Today’s Lecture
Waves on a String
Wave Power
Interference
Another Example
The figure shows a harmonic wave
as a function of x at t=0, and the
same wave as a function of t at
x =λ/2. Write a mathematical
description of the wave including
the sign of the velocity.
The resulting wave form is:
Wave on a String
Any way to calculate the wave speed? What is it likely to depend on?
Amplitude of the wave? Wave length? Mechanical properties of the
string?
All of those options are plausible, but it turns out the wave speed
only depends on mass of the string (rope) and its tension.
Wave on a String
Waves on a string resemble
harmonic oscillations of a mass
on a spring.
Tension provides the restoring
force, which wants to make the
oscillations more frequent.
Mass of the string provides the
inertia, which slows down
oscillations and wave propagation.
Wave Equation for a Wave on a String
F is the tension in the string
x+dx
x
μ is the mass per unit length
Summing forces in the y direction
Assume that θ <<1 so that
Wave Equation for wave on a string:
Speed and Frequency of a Wave on a String
In general
From the wave equation
The frequency of oscillations is:
f =
v
λ
=
1
F
λ
μ
Increasing tension by a factor of 2?
http://phet.colorado.edu/simulations/sims.php?sim=Wave_on_a_String
Speed and Frequency of a Wave on a String
A mass m1 is attached to a wire run over a
pulley. The wave speed is v = 20m/s. If a
second mass, m2, is added to the first the
wave speed becomes v = 45m/s. (a) Find
the ratio of the second mass to the first.
Since v2 = F/μ, we have:
(b) Find the frequency if the wavelength is 2.25m.
Frequency a Wave on a Guitar String
f =
v
λ
=
1
F
λ
μ
Wavelength, λ = 2L, where L is the length
of the string.
If the string has a mass m, μ = m/L, and we have
1
F
1 F
f =
=
2 L m / L 2 Lm
Thick and heavy strings - long
piano strings – low pitch.
Tight strings – high pitch.
Compare:
a mass on
a spring.
Spring constant
(restoring force)
F
4L
vs.
Frequency of a Wave on a Piano Wire
A 4.5g piano wire is under 680N tension.
Waves propagate at 320m/sec.
(a) What is the mass/length of the wire?
(b) What is the length of the wire?
(c) If the fundamental frequency has a wave length of λ=2L,
what is this frequency?
A propagating wave communicates motion and carries away
energy.
We can define wave power as amount of energy carried
away from the source (and transported through the medium)
per unit time
Wave Power
Defining u to be the vertical velocity of
small region of the string, the rate that
work is done by the tension in the string
is:
For a harmonic wave propagating along a string:
Substituting into the equation for the power, P:
Wave Power
P = FωkA sin (kx − ωt )
2
2
If we average P over time, at any position, x, we find that
< sin (kx − ωt ) > = 1 / 2
2
Therefore the power averaged over time
We can plug in
1
2
P = FωkA
2
k = ω / v and F = μv 2
1
2 2
P = μω A v
2
to get
Let’s take a closer look at the power
equation.
What kind of sense does it make?
1
2 2
P = μω A v
2
μ is the mass per unit length;
ω is the angular frequency of oscillations in the wave
A is the amplitude of the oscillations
v is the wave speed
ωA is the amplitude of velocity of the oscillations of the
material of the string
The power is proportional to the mass per unit length, to the
square of the amplitude of oscillations of the velocity of string
material and to the wave speed.
Wave front is a continuous line or a surface connecting
nearby wave crests.
Plane waves
Wave fronts are flat surfaces for
well directed light, radar or
sound beams, which propagate
in one direction without
spreading
Wave fronts are straight lines for
ripples on water surface at shore
line.
Why do we draw the lines
through troughs rather than
crests? It does not really matter.
Only shape matters.
“Straight” waves
wave fronts
Wave front is a continuous line or a surface connecting
nearby wave crests.
Wave fronts are spherical
surfaces for spherical waves
originating from a point source
and propagating in 3D space.
Wave fronts are circles for
waves on water surface
originating from a point source.
Spherical and circular waves becomes less intense as they
travel further away from their source, because the same
power emitted by the source is spread over larger area
(circumference).
Wave intensity is the wave
power per unit area
P
I=
A
Measured in
J
W
or 2
2
s⋅m
m
For a plane wave the
intensity remains constant.
For a spherical wave it
decreases with the distance,
r, from the source, like
P
I=
2
4πr
Does it mean that the power is lost as the wave propagates?
Power P of the source is 1000 watts.
The energy spreads outward through
the air. At a distance r from the source
the area A of the spherical front is 4πr2
Intensity = number of joules per
second per square meter
= power/area = P/A
Example: r = 10 m
1000 Joules per second of sound
energy is generated.
Spherical wave fronts spread
outward from sound source at the
center.
P
1000 W
2
I=
=
≈
0
.
8
W
/
m
4πr 2 4π ⋅ 100 m 2
Is this a rock concert or a whisper?
From the table we see that .8W/m2 is similar to that for a loud rock band.
Is the light coming from the Sun a spherical or a plane wave?
At the Earth surface the radius of curvature of the wave front is 150,000,000km,
The intensity is the same everywhere, and the wave is practically flat.
It is a spherical wave on the scale of the Solar system, though.
At the Earth surface the radius of
curvature of the wave front is
150,000,000km,
The intensity is the same everywhere,
and the wave is practically flat.
It is a spherical wave on the scale of
the Solar system, though.
Remember:
P
I=
4πr 2
A light bulb is a source of a
spherical wave.
If you are 5ft away from the bulb
and walk another 5ft away the light
intensity decreases by a factor of
22 = 4.
In terms of intensity of sunlight, it
is about the same as going from
Venus to mars
(a) If Intensity (flux) of the energy from the
Sun is 1.3kW/m2 at the radius of the Earth,
what is the power output of the Sun?
At the Earth surface the radius of curvature
of the wave front is 150,000,000km,
(b) At what distance is the
intensity of a 40W bulb the same
as that for a 75W bulb at 1.9m?
If there are big waves and ripples on the water surface, which are
going to run faster?
For some kinds of waves the wave speed of a simple harmonic
wave depends on the wave length.
This phenomenon is called dispersion.
Example – for large waves on the surface of deep water
λg
v=
2π
When two (or more) waves of the same kind propagate through the
same region produce a composite wave.
This phenomenon is called interference.
It is constructive, when the waves reinforce each other.
It is destructive, when they reduce each other’s amplitude.
Usually the disturbances (displacements) the waves produce are
added algebraically. This is called superposition principle.
Superposition of pulses:
Constructive Interference
Superposition of pulses: Destructive Interference
For a moment it the string becomes a straight line – no disturbance
is seen. Does the energy of wave motion disappear? Where does it
go? How can the two waves go on after that?
Two kinds of energy in the wave motion: potential – depends on deflection of
the string from the straight line; kinetic – depends on velocity of motion of the
string. In the very moment, when the string becomes straight, it is actually
moving very fast - high K.
Reflection of a wave on string.
Full or partial reflection always occurs at a boundary the wave cannot cross
For example, with a medium, which does not support the waves of that kind.
Reflection depends a lot on the boundary conditions: a clamped string vs. a
freely sliding string.
http://phet.colorado.edu/simulations/sims.php?sim=Wave_on_a_String
Reflection of a wave on string.
A partial reflection always occurs if properties of the medium change abruptly.
Abruptly compared to what?
Examples: reflection of sound off a rock, light off a glass window…
Two connected strings. What is different between them, F? μ?
How do the wave speeds in them compare? What about the amplitudes?
v=
F
μ
1
1
2 2
2
2
P = μω A v =
Fω μ A
2
2
When there are two interfering waves with close but different
frequency the result of the interference are beats.
They are perceived as a wave with an average frequency, but
with a slowly oscillating amplitude.
y1 (t ) = A cos(ω1t )
y2 (t ) = A cos(ω2t )
Resulting composite wave:
y (t ) = y1 (t ) + y2 (t ) = A cos(ω1t ) + A cos(ω2t )
After a little trigonometry or making use of Euler’s formula :
1
1
y (t ) = 2 A cos[ (ω1 − ω2 )t ] ⋅ cos[ (ω1 + ω2 )t ]
2
2
When there are two interfering waves with close but different
frequency the result of the interference are beats.
They are perceived as a wave with an mean frequency, but with a
slowly oscillating amplitude.
1
1
y (t ) = 2 A cos[ (ω1 − ω2 )t ] ⋅ cos[ (ω1 + ω2 )t ]
2
2
slowly oscillating amplitude
mean frequency
When there are two interfering waves with close but different frequency the
result of the interference are beats, with beat frequency ω = ω1 –ω2.
They are perceived as a wave with an average frequency, but with a
slowly oscillating amplitude.
http://www.kettering.edu/~drussell/Demos/superposition/superposition.html