Waves on a String

Waves on a String
In the reference frame of the string:
v
– Typeset by FoilTEX –
1
Waves on a String
In the reference frame of the pulse:
– Typeset by FoilTEX –
2
Waves on a String
Consider a small displaced segment of the string:
R
θ
– Typeset by FoilTEX –
3
Waves on a String
Consider a small displaced segment of the string:
FT
– Typeset by FoilTEX –
FT
4
Clicker question
Two strings, one thick and the other thin, are connected to form one
long string. A wave travels along the string and passes the point where the
two strings are connected. Which of the following change(s) at that point:
A. frequency
B. period
C. propagation speed
D. wavelength
E. more than one of the above
– Typeset by FoilTEX –
5
Clicker question
By shaking one end of a stretched string, a single pulse is generated.
The traveling pulse carries
A. energy
B. momentum
C. energy and momentum
D. neither of the two
– Typeset by FoilTEX –
6
Wave Power
P =
– Typeset by FoilTEX –
7
Wave Intensity
Intensity– rate at which a wave carries energy across a unit area
perpendicular to the direction of wave propagation.
I =power/unit area (W/m2)
wave fronts– surfaces on which wave phase is constant
plane wave– wave fronts are planes (wave doesn’t spread out, so intensity
is constant)
spherical wave– wave fronts are spheres
– Typeset by FoilTEX –
8
Interference
principle of superposition –displacements of two (or more) waves add
together to give net displacement
– Typeset by FoilTEX –
9
Interference–superposition
– Typeset by FoilTEX –
10
Interference–superposition
– Typeset by FoilTEX –
11
Clicker question
Two identical pulses of opposite amplitude travel along a stretched
string and interfere destructively. Which of the following is/are true?
A. There is an instant at which the string is completely straight.
B. When the two pulses interfere, the energy of the puses is momentarily
zero.
C. There is a point on the string that does not move up or down.
D. There are several points onthe string that do not move up or down.
E. More than one of the above is true.
– Typeset by FoilTEX –
12
Superposition–Fourier analysis
The principle of superposition makes it possible to construct complex shapes
by adding waves of different amplitudes, wavelengths & phases
3
position (m)
2
1
0
-1
-2
-3
– Typeset by FoilTEX –
13
Superposition–beats
Consider the superposition of two waves with slightly different frequencies.
3
position (m)
2
1
0
-1
-2
-3
– Typeset by FoilTEX –
14
Superposition–beats
Superposing two waves with slightly different frequencies gives constructive
interference in some places and destructive in others, giving rise to beats
3
position (m)
2
1
0
-1
-2
-3
– Typeset by FoilTEX –
15
Superposition–beats
3
position (m)
2
1
0
-1
-2
-3
– Typeset by FoilTEX –
16
Clicker Question
CT page 121
– Typeset by FoilTEX –
17
Dispersion
dispersion –phenomena that occurs when the wave speed depends on the
wavelength. The result is that the individual waves that make up a wave
form travel at different speeds and the waveform distorts.
non-dispersive wave –all components of the waves travel at the same
speed and the wave form retains shape.
Example of a dispersive wave: waves on the surface of deep water:
q
λg
v = 2π
– Typeset by FoilTEX –
18
Sound waves
sound waves are longitudinal waves that propagate through gases, liquids
& solids. Air molecules are displaced back and forth as pressure and
density is changed.
sound intensity level , β, quantifies sound levels. β = 10 log
I
I0
, and is
measured in decibels. I=intensity, I0 = 10−12W/m2 is a reference level.
Speed of sound in gas: v =
q
γP
ρ
P =pressure, ρ=density, γ=constant that characterizes gas
– Typeset by FoilTEX –
19