SAT Physics Waves - Doppler Effect

SAT Physics Waves - Doppler Effect

Wave Front Model of Sound
Diagrams of sound waves in exams often show the source of sound as a dot and the waves of sound as expanding circles, similar to the result seen when dropping a rock into a pond. When a rock is dropped into a still pond, waves move outward in every direction from the point of impact. The crests of the waves appear as expanding circles when viewed from above. This view of waves is known as the wave front model. The expanding circles represent the expanding wave crests. The distance between the circles (crests) is the wavelength. Although drawn as circles on paper, sound wave fronts actually form expanding three-dimensional spheres. A stationary sound source would emit spherical wave fronts as depicted in Figure 15.4.
Figure 15.4. Wave front model

The Doppler Effect and Sound
When a wave is produced by a source in a medium, the wave propagates through the medium at a constant speed, v. Once produced by the source, the speed of the wave in that particular medium will not change, regardless of the speed of the source.
    An interesting phenomenon occurs when the source of a sound is moving with respect to a stationary observer. Figure 15.5 shows a car moving to the right at constant speed. If the driver presses the horn continuously, then sound waves will leave the car and travel outward in expanding spheres.
Figure 15.5 The Doppler effect

To make it simple, assume one sound wave is emitted every second. Each sound wave front moves outward from the location where the car was at the time the wave was emitted. The first sound wave, I, was emitted when the car was at position I. This wave has been expanding for 3 seconds. The second wave, II, was emitted a second later when the car was at position II. This wave has been moving for 2 seconds. The third wave, III, was emitted when the car was at position III. This wave has been traveling for only 1 second.
    The sound of the horn will differ depending on the location of an observer, and the effects observed are known as the Doppler effect. For an observer in front of the car, the wavelengths appear to be shorter than they actually are. Since the speed of sound is constant, the shorter wavelengths create a higher frequency, and the horn will sound as though it has a higher pitch than it actually does. For an observer behind the car, the wavelengths appear to be longer than they actually are. The observer hears a lower frequency with a lower pitch than the horn actually makes. Note that the sound of the horn is not changing at all. It merely appears to have a higher frequency and shorter wavelengths when the sound source is moving toward the observer, It also apparently has a lower frequency and longer wavelength when the sound source is moving away from the observer.
Doppler Effect
(A) A sound source emits the wave fronts shown in the diagram above. In what direction is the sound source traveling compared to a stationary observer?
The wave fronts pile up in front of the sound source, so the sound source moves toward the region where the waves are closer together. The sound source is moving toward position C in the diagram.
(B) In the diagram above, where does an observer need to be positioned to hear a higher-pitch sound?

If the medium is not changing, the speed of sound is constant. Frequency and wavelength are inversely proportional. This means that the higher frequency will be heard where the wavelengths are shorter. This occurs in position C in the diagram.
    The previous explanations and examples address the most common problems involving moving sound sources and stationary observers. In some problems, the observer may be moving or both the source and observers may be moving. The key to any Doppler shift problem is the relative motion between the source and the observer. Whether the source moves toward the observer, the observer moves toward the source, or they both move toward each other, the observed effect is the same.     
   When a sound source and observer approach each other, the perceived wavelength of sound decreases and the observed frequency increases (high pitch). If the distance between the source and observer is increasing, then the perceived wavelength increases and the observed frequency decreases (low pitch).

Speed of the Source and the Speed of Sound
Several scenarios are possible depending on the speed of the source of sound. Figure 15.6 depicts four possible Doppler effect diagrams for moving sound sources.
Figure 15.6. The Doppler effect for moving sound sources

A stationary object will create concentric circular wave fronts as seen in Figure 15.6(a). A subsonic object, as seen in Figure 15.6(b), is an object moving with a speed less than the speed of sound (about 340 m/s in air). The best examples are a car continuously honking its horn or a siren on an emergency vehicle. When the sound source reaches the speed of sound, as shown in Figure 15.6(c), the wave fronts pile up on one another. These sound waves constructively interfere (add up) with each other to create a phenomenon known as the sound barrier. If a sound source exceeds the speed of sound, it is said to be supersonic, which is shown in Figure 15.6(d). When this occurs, the wave fronts constructively interfere in a manner that creates a wake of sound that is similar to the wake of a boat. When this wake of sound passes by a person, the compression of waves sounds like a boom. This is known as a sonic boom.

The Doppler Effect and Light
The Doppler effect for light waves is best known for the redshift of the absorption spectrum of a star. If a star is moving toward Earth, the wavelengths of light will seem shorter. The resulting dark lines on an absorption spectrum will appear shifted toward the blue end of the spectrum. If a star is moving away from Earth, the wavelengths of light will seem longer. The dark lines on an absorption spectrum will appear shifted toward the red end of the spectrum (redshift). Scientists observing the stars in the universe have noticed that the stars appear redshifted. This leads to the conclusion that the universe is expanding.

Superposition is when two or more waves occupy the same point in space, at a given moment, and their combination displaces the medium to reflect the sum of their individual displacements. In Figure 15.7, two wave pulses, A and B, generated in a string travel toward one another. In Figure 15.7(a), the pulses are the same size and are on the same side of the string. In Figure 15.7(b), the pulses are the same size, but pulse B is inverted.
Figure 15.7. Constructive (a) and destructive (b) wave pulses

When pulses A and B superimpose (occupy the same location), their combined displacements add to create a composite pulse. After superimposing for an instant, the wave pulses continue on their way in their original directions. In Figure 15.7(a), the wave pulses add to create a larger wave. This is an example of constructive interference. The waves overlap. The medium displacement is larger than it was for the waves individually. Figure 15.7(b) is an example of destructive interference, which occurs when waves overlap and cause a smaller displacement of the medium. When opposite waves are exactly the same size, they will cancel entirely, as shown in Figure 15.7(b).

Standing Waves
When a wave is trapped between two boundaries, the individual points move up and down but do not travel. They appear to stay in one place. This is known as a standing wave. An example can be seen on a guitar in which the guitar strings are attached at one end to the headstock and at the other end to the bridge. Plucking the string will create a standing wave. As one wave strikes a boundary, it bounces back and interferes with the next incoming wave. This creates an alternating pattern where part of the standing wave moves back and forth while several points do not appear to move at all. Figure 15.8 shows the incoming wave as a solid line and the reflected wave as a dashed line.
Figure 15.8. A standing wave pattern

The nodes are where the superposition of two waves creates destructive interference. The antinodes are the locations of greatest constructive interference. The amplitude, A, is the distance from the equilibrium line to the maximum displacement of the string. The maximum amplitude occurs at the antinodes. The wavelength of the wave is the distance between three successive nodes or three successive antinodes. The nodes are spaced exactly k/2 away from each other, and so are the antinodes. The wavelength of a standing wave is 2 times the distance between two successive nodes or antinodes.

Fundamental Frequency and Harmonics
When a simple instrument such as a guitar string or a flute is played, standing waves occur in the string or in the air inside the flute. The simplest standing waveform that can be produced in the string or between the ends of the flute produces a frequency known as the fundamental frequency, f1. The fundamental frequency is also known as the first harmonic. In a string and in an open tube (open at both ends), the fundamental frequency is associated with a waveform consisting of half of a wavelength. For closed tubes (closed at one end), the waveform consists of a quarter wavelength. Figure 15.9 shows the standing waveform associated with each of these simple instruments.
Figure 15.9. First harmonic waveforms

The values for wavelengths determined in Figure 15.9 can now be substituted into the wave speed equation in order to determine the fundamental frequency, f
  • Strings and open tubes: v = f1 (2L)
  • Open tubes: v = f1 (4L)
Other standing waves can be created in strings and in tubes. All of these patterns are known as the harmonics. They have a variety of frequencies (f1,f2, etc.) associated with the number of wavelengths fitting along the string or between the ends of the tubes. Usually questions regarding the harmonics are associated with the difference in frequency between the fundamental frequency, f1 (the first harmonic), and the second harmonic, f2. More than likely, these questions will focus on either a string or an open tube, which are mathematically the same. To create the second harmonic, the next complete standing wave must fit along the string or inside an open tube. Figure 15.10 shows the next wavelength capable of creating a standing wave in strings and tubes. The wavelengths depicted are associated with the frequencies of the second harmonic, f2.
Figure 15.10. Second harmonic waveforms

For strings and open tubes, the wavelength for the second harmonic is half of the wavelength associated with the fundamental. The wavelength of the harmonics can be easily determined if the wavelength, λ1 associated with the fundamental frequency is known. Simply multiply this key wavelength by the inverse of the harmonics number, n.
λn = (1/n)λ1
The wavelength of the second harmonic can be calculated as follows:
λ2 = (1/2)λ1
Frequency and wavelength are inversely proportional. As a result, the frequencies can be found by multiplying the fundamental frequency by n.
fn = nf1
The pattern is a bit more complicated for closed tubes and will probably not be tested.

So far, the examples of superposition are for waves traveling along the same medium at the same frequency and wavelength. However, there can also be superposition of waves that do not have the same frequency. When this occurs, the resulting superposition does not reflect a perfect sinusoidal pattern. A good example of this is the superposition of sound waves produced by two musical instruments at slightly different pitches (musical notes).
    Imagine two violinists playing the same note. A sound wave is produced from each violin. As the waves propagate through the medium of air, they will undergo superposition. If the waves are perfectly identical, they will complement each other and add constructively. If, however, the waves are slightly different, there will be both constructive and destructive portions. The destructive portions decrease the amplitude (volume) at a regular rate and cause what are known as beats.
In Figure 15.11, two completely different waves are added to create a third wave pattern, which displays the characteristic known as beats. Figure 15.11(a) and Figure 15.11(b) display the amplitude vs. time graphs for two sound waves, which have different frequencies and wavelengths. Figure 15.11(c) illustrates the superposition of these two waves.
Figure 15.11. Beats

Three distinct beats occur during the graphed time interval. This is known as the beat frequency. The beat frequency created by any two waves can be quickly determined by the absolute value of the difference between their frequencies.
Beats are detected by the human ear as quick outbursts of loud and quiet because amplitude (volume) is being affected. Most people can detect seven or fewer beats per second. For  two waves that are very dissimilar in frequency, there are too many beats per second to be detected by the human ear.


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