Test 2 - Today

1
Test 2 - Today

• 930 am in CNH-104
• Class begins at 11am

2
Wave Motion

• Sinusoidal waves
• Standing Waves

3
Sine Waves
For sinusoidal waves, the shape is a sine
function, eg.,
f(x) y(x,0) A sin(kx)
(A and k are constants)
y
A
x
-A
Then, at any time y (x,t) f(x vt) A
sink(x vt)
4
y (x,t) A sinkx wt
since kv?
Sine wave
y
l
A
v
x
-A
l (lambda) is the wavelength (length of one
complete wave) and so (kx) must increase by 2p
radians (one complete cycle) when x increases by
l. So kl 2p, or
k 2p / ?
5
Rewrite y A sin kx kvtA 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
wkv the initial phase is kx (different for
different particles).
Review SHM is described by functions of the form
y(t) A cos(wtf) A sin(p/2 f
wt), etc., with
? 2pf
angular frequency radians/sec
frequency cycles/sec (hertz)
6
Example
y
a
A
e
b
x
d
c
-A
Shown is a picture of a wave, yA sin(kx- wt),
at time t0 .
i) Which particle moves according to yA cos(wt)
? A B C D E
ii) Which particle moves according to yA sin(wt)
? A B C D E
iii ) Sketch a graph of y(t) for particle e.
7
The most general form of sine wave is y
Asin(kx ?t f)
amplitude
phase
y(x,t) A sin (kx wt f )
phase constant f
angular wave number k 2p / ? (radians/metre)
angular frequency ? 2pf (radians/second)
The wave speed is v 1 wavelength / 1 period,
so
v f? ? / k
8
Wave Velocity
The wave velocity is determined by the properties
of the medium
Transverse waves on a string
(proof from Newtons 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 Maxwells Equations for E-M fields )
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 4B) it will decrease by
2C) it will decrease by v2D) it will stay the
sameE) it will increase by v2
10
Exercise (Wave Equation)
What are w and k for a 99.7 MHz FM radio wave?
11
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,
This is for the particles (move in y), not wave
(moves in x) !
12
Standard sine wave
maximum displacement, ymax A maximum
velocity, vmax w A maximum acceleration,
amax w 2 A
Same as before for SHM !
13
Example
y
string 1 gram/m 2.5 N tension
x
vwave
Oscillator 50 Hz, amplitude 5 mm
Find y (x, t) vy (x, t) and maximum speed
ay (x, t) and maximum acceleration
14
10 min rest
15
Superposition of Waves

• Identical waves in opposite directions
• standing waves
• 2 waves at slightly different frequencies
• beats
• 2 identical waves, but not in phase
• interference

16
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.
example
L
(fundamental mode)
node
node
17
?2
Second Harmonic
Third Harmonic . . . .
18
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.).
19
Sine Waves In Opposite Directions
y2 Aosin(kx ?t)
y1 Aosin(kx ?t)
Total displacement, y(x,t) y1 y2
Trigonometry
Then
20
Example
wave at t0
y
8mm
x
1.2 m
f 150 Hz

1. Write out y(x,t) for the standing wave.
2. Write out y1(x,t) and y2(x,t) for two travelling
   waves which would produce this standing wave.

21
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?
22
Example
120 cm
m

1. m 150g, f1 30 Hz. Find µ (mass per unit
   length)
2. Find m needed to give f2 30 Hz
3. m 150g. Find f1 for a string twice as thick,
   made of the same material.

23
Solution
24
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
25
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)
node
antinode
antinode
graphical picture
26
Air Columns column with one closed end, one open
end
L
27

• Exercise Sketch the first three standing-wave
  patterns for a pipe of length L, and find the
  wavelengths and frequencies if
• both ends are closed
• both ends are open