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17.4 Wavelength and Frequency: (cont’d)
If the disturbance is sinusoidal, then:
y(x,t) = ym sin(k x – w t + f)
Let’s use Mathematica to try to understand the concepts of
amplitude, wavelength, [angular] wavenumber, period,
frequency, angular frequency, and phase.
waves
17.5 The Speed of a Traveling Wave:
Let’s see how fast the wave travels; e.g., how fast the crest moves.
k x – w t + f = a constant
The speed of the wave (v) is dx/dt; therefore,
v = w/k
v = l/T
v=lf
The wave moves one wavelength per period!!
17.6 Wave Speed on a Stretched String:
A string with linear mass density m, under a tension
t has a speed:
t
v
m
Use dimensional analysis and/ or see the proof on page 379
What happens to the speed when the frequency increases?
Example:
The wave on a string in the figure below, drawn at time t = 0, is moving to the
right.
1.
Determine the amplitude, wavelength, angular wavenumber and phase
angle (for the sin wave).
2.
You are told that the string has a LMD of 40 mg/cm and is under 10 N of
tension, find the period and angular frequency of the wave.
3.
Find the general mathematical expression describing the wave.
4.
How would this change of the wave were moving to the left?
y
mm
10
5
x
-1
-0.5
0.5
-5
-10
Mathematica output
1
1.5
2
m
17.7 Energy and Power of a Traveling String Wave:
Where is the kinetic energy minimum/maximum?
Where is the elastic potential energy minimum/maximum?
Pavg = ½ m v w2 y2m
[see the proof on page 381-2]
17.7 Energy and Power of a Traveling String Wave: (cont’d)
Pavg = ½ m v w2 y2m
The average power transmitted in the wave depends on the linear
mass density, on the speed, on the square of the frequency and on
the square of the amplitude.
In exams, we play games with the students.
For example, what happens if we increase the tension on the
string by a factor of 9?
Interaction question:
What happens if we increase the wavelength by a factor of
10, keeping the tension constant?
17.8 The Principle of Superposition of Waves:
yres(x,t) = y1(x,t) + y2(x,t)
Two (or more) overlapping waves algebraically add to produce a
resultant (or, net) wave. The overlapping waves do not alter the
motion of each other.
Let’s see this superposition
Mathematica code
17.9 Interference of Waves:
Two waves propagating along the same direction with the same
amplitude, wavelength and frequency, but differing in phase angle
will interfere with each other in a nice way.
Let’s see the (same) waves
Mathematica code
Checkpoint #5:
17.11 Standing Waves:
What happens when two waves propagating in opposite directions
with the same amplitude, wavelength and frequency will interfere
with each other such as to create standing waves!!
Let’s see the (same) waves
Mathematica code
17.11 Standing Waves: (cont’d)
Reflection at a Boundary:
1- Hard Reflection
2- Soft Reflection
Reflection-Transmission
Checkpoint #6:
You’re going to love
Mathematica; see this code.
17.12 Standing Waves and Resonance:
When a string of length L is clamped between two points, and
sinusoidal waves are sent along the string, there will be many
reflections off the clamped ends.
At specific frequencies, interference will produce nodes and
large anti-nodes.
We say we are at resonance, and that the string is resonating at
resonant frequencies.
Let’s see the (same) waves
Mathematica code
17.12 Standing Waves and Resonance: (cont’d)
The fundamental mode (n=1) has a fundamental frequency: f1 = v/(2L)
The second harmonic (n=2) has a frequency: f2 = 2 f1 = v/(L)
The nth harmonic has a frequency: fn = n f1 = nv/(2L)
The wavelength of the nth harmonic is: ln = 2L/n
What is the distance between two adjacent nodes
(or anti-nodes)?
What is the distance between a node and its
neighboring anti-node?
Example: A 75 cm long string has a wave
speed of 10 m/s, and is vibrating in its third
harmonic. Find the distance between two
adjacent anti-nodes.