 4-0,5

-0,4

-0,3

-0,2

-0,1

0,0

0,1

0,2

0,3

0,4

0,5

t = 0

t = 2

t = 4

Each of those profiles indicates the shape of the

pulse for the given time.

Let us to write the wave equation in such a way that the group x+v·t appears explicitly.

2

241

22sen

t

x

tx

y

This pulse moves to the left (negative direction of X axis) with a velocity of 0.50 m/s. See that vt = t/2.

SOME EXAMPLES / 2

9

Harmonic wave txy cos

Exemple 3: harmonic traveling wave

where x, y are in meter, t in seconds

Compare with

0 1 2 3 4 5 6 7 8 9 10-1,2

-1,0

-0,8

-0,6

-0,4

-0,2

0,0

0,2

0,4

0,6

0,8

1,0

1,2

x (m)

y (m)

0 1 2 3 4 5 6 7 8 9 10-1,2

-1,0

-0,8

-0,6

-0,4

-0,2

0,0

0,2

0,4

0,6

0,8

1,0

1,2

t = 0

0 1 2 3 4 5 6 7 8 9 10-1,2

-1,0

-0,8

-0,6

-0,4

-0,2

0,0

0,2

0,4

0,6

0,8

1,0

1,2

t = 2

t = 1

Hz s 2

11 1-

Tf

s 2T

m 2

SOME EXAMPLES / 3

This wave moves to the right (positive direction of X axis) with a velocity of 1.00 m/s

m/s 1m 1

rad/s 11-

k

v

txkAy cos 2

m 1 1- k

T

2rad/s 1

m 1A m/s 1m 2

m 2

Tv

10

Harmonic wave txtxy 2sin2cos

Exemple 4

where x, y are in meter, t in seconds

SOME EXAMPLES / 4

x (m)

y (m)

2t0t 4t

This wave moves to the right (positive direction of X axis) with a velocity of 0.50 m/s

Wavenumber and angular frequency

rad/s 1

tkxtkxy sincos

-1m 2k

m 2 k

s 22

T

1-s 2

11

Tf

m/s 5.0m 2

rad/s 11-

k

v

Phase velocity

Comparing A = 1 m, and

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VELOCITY OF MECHANICAL WAVES

T

v

B

v

Y

v

LL

AFY

/

/

strain

stress

VV

PB

/increment volume

pressure

Mechanical waves need a material medium to propagate.Its velocity of propagation depends upon the properties of the medium.

Fluids density of the fluid (kg/m3)

Compressibility modulus

Solids density of the solid (kg/m3)Young modulus

String linear density of the string (kg/m) (N) string theoftension T

VELOCITY AND ACCELERATION OF THE PARTICLES OF THE MEDIUM

txkAy cos

txkAt

yy sin

yAtxkAt

yy cos 22

2

2

Maximum velocity Ay max

Maximum acceleration Ay 2

max

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WAVES CARRY ENERGY

Every section of the string (mass m) moves up and down because the energy carried by the wave.

Let us consider a transverse wave in a tensestring.We’ll see that as the wave passes through, every point of the string describes a harmonic motion

x x

mA

From the wave equation we obtain for the element m in the fixed position x0

txkAy cos 0

Taking into account that k.x0 is constant, this can be rewritten as

tAy cos

This is the equation of the harmonic motion described by the mass element m. The angular frequency of that motion is .

Let us remind that the energy of the mass m in a harmonic motion (angular frequency , amplitude A) is given by

0x

2 2

1 AmE

Maximum velocity

Let be the mass per unit of lenght x of the string xm

xAE 2

1 22

tvx

tvAE 2

1 22

Power transmitted by the wave

2

1 22 vAt

EE

Units: Joule/second = watt

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STANDING WAVES

A standing wave is the result of the superposition of two harmonic wave motions of equal amplitude and equal frequency which propagate in opposite directions through a medium.

However the standing wave IS NOT a traveling wave, since its equation does not contain terms of the form (k x - t).

For simplicity, we will take as an example to illustrate the formation of standing waves a transverse wave that propagates towards the right () on a string attached at its ends. This wave, reflected on the right end, arises a new wave propagating in the left direction ()

Incident wave, direction (): )cos(1 tkxAy

When the traveling wave (towards the right) is reflected at the end, its phase changes radians (it is inverted).

Reflected wave, direction (): )cos(2 tkxAyT

fk

2

2 2

)cos(sin)sin(cos)cos()cos(2 tkxAtkxAtkxAtkxAy

)cos(1 tkxAy

)cos(2 tkxAy

tkxAtkxA sinsincoscos

tkxAtkxA sinsincoscos

tkxAtkxAtkxAyyy sinsin2)cos()cos(21

Every point of the string vibrates with harmonic motion of amplitude 2A sen kx: see that the amplitude depens upon the position, but the group kx-t does not appear. This is to say, the result is not a traveling wave.

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As the ends of the string are fixed, the vibration amplitude at those points must be zero. If we call L the length of the string, at any time the following conditions must be verified:

Does any pair of incident and reflected waves arise standing waves in a string, does not matter which the frequency or the wavenumber are? NO!

00sin20

Ayx

0sin2

kLAyLx

nL 2

2

nL

The equation L = n/2 means that standing waves only appear when the length L of the string is an integer multiple of a half-wavelength.

T

L

nfn 2

n

Ln

2

STANDING WAVES / 2

tkxAyyy sinsin221

,...3,2,1nnkL

For a given lenght L, the standing waves appears only when the frequencies satisfy that condition.

n

Ln

2

L

vnfn 2

From the relationship among frequency and wavelength (f = v/, where v is the propagation velocity)

nn

vf

T

v Velocity is given by ...3 ,2 ,1n

n = 1 f1 fundamental frequency

n > 1 fn higher harmonics

NodeNode Node Node Node

Anti-node Anti-node Anti-node Anti-node

This exemple:4th harmonics

n = 4n+1 nodesn antinodes

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A standing wave on a string

7th HARMONIC

Weights to tense the string

n = 1 f1 fundamental frequency

n = 2 f2 2nd harmonic

n = 3 f3 3rd harmonic

STANDING WAVES / 3

16

STANDING WAVES / EXEMPLE

Two traveling waves of 40 Hz propagate in opposite directions along a 3 m-lenght tense string given rise to the 4th harmonic of a standing wave. The mass of the string is 510-3 kg/m.

nn

vf

T

v

m 5.14

3224

n

Ln

4th harmonic means n = 4 from L = n/2 we obtain

a) Find the tension of the string

m/s 605.14044 fv

N 1860105 232 vT

b) The amplitude of the antinodes is

4 sinsin2 ntxkAy nnn

3.25 cm. Write the equation of this harmonic of the standing wave

1-

44 m

5.1

22

k

rad/s 80 2 nn f

cm 25.32 A

(cm) 80sin 5.1

2sin25.3 txy

c) Find the fundamental frequency for this tense string.

11

vf

m 61

321

The velocity of propagation is constant, and we have the fundamental frequency when

Hz 106

60

11

v

f (All harmonics are integer multiples of the fundamental frequency, so f4 = 4 f1)