MCAT Physics and Math Review
Chapter 7: Waves and Sound
As a species, our interactions with sound are amazingly complex. The human ear developed as a means of detecting longitudinal waves carried in the air—this likely served an evolutionary purpose. A rustle in the leaves could indicate not only a potential meal, but also a potential predator. Our brains are highly attuned to analyze the sounds around us, as discussed in Chapter 2 of MCAT Behavioral Sciences Review. This includes not only the normal auditory pathway from the pinna through the tympanic membrane, ossicles, cochlea, and vestibulocochlear nerve to the temporal lobe, but also secondary structures such as the superior olive, which helps localize sound, and the inferior colliculus, which is involved in the startle reflex.
Language is also inextricably linked to sound. Through changes in pitch and timbre, we can imply or evoke dozens of complex feelings. Through music, our relationship with sound becomes even more profound. As E.T.A. Hoffman, a musicologist and pedagogue, wrote in his vivid description of Beethoven’s opening motif for Symphony No. 5 in C minor, op. 67:
Radiant beams shoot through this region’s deep night, and we become aware of gigantic shadows which, rocking back and forth, close in on us and destroy everything within us except the pain of endless longing—a longing in which every pleasure that rose up in jubilant tones sinks and succumbs, and only through this pain, which, while consuming but not destroying love, hope, and joy, tries to burst our breasts with full-voiced harmonies of all the passions, we live on and are captivated beholders of the spirits.
Indeed, sound can create entire worlds that we can explore. This chapter, however, aims only to lay the foundation for understanding wave phenomena. The general properties of waves will be introduced, including a discussion of wavelength, frequency, wave speed, amplitude, and resonance. We will also review the interactions of waves meeting at a point in space through constructive and destructive interference and examine the mathematics of standing waves—the means by which musical instruments produce their characteristic sounds. The subject of sound is reviewed as a specific example of the longitudinal waveform with a focus on wave phenomena like the Doppler effect. Finally, we provide a brief discussion of the use of ultrasound and shock waves in medicine.
7.1 General Wave Characteristics
It is important to use a common language when describing waves. We’ll establish the terminology associated with wave phenomena, and then spend the rest of this chapter looking at the application of wave principles to sound. In the next chapter, we’ll shift our focus to electromagnetic waves.
TRANSVERSE AND LONGITUDINAL WAVES
The MCAT is primarily concerned with sinusoidal waves. In these waves, which may be transverse or longitudinal, the individual particles oscillate back and forth with a displacement that follows a sinusoidal pattern. Transverse waves are those in which the direction of particle oscillation is perpendicular to the propagation (movement) of the wave. To visualize this, consider “The Wave” in a stadium. While “The Wave” moves around the stadium, individuals in the stands do not run around the stadium themselves. Rather, they move perpendicular to the direction of “The Wave”—by standing up and sitting down. More common examples on the MCAT include electromagnetic waves, such as visible light, microwaves, and x-rays. You could also form a transverse wave by attaching a string to a fixed point, and then moving your hand up and down, as is demonstrated in Figure 7.1a. In any waveform, energy is delivered in the direction of wave travel, so we can say that for a transverse wave, the particles are oscillating perpendicular to the direction of energy transfer.
Longitudinal waves are ones in which the particles of the wave oscillate parallel to the direction of propagation; that is, the wave particles are oscillating in the direction of energy transfer. Sound waves are the classic example of longitudinal waves, but because we can’t see sound, this waveform is a little more difficult to picture. Figure 7.1b helps us visualize what a longitudinal waveform traveling through air would look like. In this case, the longitudinal wave created by the person moving the piston back and forth causes air molecules to oscillate through cycles ofcompression and rarefaction (decompression) along the direction of motion of the wave. You could also form a longitudinal wave by laying a slinky flat on a table top and tapping it on the end.
Figure 7.1. Wave Types (a) Transverse: particles oscillate perpendicular to the direction of propagation; (b) Longitudinal: particles oscillate parallel to the direction of propagation.
Transverse waves have particle oscillation perpendicular to the direction of propagation and energy transfer. Longitudinal waves have particle oscillation parallel to the direction of propagation and energy transfer.
Waves can be described mathematically or graphically. To do so, we must first assign meaning to the physical quantities that waves represent. The distance from one maximum (crest) of the wave to the next is called the wavelength (λ). The frequency (f) is the number of wavelengths passing a fixed point per second, and is measured in hertz (Hz) or cycles per second (cps). From these two values, one can calculate the speed (ν) of a wave:
ν = fλ
If frequency defines the number of cycles per second, then its inverse—period (T)—is the number of seconds per cycle:
Frequency is also related to angular frequency (ω), which is measured in radians per second, and is often used in consideration of simple harmonic motion in springs and pendula:
Waves oscillate about a central point called the equilibrium position. The displacement (x) in a wave describes how far a particular point on the wave is from the equilibrium position, expressed as a vector quantity. The maximum magnitude of displacement in a wave is called itsamplitude (A). Be careful with the terminology: note that the amplitude is defined as the maximum displacement from the equilibrium position to the top of a crest or bottom of a trough, not the total displacement between a crest and a trough (which would be double the amplitude). These quantities are shown in Figure 7.2.
Figure 7.2. Anatomy of a Wave
Even if simple harmonic motion in springs and strings (pendula) are not on the formal content lists for the MCAT, it is still important to be familiar with the jargon of wave motion because sound and light (electromagnetic radiation) are on those content lists!
When analyzing waves that are passing through the same space, we can describe how “in step” or “out of step” the waves are by calculating the phase difference. If we consider two waves that have the same frequency, wavelength, and amplitude and that pass through the same space at the same time, we can say that they are in phase if their respective crests and troughs coincide (line up with each other). When waves are perfectly in phase, we say that the phase difference is zero. However, if the two waves travel through the same space in such a way that the crests of one wave coincide with the troughs of the other, then we would say that they are out of phase, and the phase difference would be one-half of a wave. This could be expressed as or, if given as an angle, 180° (one cycle = one wavelength = 360°). Of course, waves can be out of phase with each other by any other fraction of a cycle, as well.
PRINCIPLE OF SUPERPOSITION
The principle of superposition states that when waves interact with each other, the displacement of the resultant wave at any point is the sum of the displacements of the two interacting waves. When the waves are perfectly in phase, the displacements always add together and the amplitude of the resultant is equal to the sum of the amplitudes of the two waves. This is called constructive interference. When waves are perfectly out of phase, the displacements always counteract each other and the amplitude of the resultant wave is the difference between the amplitudes of the interacting waves. This is called destructive interference.
If two waves are perfectly in phase, the resultant wave has an amplitude equal to the sum of the amplitudes of the two waves. If two equal waves are exactly 180 degrees out of phase, then the resultant wave has zero amplitude.
If waves are not perfectly in phase or out of phase with each other, partially constructive or partially destructive interference can occur. As shown in Figure 7.3a, two waves that are nearly in phase will mostly add together. While the displacement of the resultant is simply the sum of the displacements of the two waves, the waves do not perfectly add together because they are not quite in phase. Therefore, the amplitude of the resultant wave is not quite the sum of the two waves’ amplitudes. In contrast, Figure 7.3b shows two waves that are almost perfectly out of phase. The two waves do not quite cancel, but the resultant wave’s amplitude is clearly much smaller than that of either of the other waves.
Figure 7.3. Phase Difference (a) In phase with a difference of almost zero; (b) Out of phase with a difference of almost 180 degrees
In noise-canceling headphones, pressure waves from noise are canceled by destructive interference. The speaker creates a wave that is 180 degrees out of phase and of similar amplitude. Many frequencies are usually present in the noise, so it is difficult to get perfect noise cancellation.
Noise-canceling headphones operate on the principle of superposition. They do not simply muffle sound, but actually capture the environmental noise and, using computer technology, produce a sound wave that is almost perfectly out of phase. The combination of the two waves inside the headset results in destructive interference, thereby canceling—or nearly canceling—the ambient noise.
TRAVELING AND STANDING WAVES
If a string fixed at one end is moved up and down, a wave will form and travel, or propagate, toward the fixed end. Because this wave is moving, it is called a traveling wave. When the wave reaches the fixed boundary, it is reflected and inverted, as shown in Figure 7.4. If the free end of the string is continuously moved up and down, there will then be two waves: the original wave moving down the string toward the fixed end and the reflected wave moving away from the fixed end. These waves will then interfere with each other.
Figure 7.4. Traveling Wave
Now consider the case when both ends of the string are fixed and traveling waves are excited in the string. Certain wave frequencies will cause interference between the traveling wave and its reflected wave such that they form a waveform that appears to be stationary. In this case, the only apparent movement of the string is fluctuation of amplitude at fixed points along the length of the string. These waves are known as standing waves. Points in the wave that remain at rest (where amplitude is constantly zero) are known as nodes. Points midway between the nodes fluctuate with maximum amplitude and are known as antinodes. In addition to strings fixed at both ends, pipes that are open at both ends can support standing waves, and the mathematics relating the standing wave wavelength and the length of the string or the open pipe are similar. Pipes that are open at one end and closed at the other can also support standing waves, but because the closed end contains a node and the open end contains an antinode, the mathematics are different. Standing waves in strings and pipes are discussed in more detail later, within the context of sound, because standing wave formation is integral to the formation of sound in certain contexts.
Why are clarinets, pianos, and even half-filled wine glasses considered musical instruments, but not pencils, chairs, or paper? This discrepancy has much to do with the natural (resonant) frequencies of these objects. Any solid object, when hit, struck, rubbed, or disturbed in any way will begin to vibrate. Tapping a pencil on a surface will cause it to vibrate, as will hitting a chair or crumpling a piece of paper. Blowing air pressure between a clarinet reed and a mouthpiece, striking a taut piano string, and creating friction on a wine glass’s surface will also cause vibration. If the natural frequency is within the frequency detection range of the human ear, the sound will be audible. The quality of the sound, called timbre, is determined by the natural frequency or frequencies of the object. Some objects vibrate at a single frequency, producing a pure tone. Other objects vibrate at multiple frequencies that have no relation to one another. These objects produce sounds that we do not find particularly musical, such as tapping a pencil, hitting a chair, or crumpling paper. These sounds are called noise, scientifically. Still other objects vibrate at multiple natural frequencies (a fundamental pitch and multiple overtones) that are related to each other by whole number ratios, producing a richer, more full tone. The human brain perceives these sounds as being more musical, and all nonpercussion instruments produce such overtones. Of note for the MCAT, the frequencies between 20 Hz and 20,000 Hz are generally audible to healthy young adults, and high-frequency hearing generally declines with age.
The natural frequency of most objects can be changed by changing some aspect of the object itself. For example, a set of eight identical glasses can be filled with different levels of water so that each vibrates at a different natural frequency, producing the eight notes of a diatonic musical scale. Strings have an infinite number of natural frequencies that depend on the length, linear density, and tension of the string.
If a periodically varying force is applied to a system, the system will then be driven at a frequency equal to the frequency of the force. This is known as forced oscillation. If the frequency of the applied force is close to that of the natural frequency of the system, then the amplitude of oscillation becomes much larger. This can easily be demonstrated by a child on a swing being pushed by a parent. If the parent pushes the child at a frequency nearly equal to the frequency at which the child swings back toward the parent, the arc of the swinging child will become larger and larger: the amplitude is increasing because the force frequency is nearly identical to the swing’s natural frequency.
If the frequency of the periodic force is equal to a natural (resonant) frequency of the system, then the system is said to be resonating, and the amplitude of the oscillation is at a maximum. If the oscillating system were frictionless, the periodically varying force would continually add energy to the system, and the amplitude would increase indefinitely. However, because no system is completely frictionless, there is always some damping, which results in a finite amplitude of oscillation. In general, damping is a decrease in amplitude of a wave caused by an applied or nonconservative force. Furthermore, many objects cannot withstand the large amplitude of oscillation and will break or crumble. A dramatic demonstration of resonance is the shattering of a wine glass by loudly singing the natural frequency of the glass. This is actually possible with a steady, loud tone—the glass will resonate (oscillate with maximum amplitude) and eventually shatter.
MCAT Concept Check 7.1:
Before you move on, assess your understanding of the material with these questions.
1. Define the following terms:
· Wave speed:
· Angular frequency:
· Equilibrium position:
· Traveling wave:
· Standing wave:
2. If two waves are out of phase at any interval besides 180 degrees, how does the amplitude of the resultant wave compare to the amplitudes of the two interfering waves?
3. True or False: Sound waves are a prime example of transverse waves.
4. How does applying a force at the natural frequency of a system change the system?