Chapter 15
Any Picture you think it can represent this chapter
Waves
15.3
15.1
15.4 Chapter
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MATHEMATICAL DESCRIPTION OF WAVES DERIVATION OF
WAVE EQUATION
WAVE MOTION
15.2
COUPLED OSCILLATORS
Outline
After studying this chapter, you will be able to:
1- Define waves.
2- Classify waves according to medium and propagation.
3- Differentiate between different wave types.
4- Apply the wave equations with different examples.
Learning Objectives
According to medium According to propagation
A wave is an excitation that propagates through space or some medium as a function of time but does not generally transport matter with it.
Waves classification
Waves
1- Electromagnetic waves 2- Mechanical waves
1- Transverse waves 2- Longitudinal waves
Waves
(1) Electromagnetic waves (light, radio waves, microwaves, X-ray…..) do not
need a medium in which to propagate. They can move through empty space (vacuum).
(2) Mechanical waves (sound wave, water wave….) need a medium in which to propagate.
Many examples of waves surround us in everyday life.
• Light energy moving from the computer screen to your eye moves as light waves.
• Sound energy moving from a radio to your ear moves as sound waves.
In this chapter, we will study the mechanical waves .
Types of waves according to medium:
Coupled Oscillators
(1) Transverse Waves:
■ Move in a direction perpendicular to the direction in which the oscillators move.
■ Travel through a medium in the form of crests and troughs.
Types of waves according to propagation:
Crest Crest Crest
Trough Trough
The up-and- down motion of the rope is perpendicular to
the direction of the wave. Direction of wave Trough
Coupled Oscillators
Types of waves according to propagation:
(2) Longitudinal waves:
Propagate along the direction in which the oscillators move.
Travel through a medium in the form of compression and rarefaction.
Mathematical Description of Waves
A wave can be represented as a function of position (x) and time (t).
The figure represents a wave function.
We can write the mathematical description of a sinusoidal wave as
𝑦 𝑥, 𝑡 = 𝐴𝑠𝑖𝑛(𝑘𝑥 − 𝜔𝑡 + ∅
0)
Amplitude Angular frequencyPhase difference Wave number
Vertical position
Mechanical Description of Waves
A number of parameters can be defined to describe a periodic wave:
■ The amplitude A :
Is the maximum displacement of points on a wave from the equilibrium position in units of 𝑐𝑚 or 𝑚 .
The wavelength λ :
Is the length of one oscillation.
It is the distance between two identical adjacent points in a wave, such as two adjacent crests or troughs in a waveform (in units of 𝑐𝑚 or 𝑚).
■ The wave number k :
Is the number of wavelengths that fit in a distance of 2π (in units of rad/cm or
rad/m).
𝑘 =
2𝜋λ
x
Mechanical Description of Waves
The period T :
Is the time of one oscillations (in units of s).
It is the time between two identical adjacent points in a wave, such as two adjacent crests or troughs in a waveform (in units of 𝑠).
The frequency f :
Is the number of oscillations per second in the wave in units of 𝑠−1 or 𝐻𝑧 . 𝑓 = 1
𝑇 The angular frequency ω :
counts the number of oscillations in each time interval of 2π (in units of rad/s).
𝜔 = 2𝜋𝑓 = 2𝜋𝑇
t 10
Mechanical Description of Waves
The velocity of a wave 𝒗 :
it is measured by ( m/s ).
𝒗 =
∆𝒙∆𝒕
= λ
𝑻
= λf 𝒗 = λf =
𝟐𝝅𝑲 𝟐𝝅𝝎=
𝝎Phase difference ∅
𝟎:
𝑲 Is used to describe the difference in degrees or radians when two or more alternating quantities reach their maximum or zero values.
Phase ∅ :
it is measured by ( rad ).
Is given by
Mechanical Description of Waves
Notice that :
A sin(+ kx – ωt + ϕ0) represents a wave that travels in the positive x- direction, as the signs of the (kx) and (ωt) terms are opposite.
A sin(+ kx + ωt + ϕ0) represents a wave that travels in the negative x- direction, as the signs of the (kx) and (ωt) terms are the same.
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Sample problem 15.1
■ A wave on a string is given by:
where y and x are in meters and t is in seconds.
a) What are the angular frequency and the wave number of this wave?
b) What is the amplitude?
c) What is the velocity of this wave?
d) What is the wavelength of this wave?
e) What is its period?
f) In which direction does the wave travel?
g) What is the phase difference?
h) What is the phase at ( x = 2.3 × 10
-3m and t = 2.1 s)?
𝑦 𝑥, 𝑡 = 0.002𝑚 𝑠𝑖𝑛 78.8𝑟𝑎𝑑
𝑚 𝑥 + 346𝑟𝑎𝑑 𝑠 𝑡
SKETCH
■ The sketch shows the wave function t and x.
Research
■ Look at the given wave function:
𝑦 𝑥, 𝑡 = 0.002 𝑚 𝑠𝑖𝑛 78.8𝑟𝑎𝑑
𝑚 𝑥 + 346𝑟𝑎𝑑 𝑠 𝑡
■ Comparing with a wave traveling in the negative x-direction 𝑦 𝑥, 𝑡 = 𝐴 sin(𝑘𝑥 + 𝜔𝑡 + ∅0)
Sample problem 15.1 Sample problem 15.1
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Sample problem 15.1 Sample problem 15.1
SOLUTION:
By comparison we can find:
a) The angular frequency and the wave number:
ω = 346 rad/s k = 78.7 rad/m
b) The amplitude A= 0.002 m We can then find:
c) the velocity of the wavefrom:
𝑣 = 𝜔
𝑘 = 346
78.7 = 4.4 𝑚/𝑠 d) the wavelength from the wave number:
λ=2𝜋𝑘 = 78.72𝜋 = 0.08 𝑚 e) the period from the angular frequency:
𝑇 = 2𝜋
𝜔 = 2𝜋
346 = 0.012 s
Sample problem 15.1 Sample problem 15.1
SOLUTION:
f) The wave traveling in the negative x -direction as the signs of the (kx) and ( ωt) terms are the same.
g) the phase difference of this wavefrom is:
∅
0= 0
h) the phase at (x = 2.3 × 10
-3m and t = 2.1 s) is found from:
∅ = 𝑘𝑥 + 𝜔𝑡 + ∅
0∅ = (78.8 x 0.0023) + (346 x 2.1) +0=726.8 rad
Sample problem 15.1 Sample problem 15.1
Double-check
■ To double-check our work, we look at our sketch:
■ We see that the wavelength is approximately 0.08 m, in agreement with our result.
■ We also see that the period is approximately 0.018 s, also in agreement with our result.
𝛌 T
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Concept check
■ Which of the following waves is traveling in the positive x-direction?
A. y(x,t) = (−0.002 m) sin[(78.8 m-1)x + (346 s-1)t]
B. y(x,t) = (0.002 m) sin[(78.8 m-1)x + (346 s-1)t − 1]
C. y(x,t) = (−0.002 m) sin[(−78.8 m-1)x + (346 s-1)t]
D. y(x,t) = (0.002 m) sin[(−78.8 m-1) + (346 s-1)]
Derivation of the wave equation
■ String instruments form a large class of musical instruments.
■ Suppose a string has a mass M, a length L and a radius r.
■ 𝜇 = 𝑀𝐿 = 𝜌𝐴 = 𝜌(𝜋𝑟2) Where:
𝜇 is the linear mass density (kg/m) 𝜌 is the density (kg/m3)
A is the cross section area, (𝐴 = 𝜋𝑟2 𝑜𝑓 𝑠𝑡𝑟𝑖𝑛𝑔) (m2)
■ We can get the wave velocity on a string in terms of the tension:
𝑣 = 𝑡𝑒𝑛𝑠𝑖𝑜𝑛
𝑙𝑖𝑛𝑒𝑎𝑟 𝑚𝑎𝑠𝑠 𝑑𝑒𝑛𝑠𝑖𝑡𝑦 = 𝑇 𝜇
■ Some implications of this equation are:
– Increasing μ reduces v.
Waves on a String
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Example 15.1
Elevator Cable
■ An elevator repairman (mass 73 kg) sits on top of an elevator cabin of mass 655 kg inside a shaft of a skyscraper.
■ The cabin is suspended by a 61 m long steel cable of mass 38 kg.
■ He sends a signal to his colleague at the top
of the elevator shaft by tapping the cable with his hammer.
Find
a) The linear mass density.
b) The velocity of the wave on the string.
c) How long will it take for the wave pulse generated by the hammer tap to travel up the cable?
38 Kg
655 Kg
61 m
SOLUTION:
■ The linear mass density of the steel cable is:
■ The tension in the cable from the weight of the elevator + man is:
■ The wave speed is:
■ The pulse travels up the 61 m long cable in:
Example 15.1
T
73 655
Fg= mg
38 Kg
61 m
T = F
gT
A copper wire has a density of ρ = 8920 kg/m3, a diameter of 2.40 mm, and a length L. The wire is held under a tension of 10.00 N. Transverse waves are sent down the wire.
(a) What is the linear mass density of the wire?
(b) What is the speed of the waves through the wire?
Extra Exercise
SOLUTION:
a) Diameter = 2𝑟 𝑟 = 2.4
2 = 1.2 𝑚𝑚 𝑟 = 1.2 x10−3 𝑚
𝜇 = 𝜌 𝜋𝑟2 = 8920 x 3.14 x (1.2 x 10−3)2 = 0.04 kg/m
b)
v =
𝑇𝜇
=
100.04
= 15.8 m/s
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When the horn on a ship is sounded, the passengers hear an echo from a cliff after 4s. If the speed of sound is 340 m/s, how far away is the cliff?
A) 170 m B) 340 m C) 680 m D) 1360 m
Extra Exercise
SOLUTION:
As its an Echo, it is meaning the sound wave has traveled double the distance:
v = 2𝑑𝑡 𝑑 = 𝑉𝑡
2 = 340 𝑋 4
2 = 680 m
The END OF
CHAPTER
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