Chapter 1 Wave Motion
Dr Mohamed Saudy 1
Why learn about waves?
Waves carry useful information and energy. Waves are all around us:
light from the stoplight electricity flowing in wires radio and television and cell phone transmissions 2
Wave Motion : Basic Concept Definition: A wave is a traveling disturbance in some physical system. Alternatively, a periodic disturbance that travels through space and time
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Wavelength visualized
Parameters of a Wave
Characteristics: –Wavelength l (e.g., meters) –Frequency f
(cycles per sec Hertz)
–Propagation speed c (e.g., meters / sec)
C= f l
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Wave Motion
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Wave Motion : Classification WAVE MOTION Mechanical Waves
E. M. WAVES
Traveling Wave
Longitudinal Wave
Standing/Stationary Wave
Transverse Wave
Traveling waves – Disturbance moves along the direction of wave propagation Waves can be characterized as
Standing waves - Disturbance
Transverse or Longitudinal.
oscillates about a fixed point. 6
Types of Waves
There are two main types of waves
Mechanical waves
Some physical medium is being disturbed The wave is the propagation of a disturbance through a medium (such as water, air and rock) Examples: water waves, and sound waves
Electromagnetic waves (E.M Waves)
No medium required Examples are light, radio waves, x-rays All e.m waves travel through the vacuum at the same speed.
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General Features of Waves
In wave motion, energy is transferred over a distance Matter is not transferred over a distance All waves carry energy
The amount of energy and the mechanism responsible for the transport of the energy differ
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Mechanical Wave Requirements
Some source of disturbance A medium that can be disturbed Some physical mechanism through which elements of the medium can influence each other
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Transverse Wave
A traveling wave or pulse that causes the elements of the disturbed medium to move perpendicular to the direction of propagation is called a transverse wave The particle motion is shown by the blue arrow The direction of propagation is shown by the red arrow
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Longitudinal Wave
ď Ź
ď Ź
A traveling wave or pulse that causes the elements of the disturbed medium to move parallel to the direction of propagation is called a longitudinal wave The displacement of the coils is parallel to the propagation
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Complex Waves
Some waves exhibit a combination of transverse and longitudinal waves Surface water waves are an example Use the active figure to observe the displacements 12
Example: Earthquake Waves (Complex wave)
P waves
S waves
“P” stands for primary Fastest, at 7 – 8 km / s Longitudinal “S” stands for secondary Slower, at 4 – 5 km/s Transverse
A seismograph records the waves and allows determination of information about the earthquake’s place of origin 13
Water Waves • An ocean wave is a combination of transverse and longitudinal. • The individual particles move
in ellipses as the wave disturbance moves toward the shore. 14
Sinusoidal Waves
The wave represented by the curve shown is a sinusoidal wave It is the same curve as sin q plotted against q This is the simplest example of a periodic continuous wave
It can be used to build more complex waves
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Terminology: Amplitude and Wavelength
The crest of the wave is the location of the maximum displacement of the element from its normal position
This distance is called the amplitude, A
The wavelength, l, is the distance from one crest to the next
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Terminology: Wavelength and Period ď Ź
ď Ź
More generally, the wavelength is the minimum distance between any two identical points on adjacent waves The period, T , is the time interval required for two identical points of adjacent waves to pass by a point
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Terminology: Frequency
The frequency, ƒ, is the number of crests (or any point on the wave) that pass a given point in a unit time interval
The time interval is most commonly the second
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Terminology: Frequency, cont
The frequency and the period are related 1 ƒ T
When the time interval is the second, the units of frequency are s-1 = Hz
Hz is a hertz
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Wave Motion : Properties period ( T )
time per wave
units - time
frequency ( f )
waves per time
units - 1/time
if
T =
10 f = sec 1 f
, then T =
f = 1 sec
1 10
=
1 T cycle sec
sec
l v = = lf T =
hz 20
Terminology, Example
The wavelength, l, is 40.0 cm The amplitude, A, is 15.0 cm The wave function can be written in the form y = A cos(kx – t)
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Wave Equations
We can also define the angular wave number (or just wave number), k 2 k l The angular frequency can also be defined 2 2 ƒ T
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Wave Equations, cont
The wave function can be expressed as y = A sin (k x – t). The speed of the wave becomes v = l ƒ. If x 0 at t = 0, the wave function can be generalized to y = A sin (k x – t + ) where is called the phase constant. 23
Speed of a Wave on a String
The speed of the wave depends on the physical characteristics of the string and the tension to which the string is subjected
tension T v mass/length
This assumes that the tension is not affected by the pulse This does not assume any particular shape for the pulse 24
Example 1: A wave pulse on a string moves a distance of 10 m in 0.05 s. (a) What is the velocity of the pulse? (b) What is the frequency of a periodic wave on the same string if its wavelength l is 0.8 m? Solution: (a) The velocity of the pulse is C=x/t, where x= 10 m, t=0.05 s, C= 200 m/s (b) The periodic wave has the same velocity 200 m/s, f=C/l=250 Hz= 250 s-1
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Example 2: The tension on the longest string of a grand piano is 1098 N, and the mass per unit length is 0.065 Kg/m. What is the velocity of a wave on this string? Solution
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Example 3: An electromagnetic vibrator sends waves down a string. The vibrator makes 600 complete cycles in 5 s. For one complete vibration, the wave moves a distance of 20 cm. What are the frequency, wavelength, and velocity of the wave? Solution
The distance moved during a time of one cycle is the wavelength; therefore: l=0.2 m
The velocity of wave
v f l 120Hz 0.2m 24 m / s 27
Energy in Waves in a String
Waves transport energy when they propagate through a medium We can model each element of a string as a simple harmonic oscillator
The oscillation will be in the y-direction
Every element has the same total energy
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Energy, final
the total kinetic energy in one wavelength is Kl = ¼2A 2l The total potential energy in one wavelength is Ul = ¼2A 2l This gives a total energy of
El = Kl + Ul = ½2A 2l
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Power Associated with a Wave
The power is the rate at which the energy is being transferred:
1 2 2 Energy E 2 A l 1 2 2 A v Time t T 2
The power transfer by a sinusoidal wave on a string is proportional to the
Frequency squared Square of the amplitude Wave speed 30
Example 4: A 2 m string has a mass of 300 g and vibrates with a frequency of 20 Hz and an amplitude
of 50 mm. If the tension in the rope is 48 N, how much power must be delivered to the string? Solution
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The Superposition Principle
• When two or more waves (blue and green) exist in the same medium, each wave moves as though the other were absent. • The resultant displacement of these waves at any point is the algebraic sum (yellow) wave of the two displacements.
Constructive Interference
Destructive Interference
Formation of a Standing Wave: Incident and reflected waves traveling in opposite directions produce nodes N and antinodes A. The distance between alternate nodes or anti-nodes is one wavelength.
Possible Wavelengths for Standing Waves Fundamental, n = 1 1st overtone, n = 2 2nd overtone, n = 3 3rd overtone, n = 4
n = harmonics
2L ln  n
n  1, 2, 3, . . .
Possible Frequencies f = v/l : Fundamental, n = 1
f = 1/2L
1st overtone, n = 2
f = 2/2L
2nd overtone, n = 3
f = 3/2L
3rd overtone, n = 4
f = 4/2L
n = harmonics
f = n/2L
nv fn  2L
n  1, 2, 3, . . .
Characteristic Frequencies Now, for a string under tension, we have:
F
FL v and m Characteristic frequencies:
nv f 2L
n fn ; n 1, 2, 3, . . . 2L
Example 5: A uniform cord has a mass of 0.3 kg and a length of 6 m. The cord passes over a pulley and supports a 2 kg object. Find the speed of a pulse traveling along this cord? Solution: The tension in the cord is equal to the weight of the suspended M=2 kg object: =Mg= (2 Kg)(9.8 m/s2)=19.6 N The mass per unit length of the cord is
Therefore, the wave speed is
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Example 6: A 9-g steel wire is 2 m long and is under a tension of 400 N. If the string vibrates in three loops, what is the frequency of the wave? Solution: For three loops: n = 3
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Summary for Wave Motion: L v m
v fl
n fn ; n 1, 2, 3, . . . 2L
1 E 2 A2l , 2
2 f
1 f T
Multiple Choice 1.
The wavelength of light visible to the human eye is on the
order of 5 10–7 m. If the speed of light in air is 3 108 m/s, find the frequency of the light wave. a. 3 107 Hz b. 4 109 Hz c. 5 1011 Hz d. 6 1014 Hz e. 4 1015 Hz 2. The speed of a 10-kHz sound wave in seawater is approximately 1500 m/s. What is its wavelength in sea water? a. 5.0 cm b. 10 cm c. 15 cm d. 20 cm e. 29 cm 40
3. If y = 0.02 sin (30x – 400t) (SI units), the frequency of the wave is a. 30 Hz b. 15/ Hz c. 200/ Hz d. 400 Hz e. 800 Hz 4. If y = 0.02 sin (30x – 400t) (SI units), the wavelength of the wave is a. /15 m b. 15/ m c. 60 m d. 4.2 m e. 30 m 5. If y = 0.02 sin (30x – 400t) (SI units), the velocity of the wave is a. 3/40 m/s b. 40/3 m/s c. 60/400 m/s d. 400/60 m/s e. 400 m/s 41
6. A piano string of density 0.005 kg/m is under a tension of 1350 N. Find the velocity with which a wave travels on the string. a. 260 m/s b. 520 m/s c. 1040 m/s d. 2080 m/s e. 4160 m/s 7. A 100-m long transmission cable is suspended between two towers. If the mass density is 2.01 kg/m and the tension in the cable is 3 ď‚´ 104 N, what is the speed of transverse waves on the cable? a. 60 m/s b. 122 m/s c. 244 m/s d. 310 m/s e. 1500 m/s
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