Lecture 05 mechanical waves. transverse waves.

1. Lecture 5
Mechanical waves.
Transverse waves.

2. What is a wave ?
A wave is a traveling disturbance that
transports energy but not matter.
–
–
–
–
Examples:
Sound waves (air moves back & forth)
Stadium waves (people move up & down)
Water waves (water moves up & down)
Light waves (what moves??)
Waves exist as excitations of a (more or less) elastic
medium.

3. Types of Waves
Transverse: The medium
oscillates perpendicular to the
direction the wave is moving.
•String
•Water
Longitudinal: The medium
oscillates in the same
direction the wave is moving
•
•
Sound
Slinky
DEMO:
Rope, slinky and
wave machines

4. Forms of waves
• Continuous or periodic: go on forever in
one direction
v
→ in particular, harmonic (sin or cos)
• Pulses: brief disturbance in
the medium
• Pulse trains, which are
somewhere in between.
v
v

5. Harmonic waves
Each point has SHM

6. A few parameters
Amplitude: The maximum displacement A of a point on the wave.
Period: The timeT for a point on the wave to undergo one complete
oscillation.
Frequency: Number of oscillations f for a point on
the wave in one unit of time.
Angular frequency: radians ω for a point on the
wave in one unit of time.
y
Amplitude A
A
1
f =
T
ω = 2πf
x

7. Connecting all these SHM
Wavelength: The distance λ between identical points on the wave.
Speed: The wave moves one wavelength λ in one period T, so its
speed is
λ
v = = λf
T
Wavelength
λ
Amplitude A
A
y
x

8. Wave speed
The speed of a wave is a constant that depends
only on the medium:
→ How easy is it to displace points from equilibrium
position?
→ How strong is the restoring force back to
equilibrium?
Speed does NOT depend on amplitude, wavelength, period or
shape of wave.

9. ACT: Frequency and wavelength
The speed of sound in air is a bit over 300 m/s, and the
speed of light in air is about 300,000,000 (3x108) m/s.
Suppose we make a sound wave and a light wave with a
wavelength of 3 m each.
What is the ratio of the frequency of the light wave to
that of the sound wave?
(a) About 106
(b) About 10−6
(c) About 1000
v
f =
λ
vlight
v sound
~ 10
6
flight
fsound
~ 10 6

10. What are these frequencies???
For sound having λ = 3 m :
For light having λ = 3 m :
v 300 m/s
(bass hum)
f = ~
= 100 Hz
λ
3m
v 3 × 108 m/s
(FM radio)
f = ~
= 100 MHz
λ
3m

11. Mathematical Description
y
Suppose we have some function y = f(x) :
x
y
f(x − a) is just the same shape moved
a distance a to the right:
x
a
Let a = vt
Then, f(x − vt) will describe the same
shape moving to the right with
speed v.
y
v
vt
x

12. Math for the harmonic wave
y
λ
Consider a wave that is harmonic
in x and has a wavelength λ:
A
If y = A at x = 0:
If this is moving to the right
with speed v :
v
x
 2π 
y ( x ) = A cos 
x÷
 λ 
 2π

y ( x ,t ) = A cos  ( x − vt ) ÷
 λ


13. Different forms of the same thing
 2π

y ( x ,t ) = A cos  ( x − vt ) ÷
 λ

λ
ω
We knew: v = = λ
T
2π
Define: k =
2π
λ
Wave number
y ( x ,t ) = A cos ( kx − ωt )
ω
v =
k

14. ACT: Wave Motion
A harmonic wave moving in the positive x direction can be described
by the equation y(x,t) = A cos ( kx - ωt )
Which of the following equations describes a harmonic wave moving in
the negative x direction?
(a) y(x,t) = A cos (kx + ωt)
(b) y(x,t) = A cos (−kx + ωt)
(c) Both
y ( x ,t ) = A cos ( kx − ωt )
came from
y ( x ,t ) = A cos ( kx + ωt )
cos ( −kx + ωt ) = cos ( kx − ωt )
↔
↔
 2π

y ( x ,t ) = A cos  ( x − vt ) ÷
 λ

− x direction
+ x direction
It’s the relative sign that matters.

15. The wave equation
General wave: y = f ( x −vt )
∂y
∂f
=v
∂t
∂u
2
∂2y
2 ∂ f
=v
2
∂t
∂u 2
Let u = x −vt
∂y
∂f
=
∂x
∂u
∂2y
∂2f
=
∂x 2 ∂u 2
2
∂2y
2 ∂ y
=v
2
∂t
∂x 2
∂2y
1 ∂2y
− 2
= 0 Wave equation
2
2
∂x
v ∂t

16. ACT: Waves on a string
A single pulse is sent along a stretched rope.
What can the person do to make the start of
the pulse arrive at the wall in a shorter time?
A. Flick hand faster
B. Flick hand further up and down
C. Pull on rope before flicking hand
Pulling on rope increases tension, and propagation speed depends
only on medium, not on how you start the wave.

17. Faster flick up/down
→ narrow pulses
Slower flick up/down → wider pulses
Large flick up/down → higher pulses
Once pulse leaves your hand, you cannot influence it.
Propagation speed down string is ~ same for all these
pulses

18. What determines the wave speed?
Back to 221…
Problem: A pulse travels in the +x direction in a
string with mass per unit length of the string is
µ (kg/m) subject to a uniform tension F .
What is the speed of the pulse?

19. Consider the segment of length ∆x when the string is relaxed: ∆m = µ∆x
y
F2
F2y
F
F1x
∆m
F2x
F
µ
F1
x
x + ∆x
F1y
|F1x | = |F2x| because ax = 0 (transversal wave, no displacement in the x
direction)
Fx must also be equal to the tension in the string when there is no wave,
ie, |F1x | = |F2x| = F

20. y
F2
F2y
F
∆m = µ∆x
F
µ
F1
At x :
At x + ∆x :
F1y
F
F2y
F
x
x + ∆x
F1y
 ∂y 
= −
÷
∂x x

Fy
net
= F2 y − F1y
 ∂y 
=
÷
∂x x +∆x

∂2 y
 ∂y 
 ∂y  
F 
−
÷
÷  = µ∆x
∂t 2
 ∂x x +∆x  ∂x x 
 ∂y 
 ∂y  
= F 
−
÷
÷ 
∂x x +∆x  ∂x x 



= ∆m ay
∂2 y
= µ∆x
∂t 2

21. ∂2 y
 ∂y 
 ∂y  
F 
−
÷
÷  = µ∆x
∂t 2
 ∂x x +∆x  ∂x x 
 ∂y 
 ∂y 
−

÷
÷
µ ∂2 y
 ∂x x +∆x  ∂x x
=
∆x
F ∂t 2
 ∂y 
 ∂y 
−

÷
÷
∂2 y
 ∂x x +∆x  ∂x x
lim
=
∆x →0
∆x
∂x 2
Wave equation!
∂2 y
µ ∂2 y
=
∂x 2 F ∂t 2
1
v2
Wave speed
in a string
F
v =
µ

22. In-class example: Wave speed
A string under a certain tension has transverse waves with a wave
speed v. A second string made of the same material but half the
cross-sectional area is under twice the tension of the first string.
What is the speed of transverse waves in the second string?
A. v/4
B. v/2
C. v
F ' = 2F
µ
µ' =
(half the mass in the same length)
2
D. 2v
E. 4v
v'=
F'
2F
F
=
=2
= 2v
µ'
µ
µ
2