Wave motion chapter 1

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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 = ¼2A 2l The total potential energy in one wavelength is Ul = ¼2A 2l This gives a total energy of 

El = Kl + Ul = ½2A 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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