College Physics 140 Chapter 11: Waves
We will be investigating the physics behind waves!
Chapter 11: Waves
• Energy Transport by Waves
• Longitudinal and Transverse Waves • Transverse Waves on Strings
• Periodic Waves
• Mathematical and Graphical Descriptions of Waves
• Reflection and Refraction of Waves
• Interference and Diffraction
• Standing Waves on a String
Waves and Energy Transport
A wave is a disturbance that travels outward from its source. Waves carry energy. The energy is transported outward from the source; maOer is not.
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.
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
2
4π r
4π r
Intensity has units of waOs/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 =
4π res2
Im =
Psun
4π rms2
Divide one equation by the other:
Psun
2
2
11
2
!
$
!
$
I m 4π rms
res
1.50 ×10 m
=
=# & =#
& = 6.57
10
P
Ie
sun
" rms % " 5.85 ×10 m %
4π res2
I m = 6.57I e = 9200 W/m 2
Transverse and Longitudinal Waves
A transverse wave is where the motions of the particles are transverse (perpendicular) to the direction of wave travel. A longitudinal wave is where the motions of the particles are along the same direction as the wave propagation.
Compression, a region of high density
Rarefaction, a region of low density
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.
Transverse Waves on a String
AOach a mass to a string to provide tension. The string is then shaken at one end with a frequency f.
L
AOach a wave driver here
A wave traveling on this string will have a speed of v =
F
µ
M
where F is the force applied to the string (tension) and µ is the mass/unit length of the string (linear mass density).
Example (text problem 11.8): 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?
F
The speed of a wave on a string isv =
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 paOern 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
The period T is measured by the amount of time it takes for a point on the wave to go through one complete cycle of oscillations. The frequency is then f = 1/T.
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.
Example (text problem 11.13): What is the wavelength of a wave whose speed and period are 75.0 m/s and 5.00 ms, respectively?
λ
v=λf =
T
Solving for the wavelength:
λ = vT = ( 75.0 m/s) ( 5.00 ×10 −3 s) = 0.375 m
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 (ω t ± kx )
+ is used for a wave traveling in the -­‐‑x direction, and -­‐‑ is used for a wave traveling in the +x direction.
2π
k=
λ
(ω t ± 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.
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,…)
Example (text problem 11.21): 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 (ω t − 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
Example (text problem 11.21): 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 (ω t − kx )
(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
v = λ f = # & ( 2π f ) = =
= 100 m/s
" 2π %
k 6.00 rad/m
(e) What direction is the wave traveling.
Along the +x direction.
The Principle of Superposition
For small amplitudes, waves will pass through each other and emerge unchanged.
Superposition Principle: When two or more waves overlap, the net disturbance at any point is the sum of the individual disturbances due to each wave.
Two traveling wave pulses: left pulse travels right; right pulse travels left.
Reflection and Refraction At an abrupt boundary between two media, a reflection will occur. A portion of the incident wave will be reflected backward from the boundary.
When you have a wave that travels from a “low density” medium to a “high density” medium, the reflected wave pulse will be inverted. The frequency of the reflected wave remains the same.
When a wave is incident on the boundary between two different media, a portion of the wave is reflected, and a portion will be transmiOed into the second medium.
The frequency of the transmiOed wave also remains the same. However, both the wave’s speed and wavelength are changed such that:
v1 v2
f= =
λ1 λ2
The transmiOed wave will also suffer a change in propagation direction (refraction).
Example (text problem 11.36): Light of wavelength 0.500 µm in air enters the water in a swimming pool. The speed of light in water is 0.750 times the speed in air. What is the wavelength of the light in water?
Since the frequency is unchanged in both media:
vair vwater
f=
=
λair λwater
! vwater $
λwater = #
& λair
" vair %
! 0.750vair $
=#
& 0.500 µ m = 0.375µ m
" vair %
Interference and Diffraction
Two waves are considered coherent if they have the same frequency and maintain a fixed phase relationship.
Two waves are considered incoherent if the phase relationship between them varies randomly.
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.
When two waves travel different distances to reach the same point, the phase difference is determined by:
d1 − d2 phase difference
=
λ
2π
Diffraction is the spreading of a wave around an obstacle in its path.
2π ( d1 − d2 )
phase difference =
λ
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).
Apply the superposition principle to the two waves on the string:
y(x, t) = y1 (x, t) + y2 (x, t)
= A (sin (ω t + kx ) − sin (ω t − kx ))
= ( 2A cosω t ) sin kx
The previous expression is the mathematical form of a standing wave. A
A
A
N
N
N
N
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.
The nodes occur where y(x,t) = 0.
y ( x, t ) = 2A cosω t 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
π 3π
kx = , ,…
2 2
2n +1) π
(
kx =
and n = 0, 1, 2,…
2
The previous expression is the mathematical form of a standing wave. A
A
A
N
N
N
If the string has a length L, and both ends are fixed, then y(x = 0, t) = 0 and y(x = L, t) = 0.
The wavelength of a standing wave:
N
y ( x = 0, t ) ∝ sin k ( 0 ) = 0
y ( x = L, t ) ∝ sin kL = 0
kL = nπ
2π
L = nπ
λ
2L where n = 1, 2, 3,…
λ=
n
The previous expression is the mathematical form of a standing wave. A
A
A
N
N
N
N
2L
λn =
n
These are the permiOed wavelengths of standing waves on a string; no others are allowed.
The speed of the wave is:v =
λn f n
The allowed frequencies are then:
fn =
v nv
=
λn 2L
n =1, 2, 3,…
The n = 1 frequency is called the fundamental frequency.
! v $
v nv
fn = =
= n # & = nf1
" 2L %
λn 2L
All allowed frequencies (called harmonics) are integer multiples of f1.
Example (text problem 11.51): 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
2
v
λ
f
f
2L
( 1 1) 1 ( )
82 N )
(
−4
=
=
4.5
×10
kg/m
2
2
(329.63 Hz) (2 * 0.65 m )