transverse and longitudinal waves · 2010-11-17 · wave speed the speed of transverse waves on a...

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Copyright © 2008 Pearson Education, Inc., publishing as Pearson Addison-Wesley.

You may not realize it, but you are surrounded by waves. The “waviness” of a water wave is readily apparent, from the ripples on a pond to ocean waves large enough to surf. It’s less apparent that sound and light are also waves. Chapter Goal: To learn the basic properties of traveling waves.


Topics: •  The Wave Model •  One-Dimensional Waves •  Sinusoidal Waves •  Waves in Two and Three Dimensions •  Sound and Light •  Power, Intensity, and Decibels •  The Doppler Effect


Transverse and Longitudinal Waves


Transverse and Longitudinal Waves


Wave Speed

The speed of transverse waves on a string stretched with tension Ts is

where µ is the string’s mass-to-length ratio, also called the linear density.


EXAMPLE 20.1 The speed of a wave pulse

QUESTION:

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One-Dimensional Waves

•  To understand waves we must deal with functions of two variables, position and time. •  A graph that shows the wave’s displacement as a function of position at a single instant of time is called a snapshot graph. For a wave on a string, a snapshot graph is literally a picture of the wave at this instant. •  A graph that shows the wave’s displacement as a function of time at a single position in space is called a history graph. It tells the history of that particular point in the medium.


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The graph at the top is the history graph at x = 4 m of a wave traveling to the right at a speed of 2 m/s. Which is the history graph of this wave at x = 0 m?


The graph at the top is the history graph at x = 4 m of a wave traveling to the right at a speed of 2 m/s. Which is the history graph of this wave at x = 0 m?


EXAMPLE 20.2 Finding a history graph from a snapshot graph

QUESTION:







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Sinusoidal Waves •  A wave source that oscillates with simple harmonic motion (SHM) generates a sinusoidal wave. •  The frequency f of the wave is the frequency of the oscillating source. •  The period T is related to the wave frequency f by

•  The amplitude A of the wave is the maximum value of the displacement. The crests of the wave have displacement Dcrest = A and the troughs have displacement Dtrough = −A.


Sinusoidal Waves


Sinusoidal Waves •  The distance spanned by one cycle of the motion is called the wavelength λ of the wave. Wavelength is measured in units of meters. •  During a time interval of exactly one period T, each crest of a sinusoidal wave travels forward a distance of exactly one wavelength λ. •  Because speed is distance divided by time, the wave speed must be

or, in terms of frequency


Sinusoidal Waves •  The angular frequency of a wave is

•  The wave number of a wave is

•  The general equation for the displacement caused by a traveling sinusoidal wave is

This wave travels at a speed v = ω/k. Copyright © 2008 Pearson Education, Inc., publishing as Pearson Addison-Wesley.

Waves in Two and Three Dimensions

•  Suppose you were to take a photograph of ripples spreading on a pond. If you mark the location of the crests on the photo, these would be expanding concentric circles. The lines that locate the crests are called wave fronts, and they are spaced precisely one wavelength apart. •  Many waves of interest, such as sound waves or light waves, move in three dimensions. For example, loudspeakers and light bulbs emit spherical waves. •  If you observe a spherical wave very, very far from its source, the wave appears to be a plane wave.

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Copyright © 2008 Pearson Education, Inc., publishing as Pearson Addison-Wesley. Copyright © 2008 Pearson Education, Inc., publishing as Pearson Addison-Wesley.


Sound Waves


Sound Waves

•  For air at room temperature (20°C), the speed of sound is vsound = 343 m/s. •  Your ears are able to detect sinusoidal sound waves with frequencies between about 20 Hz and about 20,000 Hz, or 20 kHz. •  Low frequencies are perceived as “low pitch” bass notes, while high frequencies are heard as “high pitch” treble notes. •  Sound waves exist at frequencies well above 20 kHz, even though humans can’t hear them. These are called ultrasonic frequencies. •  Oscillators vibrating at frequencies of many MHz generate the ultrasonic waves used in ultrasound medical imaging.

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The speed of sound in air is a function of (a) wavelength (b) frequency (c) Temperature (d) amplitude


EXAMPLE 20.6 Sound wavelengths

QUESTION:






Electromagnetic Waves

•  A light wave is an electromagnetic wave, an oscillation of the electromagnetic field. •  Other electromagnetic waves, such as radio waves, microwaves, and ultraviolet light, have the same physical characteristics as light waves even though we cannot sense them with our eyes. •  All electromagnetic waves travel through vacuum with the same speed, called the speed of light. The value of the speed of light is c = 299,792,458 m/s. •  At this speed, light could circle the earth 7.5 times in a mere second—if there were a way to make it go in circles!


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The Index of Refraction •  Light waves travel with speed c in a vacuum, but they slow down as they pass through transparent materials such as water or glass or even, to a very slight extent, air. •  The speed of light in a material is characterized by the material’s index of refraction n, defined as



A light wave travels through three transparent materials of equal thickness. Rank in order, from the largest to smallest, the indices of refraction n1, n2, and n3.

A.  n1 > n2 > n3 B.  n2 > n1 > n3 C.  n3 > n1 > n2 D.  n3 > n2 > n1 E.  n1 = n2 = n3


A light wave travels through three transparent materials of equal thickness. Rank in order, from the largest to smallest, the indices of refraction n1, n2, and n3.

A.  n1 > n2 > n3 B.  n2 > n1 > n3 C.  n3 > n1 > n2 D.  n3 > n2 > n1 E.  n1 = n2 = n3


Power and Intensity


EXAMPLE 20.9 The intensity of a laser beam

QUESTION:

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EXAMPLE 20.9 The intensity of a laser beam


If you want to build a solar power plant to supply energy to a city using 20% efficient photovoltaic cells, how much land in a sunny place do you need?

A. ~100,000 ft2 =~10,000 m2= about twice the size of a football field

B. ~0.5 million ft2=50,000 m2=about twenty times the size of a football field

C. ~10 million ft2=~1 km2= about 20% of the area of the Cal Poly Pomona campus

D. ~50 million ft2=~5 km2=~1200 acres=area of the Cal Poly Pomona campus

E. ~0.5 billion ft2=~20 square miles= ~area of City of Pomona


Intensity and Decibels •  Human hearing spans an extremely wide range of intensities, from the threshold of hearing at ≈ 1 × 10−12 W/m2 (at midrange frequencies) to the threshold of pain at ≈ 10 W/m2. •  If we want to make a scale of loudness, it’s convenient and logical to place the zero of our scale at the threshold of hearing. •  To do so, we define the sound intensity level, expressed in decibels (dB), as

where I0 = 1 × 10−12 W/m2. Copyright © 2008 Pearson Education, Inc., publishing as Pearson Addison-Wesley.

Intensity, Decibels, and Frequency


Four trumpet players are playing the same note. If three of them suddenly stop, the sound intensity level decreases by

A. 4 dB B. 6 dB C. 12 dB D. 40 dB


Four trumpet players are playing the same note. If three of them suddenly stop, the sound intensity level decreases by

A. 4 dB B. 6 dB C. 12 dB D. 40 dB

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Linear and Logarithmic Scales

Linear Scale Decibel 0.01 -20 0.1 -10 0.5 -3 1 0 2 3 4 6

10 10 20 13 100 20

1000 30

REFERENCE



The Doppler Effect

•  An interesting effect occurs when you are in motion relative to a wave source. It is called the Doppler effect. •  You’ve likely noticed that the pitch of an ambulance’s siren drops as it goes past you. A higher pitch suddenly becomes a lower pitch. •  As a wave source approaches you, you will observe a frequency f+ which is slightly higher than f0, the natural frequency of the source. •  As a wave source recedes away from you, you will observe a frequency f− which is slightly lower than f0, the natural frequency of the source.



The frequencies heard by an observer moving at speed v0 relative to a stationary sound source emitting frequency f0 are

The Doppler Effect The frequencies heard by a stationary observer when the sound source is moving at speed v0 are

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DOPPLER EFFECT FORMULA

f = v + vOv − vS

f0

+ toward - Away from


Doppler Effect Demo: OW-B-DB Doppler Ball

A sound source moves in a circle. At which point, the oscillation frequency detected by the observer is the highest?

A, B, C, D E) Equal at all points

observer

A

B

C

D


EXAMPLE 20.11 How fast are the police traveling?

QUESTION:





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Doppler Effect For light waves


Shock Waves

Sonic Boom Wake of a Boat Bull whip


transverse and longitudinal waves · 2010-11-17 · wave speed the speed of transverse waves on a...

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