11/17/10. Transverse and Longitudinal Waves. Transverse and Longitudinal Waves. Wave Speed. EXAMPLE 20.1 The speed of a wave pulse

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1 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 T s is EXAMPLE 20.1 The speed of a wave pulse QUESTION: where µ is the string s mass-to-length ratio, also called the linear density. 1

2 EXAMPLE 20.1 The speed of a wave pulse EXAMPLE 20.1 The speed of a wave pulse EXAMPLE 20.1 The speed of a wave pulse EXAMPLE 20.1 The speed of a wave pulse 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. 2

3 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 EXAMPLE 20.2 Finding a history graph from a snapshot graph QUESTION: EXAMPLE 20.2 Finding a history graph from a snapshot graph EXAMPLE 20.2 Finding a history graph from a snapshot graph 3

4 EXAMPLE 20.2 Finding a history graph from a snapshot graph 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 D crest = A and the troughs have displacement D trough = 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. 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. 4

5 Waves in Two and Three Dimensions Waves in Two and Three Dimensions Sound Waves Sound Waves For air at room temperature (20 C), the speed of sound is v sound = 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. 5

6 The speed of sound in air is a function of (a) wavelength (b) frequency (c) Temperature (d) amplitude QUESTION: EXAMPLE 20.6 Sound wavelengths EXAMPLE 20.6 Sound wavelengths EXAMPLE 20.6 Sound wavelengths 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! 6

7 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 n 1, n 2, and n 3. A light wave travels through three transparent materials of equal thickness. Rank in order, from the largest to smallest, the indices of refraction n 1, n 2, and n 3. A. n 1 > n 2 > n 3 B. n 2 > n 1 > n 3 C. n 3 > n 1 > n 2 D. n 3 > n 2 > n 1 E. n 1 = n 2 = n 3 A. n 1 > n 2 > n 3 B. n 2 > n 1 > n 3 C. n 3 > n 1 > n 2 D. n 3 > n 2 > n 1 E. n 1 = n 2 = n 3 Power and Intensity EXAMPLE 20.9 The intensity of a laser beam QUESTION: 7

8 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 ft 2 =~10,000 m 2 = about twice the size of a football field B. ~0.5 million ft 2 =50,000 m 2 =about twenty times the size of a football field C. ~10 million ft 2 =~1 km 2 = about 20% of the area of the Cal Poly Pomona campus D. ~50 million ft 2 =~5 km 2 =~1200 acres=area of the Cal Poly Pomona campus E. ~0.5 billion ft 2 =~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 W/m 2 (at midrange frequencies) to the threshold of pain at 10 W/m 2. 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 Intensity, Decibels, and Frequency where I 0 = W/m 2. Four trumpet players are playing the same note. If three of them suddenly stop, the sound intensity level decreases by 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 A. 4 db B. 6 db C. 12 db D. 40 db 8

9 Linear and Logarithmic Scales Linear Scale Decibel 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 f 0, 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 f 0, the natural frequency of the source. The Doppler Effect The frequencies heard by a stationary observer when the sound source is moving at speed v 0 are The frequencies heard by an observer moving at speed v 0 relative to a stationary sound source emitting frequency f 0 are 9

10 DOPPLER EFFECT FORMULA Doppler Effect Demo: OW-B-DB Doppler Ball f = v + v O v v S f 0 + toward - Away from 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 B A C D observer EXAMPLE How fast are the police traveling? QUESTION: EXAMPLE How fast are the police traveling? EXAMPLE How fast are the police traveling? 10

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