What is a wave?
We define a wave as disturbance or variation that transfers energy
progressively from point to point. It may take the form of an elastic
deformation, a variation of pressure, electric or magnetic intensity, electric
potential, or temperature.
In short:
1. A disturbance or variation which travels through a medium or just
through space.
2. Must transfer energy from one location to another
Types of waves:
Mechanical waves
A wave which needs a medium in order to propagates
itself. Sound waves, waves in a Slinky, and water waves
are all examples of this. Sound waves need air molecules
in order to exist; the Slinky waves need the Slinky, and
the waves in the ocean need the water
It follows, then, that mechanical waves cannot exist in a
vacuum. This is the factor that distinguishes them from
electromagnetic waves
Electromagnetic Waves
Radio signals, light rays, x-rays, and cosmic rays
Matter Waves
Any moving object can be described as a wave
When a stone is dropped into a pond, the water is
disturbed from its equilibrium positions as the wave
passes; it returns to its equilibrium position after the wave
has passed.
The water moves
up and down as the
disturbance moves
outward.
…………..more mechanical waves: Shock, Sound, Earthquakes
Sonic Boom
v > vsound
Earthquakes
Matter Waves
http://phys.educ.ksu.edu/vqm/html/doubleslit/index.html
Longitudinal Waves:
In a longitudinal wave the particle displacement is parallel to the
direction of wave propagation. The figure below shows a onedimensional longitudinal plane wave propagating through air. The
particles do not move down the tube with the wave; they simply
oscillate back and forth about their individual equilibrium positions
In a transverse wave the particle displacement is perpendicular to the
direction of wave propagation. The animation below shows a onedimensional transverse plane wave propagating from left to right. The
particles do not move along with the wave; they simply oscillate up and
down about their individual equilibrium positions as the wave passes by.
Pick a single particle and watch its motion.
http://www.kettering.edu/~drussell/Demos/waves/wavemotion.html
Both types of waves can move through solids. Only longitudinal waves can
move through a fluid. A transverse wave can move along the surface of a
fluid.
Assume F1 = F2 = F (Tension)
θ + ∆θ
θ
µ (mass/unit lenght)
Fy = − F sin(θ ) + F sin(θ + ∆θ )
≈ T∆θ
θ
Wave equation
µ ∂2 y
∂2 y
= 2
2
F ∂t
∂x
d2y
(dm) 2 = F∆θ
dt
d2y
( µdx) 2 = F∆θ
dt
∂y
∂θ ∂ 2 y
1
tan(θ ) =
and
= 2
2
∂x
cos (θ ) ∂x ∂x
∂2 y
∂2 y
( µdx) 2 = F 2 dx
∂t
∂x
….. continued
∂2 y ∂2 y
F 2 = 2
∂t
∂x
Solution
f ( x ± Ct ) and C =
Single valued function!
F
µ
Dimension of C?
from x ± Ct we see that C must be a velocity v =
1 ∂ y ∂ y
= 2
2
2
∂x
v ∂t
2
2
Wave equation
For -Ct
v
F
µ
Transverse Waves on a String
Attach a mass to a string to provide tension. The string is
then shaken at one end at a frequency f.
L
Attach a
vibrator
here
M
A wave traveling on this string will have a speed of v =
F
µ
where F is the force applied to the string (tension) and µ
is the mass/unit length of the string (linear mass density).
Boundary Conditions
When you have a fixed (closed)
end or a wave that travels from a
“low density” medium to a “high
density” medium, the reflected
wave pulse will be inverted.
When you have loose (open) end
or a wave that travels from a “high
density” medium to a “low density”
medium, the reflected wave pulse
will be not inverted.
http://www.kettering.edu/~drussell/Demos/reflect/reflect.html
Exercise: When the tension in a cord is 75.0 N, the wave
speed is 140 m/s. What is the linear mass density of the
cord?
The speed of a wave on a string is
v=
F
µ
Solving for the linear mass density:
F
75.0 N
−3
µ= 2 =
=
3
.
8
×
10
kg/m
2
v
(140 m/s )
Periodic waves
A periodic wave repeats the same pattern over and over.
For periodic waves: v=λf
v is the wave’s speed
f is the wave’s frequency
λ Is the wave’s wavelength
Periodic waves – normal understanding
of waves
-The disturbance repeats
over and over
-Can be generated by SHM
resulting in SHM of the elements of the medium
The wave is characterized by
Amplitude A: The maximum displacement of the elements of a medium
Frequency f: Number of crests or dips that pass by a point per unit of time
Phase : Where is the wave at time = 0 ?
Propagation speed v
From there we can derive:
Period T: How long it takes for an entire cycle to pass by T = 1/f
Wavelength : The distance from crest to crest or the length of one
complete cycle v = f
Excersise: What is the wavelength of a wave whose
speed and period are 75.0 m/s and 5.00 ms, respectively?
v = λf =
λ
T
Solving for the wavelength:
λ = vT = (75.0 m/s )(5.00 ×10 −3 s ) = 0.375 m
v=
F
µ
y ( x, t ) = A sin
ω
λ = vT = v
2π
ω
2π
λ
→ω = v
Lets introduce a new parameter, the wavenumber, k =
y ( x, t ) = A sin (kx − ωt )
y ( x, t ) = A sin (kx + ωt )
Superposition
y ( x, t ) = y1 + y2 = 2 A sin(kx) cos( wt )
Standing wave
( x − vt )
2π
λ
2π
λ
…………mathematical description continued
A complete mathematical description will give the displacement at any position x
with any instant time t
y ( x, t ) = A cos(kx ± ωt )
y( x, t ) = A sin(kx ± ωt )
+ is used for a wave traveling in the
–x direction, and – is used for a
wave traveling in the +x direction.
k=
2π
is called the wave number.
λ
(ωt ± kx + ϕ ) is called the phase.
The above picture is a snapshot (time is frozen).
Two points on the wave are “in phase” if:
kx − kx = nλ (n= 1, 2, 3,…)
2
1
“Snapshots” of a traveling wave y(x,t) = A cos (ωt ± kx)
where A = 1.0 m, k = 1 rad/m, and ω = π rad/sec.
Wave travels
to the left
(-x-direction)
time
Wave travels
to the right
(+x-direction)
time
Exercise: A wave on a string has an equation:
y ( x, t ) = (4.00 mm )sin ((600 rad/sec)t − (6.00 rad/m )x )
(a) What is the amplitude of the wave?
A = 4.00 mm
(b) What is the wavelength?
The wave number k is 6.00 rad/m.
2π
2π
λ=
=
= 1.05 m
k
6.00 rad/m
(c) What is the period?
2π
2π
T=
=
= 1.05 ×10 − 2 sec
ω 600 rad/sec
(d) What is the wave speed?
λ
ω 600 rad/sec
(2πf ) = =
= 100 m/s
v = λf =
2π
k 6.00 rad/m
(e) What direction is the wave traveling.
Along the +x direction.
Intensity is a measure of the amount of energy/sec that
passes through a square meter of area perpendicular to the
wave’s direction of travel.
Power
P
I=
=
2
4πr
4πr 2
Intensity has units
of watts/m2 .
This is an inverse square law. The intensity drops as the
inverse square of the distance from the source. (Light
sources appear dimmer the farther away from them you are.)
Example: At the location of the Earth’s upper atmosphere,
the intensity of the Sun’s light is 1400 W/m2. What is the
intensity of the Sun’s light at the orbit of the planet Mercury?
Psun
Ie =
4πres2
Psun
Im =
2
4πrms
Divide one equation by the other:
Psun
2
r
I m 4πrms
=
= es
Psun
Ie
rms
4πres2
2
=
1.50 ×10 m
5.85 ×1010 m
∴ I m = 6.57 I e = 9200 W/m 2
11
2
= 6.57