General Physics (PHY 2130)
Lecture 27
•  Waves
  mathematical description of waves
  interference and diffraction
  standing waves
http://www.physics.wayne.edu/~apetrov/PHY2130/
Lightning Review
Last lecture:
1.  Waves
  types of the waves (transverse/longitudinal)
  speed of the wave
Review Problem: The speed of a wave on a string depends on
1.
2.
3.
4.
the amplitude of the wave
the material properties of the string
both of the above
neither of the above
3
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
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.)
Intensity has units of watts/m2 .
4
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?
Recall:
Psun
Ie =
4π res2
Psun
Im =
2
4π rms
Divide one equation by the other:
Psun
2
2
2
11
I m 4π rms ⎛ res ⎞ ⎛ 1.50 ×10 m ⎞
⎟⎟ = 6.57
=
= ⎜⎜ ⎟⎟ = ⎜⎜
10
Psun
Ie
⎝ rms ⎠ ⎝ 5.85 × 10 m ⎠
4π res2
I m = 6.57 I e = 9200 W/m 2
5
Periodic Waves
A periodic wave repeats the same pattern over and over.
For periodic waves: v = λf
v = the wave’s speed
f = the wave’s frequency
λ = the wave’s wavelength
Period T: the amount of time it
takes for a point on the wave to go
through one complete cycle of
oscillations.
frequency
f = 1/T.
Amplitude (A): The maximum
displacement from equilibrium
The intensity of a wave is proportional
to the square of its amplitude
6
One way to determine the wavelength is by measuring the distance between
two consecutive crests.
The maximum
displacement from
equilibrium is amplitude
(A) of a wave.
7
Example: What is the wavelength of a wave whose speed and period are 75.0
m/s and 5.00 ms, respectively?
Recall the formula that connects wavelength and frequency:
v = λf =
λ
T
the wavelength:
λ = vT = (75.0 m/s)(5.00 × 10 −3 s )
= 0.375 m
8
Mathematical Description of a Wave
To describe a wave, we must know the position of the particles in the medium.
This requires a function of the form y(x,t).
y( x, t ) = A cos(ωt ± kx)
+ is used for a wave traveling in the -x
direction, and - is used for a wave traveling in
the +x direction.
k=
2π
λ
(ωt ± kx)
is called the wave number.
is called the phase.
Note: it would also be valid to use the sine function in the above description.
9
The above picture is a snapshot (time is frozen). Two points on the wave are “in
phase” if:
kx2 − kx1 = 2π n
x2 − x1 = nλ
(n = 1, 2, 3,…)
10
Example: A wave on a string has an equation:
y( x, t ) = (4.00 mm)sin((600 rad/sec) t − (6.00 rad/m) x)
Compare this to
y( x, t ) = Asin(ωt − kx)
(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
11
Example continued:
(c) What is the period?
2π
2π
T=
=
= 1.05 ×10 − 2 sec
ω 600 rad/sec
(d) What is the wave speed?
⎛ λ
v = λf = ⎜
⎝ 2π
ω 600 rad/sec
⎞
= 100 m/s
⎟(2πf ) = =
k 6.00 rad/m
⎠
(e) What direction is the wave traveling.
Along the +x direction.
12
Graphing Waves
The next two slides show three “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.
13
Wave travels to the
left
(-x-direction)
time
14
Wave travels to the
right
(+x-direction)
time
15
The Principle of Superposition
When two or more waves overlap, the net disturbance at any point is the
sum of the individual disturbances due to each wave.
Animations:
http://www.acs.psu.edu/drussell/
demos/superposition/
superposition.html
16
Interference and Diffraction
Two waves are considered coherent if they have the same frequency and
maintain a fixed phase relationship.
When waves are in phase, their superposition gives constructive
interference.
When waves are one-half a cycle out of phase, their superposition gives
destructive interference.
17
When two waves travel
different distances to reach
the same point, the phase
difference is determined by:
d1 − d 2
λ
phase difference
=
2π
18
Diffraction is the spreading of a wave around an obstacle in its path.
19
Standing Waves
Pluck a stretched string such that y(x,t) = A sin(ωt + kx)
When the wave strikes the wall, there will be a reflected wave
that travels back along the string.
The reflected wave will be 180° out of phase with the wave
incident on the wall. Its form is y(x,t) = -A sin (ωt - kx).
The interference of the incident and reflected wave would result
a standing wave:
A node (N) is a point of
zero oscillation.
An
antinode (A) is a point of
maximum displacement.
All points between nodes
oscillate up and down.
20
The reflected wave will be 180° out of phase with the wave incident on
the wall. Its form is y(x,t) = -A sin (ωt - kx).
Apply the superposition principle to the two waves on the string:
y ( x, t ) = y1 ( x, t ) + y2 ( x, t )
= A(sin (ωt + kx ) − sin (ωt − kx ))
= (2 A cos ωt )sin kx
This is the mathematical form of a standing wave.
The nodes occur where y(x,t) = 0. That is when sin kx = 0
The antinodes occur when y(x,t) = 2Acosωt that is when sin kx=± 1;
21
Consider the mathematical form of a standing wave.
y( x, t ) = (2 A cosωt )sin kx
The nodes occur where y(x,t) = 0.
y(x, t ) = 2 A cosωt sin kx = 0
The nodes are found from the locations where sin kx=0, which
happens when kx = 0, π, 2π,….
That is when kx = nπ where n = 0,1,2,…
The antinodes occur when sin kx=± 1; that is where
kx =
π 3π
,
,…
2 2
(
2n + 1)π
kx =
and n = 0,1, 2,…
2
22
Standing Waves on a String fixed at both ends
If the string has a length L, and
both ends are fixed, then
y(x=0,t) = 0 and y(x=L, t) = 0.
y (x = 0, t ) ∝ sin k (0 ) = 0
y (x = L, t ) ∝ sin kL = 0
kL = nπ
2π
λ
λ=
L = nπ
2L
n
where n = 1, 2, 3,…
23
2L
λn =
n
These are the permitted wavelengths of standing waves on a
string; no others are allowed.
The speed of the wave is:
The allowed frequencies are then:
v = λn f n
v
nv
fn =
=
λn 2 L
n =1, 2, 3,…
24
The n=1 frequency is called the fundamental frequency.
v
nv
⎛ v ⎞
fn =
=
= n⎜ ⎟ = nf1
λn 2 L ⎝ 2L ⎠
All allowed frequencies (called harmonics) are integer multiples of f1.
25
Example: A Guitar’s E-string has a length 65 cm and is stretched to a
tension of 82 N. It vibrates with a fundamental frequency of 329.63 Hz.
Determine the mass per unit length of the string.
For a wave on a string:
v=
F
µ
Solving for the linear mass density:
F
F
F
µ= 2 =
= 2
2
v
(λ1 f1 ) f1 (2 L )2
(
82 N )
=
(329.63 Hz )2 (2 * 0.65 m )2
= 4.5 ×10 − 4 kg/m
26
Sound Waves
•  Sound waves are longitudinal.
•  They can be represented by either variations in pressure (gauge pressure)
or by displacements of an air element.
•  Speed of sound in air at 00C ~ 331 m/s
•  Frequencies
•  Audible range – 20 Hz to 20 kHz
•  Infrasound - below 20 Hz
•  Ultrasound - above 20 kHz
27
The middle of a
compression (rarefaction)
corresponds to a pressure
maximum (minimum).