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.
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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)
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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
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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
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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
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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
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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)