Test 2 - Today
• 9:30 am in CNH-104
• Class begins at 11am
Physics 1B03summer-Lecture 9
Wave Motion
• Sinusoidal waves
• Standing Waves
Physics 1B03summer-Lecture 9
Sine Waves
For sinusoidal waves, the shape is a sine function, eg.,
f(x) = y(x,0) = A sin(kx)
A
(A and k are constants)
y
x
-A
Then, at any time: y (x,t) = f(x – vt) = A sin[k(x – vt)]
Physics 1B03summer-Lecture 9
Sine wave: y (x,t) = A sin[kx – wt]
A
y
since kv=ω
l
v
x
-A
l (“lambda”) is the wavelength (length of one complete wave);
and so (kx) must increase by 2π radians (one complete cycle)
when x increases by l. So kl = 2p, or
k = 2π / λ
Physics 1B03summer-Lecture 9
Rewrite:
y = A sin [kx – kvt]=A sin [kx – wt]
The displacement of a particle at location x is a sinusoidal
function of time – i.e., simple harmonic motion:
y = A sin [ constant – wt]
The “angular frequency” of the particle motion is w=kv; the
initial phase is kx (different for different particles).
Review: SHM is described by functions of the form
y(t) = A cos(wt+f) = A sin(p/2 –f –wt), etc., with
ω = 2πf
“angular frequency”
radians/sec
frequency:
cycles/sec (=hertz)
Physics 1B03summer-Lecture 9
Example
y
A
a
e
b
-A
c
x
d
Shown is a picture of a wave, y=A sin(kx- wt), at time t=0 .
i) Which particle moves according to y=A cos(wt) ?
AB CDE
ii) Which particle moves according to y=A sin(wt) ?
ABCDE
iii ) Sketch a graph of y(t) for particle e.
Physics 1B03summer-Lecture 9
The most general form of sine wave is y = Asin(kx ± ωt – f)
amplitude
“phase”
y(x,t) = A sin (kx ± wt –f )
angular wave number
k = 2π / λ
(radians/metre)
phase constant f
angular frequency
ω = 2πf
(radians/second)
The wave speed is v = 1 wavelength / 1 period, so
v = fλ = ω / k
Physics 1B03summer-Lecture 9
Wave Velocity
The wave velocity is determined by the properties of the medium:
Transverse waves on a string:
v wave
FT
tension


mass/unit length

(proof from Newton’s second law – Pg.625)
Electromagnetic wave (light, radio, etc.):
v = c  2.998108 m/s (in vacuum)
v = c/n (in a material with refractive index n)
(proof from Maxwell’s Equations for E-M fields )
Physics 1B03summer-Lecture 9
Quiz
You double the diameter of a string. How will the speed of the
waves on the string be affected?
A) it will decrease by 4
B) it will decrease by 2
C) it will decrease by √2
D) it will stay the same
E) it will increase by √2
Physics 1B03summer-Lecture 9
Exercise (Wave Equation)
What are w and k for a 99.7 MHz FM radio wave?
Physics 1B03summer-Lecture 9
Particle Velocities
Particle displacement, y (x,t)
Particle velocity, vy = dy/dt (x held constant)
(Note that vy is not the wave speed v – different
directions! )
Acceleration,
d2y
ay 
 2
dt
dt
dv y
This is for the particles (move in y),
not wave (moves in x) !
Physics 1B03summer-Lecture 9
“Standard” sine wave:
y  A sin( kx  wt -  )
dy
vy 
 wA cos( kx  wt -  )
dt
dv y
ay 
 -w 2 A sin( kx  wt -  )
dt
 -w 2 y
maximum displacement, ymax = A
Same as before
maximum velocity, vmax = w A
for SHM !
2
maximum acceleration, amax = w A
Physics 1B03summer-Lecture 9
Example
y
x
string: 1 gram/m; 2.5 N tension
vwave
Oscillator:
50 Hz, amplitude 5 mm
Find: y (x, t)
vy (x, t) and maximum speed
ay (x, t) and maximum acceleration
Physics 1B03summer-Lecture 9
10 min rest
Physics 1B03summer-Lecture 9
Superposition of Waves
1) Identical waves in opposite directions:
“standing waves”
2) 2 waves at slightly different frequencies:
“beats”
3) 2 identical waves, but not in phase:
“interference”
Physics 1B03summer-Lecture 9
Practical Setup: Fix the ends, use reflections.
We can think of travelling waves reflecting back and forth from the
boundaries, and creating a standing wave. The resulting standing
wave must have a node at each fixed end. Only certain wavelengths
can meet this condition, so only certain particular frequencies of
standing wave will be possible.
fl  v 
example:
FT

L
1
2
l1  L 
l1  2 L
v
f1 
2L
node
node
(“fundamental mode”)
Physics 1B03summer-Lecture 9
λ2
l2  L
v
f 2   2 f1
L
Second Harmonic
λ3
3
2
l3  L
l3  23 L
v
v
f3 
3
l3
2L
 3 f1
Third Harmonic . . . .
Physics 1B03summer-Lecture 9
In this case (a one-dimensional wave, on a string with both ends
fixed) the possible standing-wave frequencies are multiples of
the fundamental: f1, 2f1, 3f1, etc. This pattern of frequencies
depends on the shape of the medium, and the nature of the
boundary (fixed end or free end, etc.).
Physics 1B03summer-Lecture 9
Sine Waves In Opposite Directions:
y1 = Aosin(kx – ωt)
y2 = Aosin(kx + ωt)
Total displacement, y(x,t) = y1 + y2
a b
a -b


Trigonometry : sin a  sin b  2 sin 
 cos

 2 
 2 
kx  wt  " a"
kx - wt  " b"
Then:
y( x, t )  2 A0 sin kx cos wt
Physics 1B03summer-Lecture 9
Example
wave at t=0
8mm
y
x
1.2 m
f = 150 Hz
a) Write out y(x,t) for the standing wave.
b) Write out y1(x,t) and y2(x,t) for two
travelling waves which would produce
this standing wave.
Physics 1B03summer-Lecture 9
Example
m
When the mass m is doubled, what happens to
a) the wavelength, and
b) the frequency
of the fundamental standing-wave mode?
What if a thicker (thus heavier) string were used?
Physics 1B03summer-Lecture 9
Example
120 cm
m
a) m = 150g, f1 = 30 Hz. Find μ (mass per unit length)
b) Find m needed to give f2 = 30 Hz
c) m = 150g. Find f1 for a string twice as thick, made of
the same material.
Physics 1B03summer-Lecture 9
Solution
Physics 1B03summer-Lecture 9
Standing sound waves
Sound in fluids is a wave composed of longitudinal vibrations
of molecules. The speed of sound in a gas depends on the
temperature. For air at room temperature, the speed of
sound is about 340 m/s.
At a solid boundary, the vibration amplitude must be zero
(a standing wave node).
Physical picture of
particle motions (sound
wave in a closed tube)
node
antinode node
antinode node
graphical picture
Physics 1B03summer-Lecture 9
Standing sound waves in tubes – Boundary Conditions
-there is a node at a closed end
-less obviously, there is an antinode at an open end
(this is only approximately true)
antinode
node
antinode
graphical picture
Physics 1B03summer-Lecture 9
Air Columns: column with one closed end, one open end
L  14 l1
L
L  34 l3
L  54 l5
Physics 1B03summer-Lecture 9
Exercise: Sketch the first three standing-wave
patterns for a pipe of length L, and find the
wavelengths and frequencies if:
a) both ends are closed
b) both ends are open
Physics 1B03summer-Lecture 9