This chapter will describe waves in general, but will focus on sound waves in particular. Common types of waves are water, light, sound, and radio waves (Figure 15.1). Although they are very different from one another, they share many characteristics. With the whole world going wireless, knowledge of waves and their properties is very important.
FIGURE 15.1 Water waves.
To create a wave, you must disturb the medium. In case of water, when you tap it with a stick, the disturbance will propagate outward (Figure 15.2). The actual water molecules touched by the stick do not travel outward. Disturbing the water merely pushes on the water adjacent to it, and then that water pushes on the water next to it, and so on. It is like dominoes falling over and causing the others to fall over. There is motion, but there is no transport of matter (Figure 15.3). It is just the disturbance that moves. The wave carries energy but no matter.
FIGURE 15.2 The creation of a water wave.
FIGURE 15.3 Dominoes behave in a wavelike manner, propagating a disturbance. No one domino gets transported; only the energy is transported.
15.1 VELOCITY, FREQUENCY, AND WAVELENGTH
Creating waves on water requires disturbing the water. For example, tapping the water with a stick at some rate or frequency will produce a wave with the same frequency. This wave will travel outward with some velocity producing a series of crests and troughs. The period of the wave is the time it takes for the wave to travel a distance from crest to crest (Figure 15.4). The frequency of the wave is the rate at which the wave “wiggles”. The frequency of the waves is related to the period of the wave by:
FIGURE 15.4 A wave of wavelength λ, a period T.
Frequency has units of hertz, Hz or cycles/second.
The wavelength of a wave is the distance between crests (Figure 15.4). A wave’s wavelength (λ), frequency ( f ), and speed (v) are related. All waves, including light, sound, and radio can be described by the following equation:
Example 15.1
The musical note A has a frequency of 440 Hz (Figure 15.5). If the sound wave is traveling at 343 m/s, what is its wavelength?
FIGURE 15.5 What is the wavelength of this wave?
15.2 WAVE TYPES
There are two types of waves, transverse and longitudinal.
15.2.1 Transverse Waves
Transverse waves oscillate perpendicular to the direction of travel of the wave (Figure 15.6). Light is a transverse wave (Figure 15.7).
FIGURE 15.6 A jump rope demonstrating transverse waves.
FIGURE 15.7 Light is a transverse wave. It is made up of electric and magnetic fields that oscillate perpendicular to the direction of travel.
15.2.2 Longitudinal Waves
Longitudinal waves oscillate back and forth in the direction the wave is traveling (Figure 15.8). Sound is a longitudinal wave.
Some waves can be both transverse and longitudinal, as in the case of water (see Figure 15.9).
FIGURE 15.8 Longitudinal waves oscillate back and forth in the direction the wave is traveling.
FIGURE 15.9 Water is both a longitudinal and a transverse wave. A cork will bob up and down (the transverse component) and will also move back and forth (the longitudinal component).
15.3 GENERAL WAVE PROPERTIES
15.3.1 Refraction
Refraction is the bending of a wave when it moves from one medium into another. Waves refract because the speed of the wave changes when the wave moves from one medium into another. For example, light travels slower through water than through air. A wave front, striking the surface of water from air, will bend because the speed of the wave changes when the wave enters the water. The wave front in the water will lag behind the incoming wave. As a result, the wave will bend (Figure 15.10).
Sound waves exhibit refraction when moving from warm air of relatively low density to a cooler, denser air (Figure 15.11). The speed of sound in warm air is faster than in cool air. As a result, the wave that enters the cool air falls behind the wave in the warm air. The sound wave is effectively bent because of this slowing down.
FIGURE 15.10 Light moving from air to water. (a) Light bends because it slows down in the water and cannot keep up with the incoming wave. (b) Objects appear shallower in the water because of refraction.
15.3.2 Superposition
Multiple waves can travel through the same medium without affecting each other. The overall wave will be the sum of all the waves. This is called superposition. Light, sound, and water waves all obey superposition. For example, multiple electrical signals of different frequencies can be sent down a single wire, and at the end of the wire the signal made up of all the individual signals can be filtered to isolate each individual signal. If you looked at the waveform coming out of this wire, it would look like one big messy wave (Figure 15.12). The individual waves can be separated out with a filter because they operate at different frequencies (Figure 15.13).
FIGURE 15.11 Sound waves bend when entering a different temperature region.
15.3.3 Constructive Interference
Waves that are in sync with one another combine to make a bigger wave. This is known as constructive interference (Figure 15.14).
FIGURE 15.12 A superposition of many signals on one wire.
15.3.4 Destructive Interference
Waves that are out of sync with one another subtract to make a smaller wave. This is known as destructive interference (Figure 15.15).
Noise canceling earbuds listen to incoming sound and create another sound 180° out of phase to combine with the incoming sound. The two waves destructively interfere with one another leaving the listener in silence (Figure 15.16).
FIGURE 15.13 A filter separating out the individual waves.
FIGURE 15.14 Constructive interference.
FIGURE 15.15 Destructive interference. Pictured are two waves 180° out of phase.
FIGURE 15.16 Noise canceling earbuds use destructive interference to cancel out noise.
15.3.5 Tone
Multiple waveforms, acting together, make up the tone of a sound (Figure 15.17). The waves making up one sound may have different frequencies, phases, and intensities.
FIGURE 15.17 Most sound waves are a composite of many different waves.
15.3.6 Resonance
The resonance frequency is the frequency at which if a system is stimulated with will produce a maximum response. For example, a singer can shatter a glass if the volume and frequency of his or her voice are just right (Figure 15.18). Antennas emit the maximum power at their designed resonance frequencies.
FIGURE 15.18 Example of resonance: maximum energy transfer.
15.3.7 Diffraction
Diffraction is the bending of waves around obstacles. Diffraction occurs most strongly when the size of the obstacle is the size of the wavelength of the wave (Figure 15.19).
FIGURE 15.19 An example of diffraction is sound traveling through openings such as doorways.
15.4 DISTANCE AND VELOCITY MEASUREMENTS WITH WAVES
15.4.1 Radar, Lidar, and Sonar
Radar and sonar use radio and sound waves, respectively, to determine the distance to an object (Figures 15.20 and 15.21). When a wave is emitted, a clock is started and it times how long it takes for the wave to travel, reflect off an object, and return. Radar is typically used for long-distance determinations, such as by air traffic controllers to locate airplanes. Lidar works on the same principle as a radar; it only uses light instead of radio waves. Sonar moves much more slowly than radar and attenuates over a shorter distance, which is why it is better used for short-distance applications. Underwater, radar fails because radio waves are heavily absorbed in water. Sonar is therefore used by submarines and fishermen.
FIGURE 15.20 Radar.
FIGURE 15.21 Sonar.
15.4.2 Doppler Shift
If a source of sound or light is approaching you, the frequency of the waves will increase; if the source is moving away from you, the frequency will decrease (Figure 15.22). Consider an approaching train. You hear the pitch of the engine increasing as the train approaches, but as the train passes, the pitch lowers. This is because the sound waves become compressed as the source of sound approaches you and expand when the source moves away from you. This change in pitch or frequency due to a relative motion of the source is known as a Doppler shift. How much the pitch changes depends on the speed of the source.
FIGURE 15.22 The Doppler shift of approaching and receding sources.
An application of the Doppler effect is a police radar gun, which consists of a microwave radio source emitting a beam of radio waves at a given frequency (Figure 15.23). The frequency of radio waves increases after the waves bounce off oncoming objects. This is because the wave becomes compressed. The reflected wave returns to the gun, and its frequency is compared with the frequency of the original wave. The difference in the frequencies of the two waves is a measure of how fast the car is moving. Doppler radar is used to determine the speed of clouds (Figure 15.24).
FIGURE 15.23 Police radar using the Doppler effect to determine speed.
Sound waves exhibit the Doppler effect. Ultrasonic transducers are common ways to measure the distance to an object and can also be used to determine its speed (Figure 15.25). Speed can be determined by bouncing an ultrasonic sound wave off of a moving object, then measuring the change in frequency between the outgoing and incoming reflected waves.
FIGURE 15.24 Weather forecasters use Doppler radar to determine the speed of clouds.
Mathematically, the frequency shift from the Doppler effect for light and sound is given by:
For light,
For sound,
where
fo = frequency of the observer
fs = frequency of the source
c = speed of light
v = speed of sound
vs = speed of the source
vo = speed of the observer
FIGURE 15.25 The Doppler effect can be applied to ultrasonic waves to determine speed.
Example 15.2
A train is approaching you at 22.0 m/s and is blowing a whistle at a frequency of Hz. Suppose the speed of sound is 345 m/s. What does the frequency of the whistle sound like to you? Assume you are standing still, vo = 0 (Figure 15.26).
FIGURE 15.26 What does the frequency of the whistle sound like to you?
FIGURE 15.27 What does the return frequency measure?
Example 15.3
Some police radar guns use a microwave transmitter that sends out frequencies in the X-band of 10.525 GHz or the K-band centered at 24.150 GHz. Suppose a police officer is shooting radar with a radar gun that sends out a frequency of 24.150 GHz (Figure 15.27). If a car is approaching at 100 mph, what does the return frequency measure? Use c = 186,000 miles/s for the speed of light.
Note: We have neglected the rules of significant digits in this example.
15.5 SOUND WAVES
Sound is a longitudinal wave. Sound waves need a medium through which to travel, such as air, solids, or liquids. To create a sound wave, we need to somehow disturb the medium. For example, a stereo speaker creates sound by moving its cone back and forth (Figure 15.28). As the speaker cone moves forward, it compresses the air; when it moves backward, it creates a slight vacuum. The speaker is alternately compressing and decompressing the air. The volume of air disturbed by the speaker does not travel outward. It merely pushes on the air next to it, and then that air pushes on the air next to it. It is just the disturbance that moves.
FIGURE 15.28 A speaker cone moves back and forth, creating sound. Sound is the compression and decompression of air traveling out from the source.
15.5.1 Speed of Sound
Sound typically travels fastest through solids, then liquids, and slowest through gases (Table 15.1).
The speed of sound changes with temperature, pressure, and humidity of the air. At 72°F in dry air, at 1 atmosphere (atm) of pressure, the velocity is:
Table 15.1
The Speed of Sound Through Media
In general, the speed of sound changes with temperature in air according to:
where v is in m/s and Tc is in Celsius, the speed of sound increases with temperature.
Example 15.4
What is the velocity of sound in air at 31°C?
15.5.2 Pitch
The pitch of sound is the same as the frequency. In terms of generating sound with a speaker, the pitch is determined by how fast the speaker cone is moving back and forth. If the cone is oscillating back and forth at a high frequency, the pitch will be high (Figure 15.29).
FIGURE 15.29 A speaker generating high- and low-frequency sounds.
The human ear can hear sounds that range from a frequency of 20 Hz to 20 kHz. Sounds below this range are called infrasonic. Sounds above this range are called ultrasonic.
Low-frequency sounds spread out from a source, that is, they are not very directional (Figure 15.30). They are not readily absorbed, and as a result, they travel through walls easily. This is familiar to anyone living next door to someone with a stereo.
FIGURE 15.30 Low-frequency sounds spread out from a source.
High-frequency sounds are more directional than low-frequency sounds (Figure 15.31). Because of this, in the ultrasonic region, there are many applications where the waves can be focused.
FIGURE 15.31 High-frequency sounds are directional.
15.5.3 Ultrasound Imaging
Obstetricians use ultrasound imaging to see a fetus while it is still inside the mother (Figure 15.32). Sound waves, with frequencies of 3.5–7 MHz, are sent into the mother’s abdomen. Sound travels at about 1,540 m/s in soft tissues. Quartz crystals generate the sound and also act as the receiver of the returning signals. These waves are reflected and absorbed by the soft tissues. The reflected waves returning to the ultrasound transducer are then put together to form the image of the fetus. The higher-frequency waves have higher resolution than the low-frequency waves, but they do not penetrate as deeply.
FIGURE 15.32 Ultrasound imaging.
15.5.4 Ultrasonic Tape Measure
An ultrasonic tape measure measures the distance to objects by bouncing an ultrasonic wave off the object and timing how long it takes for the wave to come back. The longer it takes, the farther away the object (Figures 15.33 and 15.34).
FIGURE 15.33 The ultrasonic tape measure works by sending a sound wave and timing how long it takes to return after bouncing off an object whose distance is being measured.
FIGURE 15.34 Ultrasonic Rangefinder courtesy of Adafruit.
15.5.5 Ultrasonic Cleaning
Ultrasonic cleaning uses sound waves above 20 kHz to vibrate dirt loose from objects being cleaned (Figure 15.35). The sound causes the object to vibrate at the same frequency, shaking the dirt and grime loose.
FIGURE 15.35 Ultrasonic cleaning.
15.5.6 Sound Pressure
A sound wave is essentially a pressure wave, that is, a compression and decompression of air that travels. When this pressure wave impacts the eardrum, the eardrum sends a signal to the brain, and this signal is interpreted as sound. There is energy in this pressure wave. The rate at which this energy passes through a given area is given by the intensity.
15.5.7 Sound Intensity
The intensity or the amount of power delivered by waves per unit area is given by (Figure 15.36):
FIGURE 15.36 Sound of power P passing through an area A.
If we consider a point source of sound radiating isotropically (equally in all directions) outward, then the sound radiates outward spherically and the intensity at a distance r from the source is given by (Figure 15.37),
FIGURE 15.37 Sound radiating isotropically (equally in all directions) outward.
Example 15.5
A 1.0 W point source of sound radiates outward isotropically. What is the intensity 3.0 m away from the source?
15.5.8 Sound Intensity Level
The threshold intensity for hearing is Io = 1 × 10−12 W/m2. A sound that has twice the intensity as another does not sound twice as loud to the human ear. The ear’s response to sound is said to be logarithmic. A sound that is twice as intense as another does not sound twice as loud but only slightly louder. The volume control on a stereo is logarithmic, which takes into account the logarithmic nature of our hearing.
Sound intensity level is measured in decibels (dB) and is given by:
Sound levels can be measured with a decibel meter (Figure 15.38). From the sound intensity level equation, we can say that a sound that is 10N (where N is an integer) times more intense than the threshold of hearing means that SL = N ×10 dB.
■ A sound that is ten times more intense than threshold means SL = 10 dB.
■ A sound that is 100 times more intense than threshold means SL = 20 dB.
■ A sound that is 1,000 times more intense than threshold means SL = 30 dB.
Another way of thinking in decibels is the rule of 3 dB, which says that for every 3 dB increase in sound intensity level, the intensity (I) doubles. See Table 15.2 for typical intensity levels.
FIGURE 15.38 A decibel meter.
Table 15.2
Typical Sound Levels
Example 15.6
Let’s compare the intensity and the sound intensity levels of two sounds. According to the sound level table, the sound level of normal conversation is 60 dB. How does the intensity of this sound compare with the intensity of a sound at sound intensity level 66 dB?
Increasing a sound from 60 to 63 dB means the intensity has doubled according to the rule of 3 dB. Increasing a sound from 63 to 66 dB means the intensity has doubled again. Therefore, the intensity must have increased 2 times 2, or 4 times, rising from 60 to 66 dB.
Example 15.7
What is the sound intensity level of a sound with intensity 105 times as intense as the threshold for hearing?
A sound that is 100,000 times more intense than threshold is only 50 dB larger!
15.5.9 Microphones
Most microphones in use are dynamic air pressure sensors. A diaphragm responds to a sound wave by moving back and forth. A sensor behind the diaphragm measures this motion and converts it into an electrical signal. Any type of sensor that is capable of measuring a displacement in a diaphragm can be used in a microphone. Listed below are four types of microphones (Figures 15.39–15.42).
FIGURE 15.39 A crystal microphone. As the diaphragm moves back and forth from a sound wave, it causes the crystal to be flexed. Deforming this piezoelectric crystal produces a voltage across it proportional to the level of the sound wave.
FIGURE 15.40 A dynamic or moving coil microphone. As the diaphragm moves back and forth from a sound wave, the attached coil moves through a magnetic field, inducing a voltage into it.
FIGURE 15.41 A condenser microphone measures sound levels using a capacitor. The diaphragm acts as a movable plate on a capacitor that responds to sound pressure.
FIGURE 15.42 An electret-condenser microphone is essentially a condenser microphone with a permanent charge placed on the diaphragm at manufacturing. A built-in preamplifier, such as a field effect transistor (FET), amplifies the small signals.
15.5.10 Gravitational Waves
Gravitational waves are ripples in space-time that travel at the speed of light caused by some violent and energetic event such as the merger of two black holes. Albert Einstein predicted such waves in 1916, in his theory of gravity, General Relativity. In 2015, approximately 100 years after Einstein’s prediction, these waves were first detected. The source of this detection turned out to be two black holes spiraling into one another. General Relativity predicts that when a mass is accelerated it will produce waves on the space surrounding it, think of ripples on a pond when a rock is tossed in. These waves are extremely weak and require extremely large accelerated masses, and a very sensitive instrument to measure them. This first detection of gravitational waves took place on twin Laser Interferometer Gravitational-wave Observatory (LIGO) detectors, located in Livingston, Louisiana and Hanford, Washington, USA (Figure 15.43). The signals measured in these detectors agree with the prediction of the merger of two black holes about 29 and 36 times the mass of the sun 1.3 billion years ago; it took 1.3 billion years traveling at the speed of light to reach us.
At each observatory, a 4 km long L-shaped set of tubes under vacuum, called an interferometer, uses laser light that travels back and forth bouncing off mirrors at each end. The laser light from the two arms are then combined together. Destructive and constructive interferences will take place between the two beams if the distance between the arms change. Einstein’s theory predicts that the distance between the mirrors will change when a gravitational wave passes through the detector. A change as small as 10−19 m can be measured with these detectors. This is smaller than the diameter of a proton!
FIGURE 15.43 Laser Interferometer Gravitational-wave Observatory (LIGO).
15.6 CHAPTER SUMMARY
Symbols used in this chapter.
All waves, including light, sound, and radio can be described by the following equation:
Transverse waves: Transverse waves oscillate perpendicular to the direction of travel of the wave. Light is a transverse wave.
Longitudinal waves: Longitudinal waves oscillate in the direction of travel of the wave. Sound is a longitudinal wave.
Refraction: Refraction means the bending of a wave when it moves from one medium into another.
Superposition: Multiple waves can travel through the same medium without affecting each other.
Constructive interference: Waves that are in step or in phase with one another add to make a bigger wave.
Destructive interference: Waves that are out of step or out of phase with one another subtract to make a smaller wave.
Tone: The tone of a sound describes the way multiple waveforms making up the sound wave act together.
Resonance: The resonance frequency is the frequency at which if a system is stimulated with will produce a maximum response.
Diffraction: Diffraction is the bending of waves around obstacles.
Doppler shift: Doppler shift is the shifting of the frequency of a wave because of a relative motion between the source and the observer. Mathematically, the frequency shift from the Doppler effect for light and sound is given by:
For light,
For sound,
where:
fo = frequency measured by the observer
fs = frequency of the source
c = speed of light
v = speed of sound
vs = speed of the source
vo = speed of the observer
In general, the speed of sound changes with temperature in air according to:
Pitch: The pitch of sound is the same as the frequency.
Sound Intensity: The intensity or the amount of power delivered by waves per unit area is given by:
If we consider a point source of sound radiating isotropically (equally in all directions) outward, then the sound radiates outward spherically and the intensity at a distance r from the source is given by,
15.6.1 Sound Intensity Level
The threshold intensity for hearing is Io = 1 × 10−12 W/m2. A sound that has twice the intensity as another does not sound twice as loud to the human ear. The ear’s response to sound is said to be logarithmic. A sound that is twice as intense as another does not sound twice as loud but only slightly louder. The volume control on a stereo is logarithmic, which takes into account the logarithmic nature of our hearing.
Sound intensity level is measured in decibels (dB) and is given by:
PROBLEM SOLVING TIPS
When solving problems involving the formula v = λ · f be sure to convert f into hertz.
PROBLEMS
If a water wave comes onto the shore every 5 s, what is the frequency of the wave?
If you are listening to a radio station that broadcasts at 96 MHz, what is the wavelength of this wave?
If a signal from a satellite takes 0.01 s to travel to earth, how far away is the satellite?
How can you create an electromagnetic wave?
How does a stereo speaker produce sound?
How are fiber optic cables able to carry much more information than a copper wire of the same size?
How can many conversations take place at the same time on a phone line and not get mixed together?
What is the difference between a transverse wave and a longitudinal wave?
A police officer is shooting his radar gun in the X-band at an oncoming car traveling at 12 mph. What is the change in frequency of the returned signal?
An ultrasonic and a laser ranger are used to determine the distance to an object 25 m away. How long does it take for each signal to go out and return after reflecting off this object?
What is the speed of sound in air at 28 ° C?
When wood wind instruments warm up they become “sharp”. This is because the air inside the instrument warms up. Why would this make the instrument sound sharp?
Is the speed of sound faster in warm air or cold air?
What is the sound intensity level of a sound that is 105 times more intense than the threshold of hearing?
A 0.010 W point source of sound radiates outward equally in all directions. What is the intensity 3.0 m away from the source?
What is the sound intensity level of the sound in problem 15?
One sound is twice as intense as another sound. What is their relative intensity in dB?
An increase in one octave means a sound has doubled in frequency. If a 1 kHz sound rises 3 octaves, what is the new frequency?
An infrared LED (light-emitting diode) emits light with a wavelength of 940 nm. What is its frequency?
A tachometer measures the speed of rotation of a shaft. A simple tachometer can be made by attaching a piece of tape to a shaft, and listening with a microphone to the tape slap a fixed object as the shaft rotates. By observing the microphone’s output on an oscilloscope, the period of the signal can be measured. The rpm (revolutions per minute) of the shaft is given by rpm = 60/period(s). If the frequency of the pattern on the scope is 50 Hz, what is the shaft’s rpm?
Electrical power in the U.S. operates at 60 Hz. What is the period of this signal?
A car playing its radio is approaching you at 80 ft/s. If the frequency of the sound as heard by the driver is 5 kHz, what frequency does a pedestrian hear as the car is approaching her? Assume the speed of sound is 1,132 ft/s.
What is the speed of a train receding from you if frequency of the whistle the engineer hears is twice as high as what you hear standing on the track? Assume the speed of sound is 343 m/s.
What is the velocity of sound inside your lungs at 37°C?
A long-range microphone uses a parabolic microphone of area 0.8 m2 to collect the sound. If it is listening to a sound with an intensity of 10−5 W/m2, how much power is being gathered?
A source of sound that radiates outward spherically has an intensity of 10−3 W/m2 at a distance of 4 m. What is its power?
What is the sound intensity level of a sound with an intensity of 10−8 W/m2?
Using Table 15.1, find out for which material is the speed of sound the fastest.
Why can’t sound travel through a vacuum?
Does the speed of sound increase with air temperature or decrease?
What is the frequency of sound waves used in ultrasound imaging?
How does a dynamic microphone work?
How does a police radar gun work?
A sound that increases by 3 dB in sound intensity level increases by what intensity?
What is the sound level of a normal conversation?
What sounds are more directional, low or high frequency sounds?
What happens at the resonance frequency of a system?