Chapter 7: Waves and Sounds

Amplitude:

Amplitude is the maximal displacement of a wave from the equilibrium position.

Angular frequency:

Angular frequency is the same as frequency, but is measured in radians per second.

Natural (resonant) frequencies

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

Speed of sound

B= bulk modulus, a measure of the medium's resistance to compression (B increases from gas to liquid to solid) p= density of the medium Because the bulk modulus increases disproportionately more than density as one goes from gas to liquid to solid, sound travels fastest through a solid and slowest through a gas. The speed of sound in air at 20°C is approximately 100 m/s

Standing waves

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.

Strings

Consider a string, such as a guitar or violin string, or a piano wire, fixed rigidly at both ends. Because the string is secured at both ends and is therefore immobile at these points, they are considered nodes. If a standing wave is set up such that there is only one antinode between the two nodes at the ends, the length of the string corresponds to one-half the wavelength of this standing wave This is because on a sine wave, the distance from one node to the next node is one-half of a wavelength. ---- If a standing wave is set up such that there are two antinodes between the ends, there must be a third node located between the antinodes. In this case, the length of the string corresponds to the wavelength of this standing wave. Again, the distance on a sine wave from a node to the second consecutive node is exactly one wavelength.

True or False: Sound waves are a prime example of transverse waves.

False. Sound waves are the most common example of longitudinal waves on the MCAT.

Frequency:

Frequency is a measure of how oen a waveform passes a given point in space. It is measured in Hz

Possible frequencies of harmonic

From the relationship f=v/wavelength that where ν is the wave speed, the possible frequencies are:

Sound level (measured in decibels)

I = intensity of wave Io = threshold of hearing (10^-12 W/m^2) which is used as the reference intensity

Forced oscillation -force frequency

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. 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.

Traveling Wave

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.

Resonating

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. 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.

Timbre

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.

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?

If two waves are perfectly in phase, the amplitude of the resulting wave is equal to the sum of the amplitudes of the interfering waves. If two waves are perfectly out of phase, the amplitude of the resulting wave is the difference of the amplitudes of the interfering waves. Therefore, if the two waves are anywhere between these two extremes, the amplitude of the resulting wave will be somewhere between the sum and difference of the amplitudes of the interfering waves.

Shock wave and Sonic boom

In a special case of the Doppler effect, an object that is producing sound while traveling at or above the speed of sound allows wave fronts to build upon one another at the front of the object. This creates a much larger amplitude at that point. Because amplitude for sound waves is related to the degree of compression of the medium, this creates a large pressure differential or pressure gradient. This highly condensed wave front is called a shock wave, and it can cause physical disturbances as it passes through other objects. The passing of a shock wave creates very high pressure, followed by very low pressure, which is responsible for the phenomenon known as a sonic boom Unlike its depiction in movies and television, a sonic boom can be heard any time that an object traveling at or faster than the speed of sound passes a detector, not just at the point that the speed of sound is exceeded (Mach 1). Once an object moves faster than the speed of sound, some of the effects of the shock wave are mitigated because all of the wave fronts will trail behind the object, destructively interfering with each other.

Pitch

Lower-frequency sounds have lower pitch, and higher-frequency sounds have higher pitch.

Doppler ultrasound

Most ultrasound transmitters and receivers are packaged in a single unit. The transmitter and receiver do not function simultaneously, however, because one of the objectives of the system is to reduce interference. In addition to the standard ultrasound, most modern ultrasound machines also have a Doppler mode. Doppler ultrasound is used to determine the flow of blood within the body by detecting the frequency shi that is associated with movement toward or away from the receiver.

Period:

Period is the time necessary to complete a wave cycle

Echolocation

The Doppler effect can be used by animals through the process of echolocation. In echolocation, the animal emitting the sound (usually a dolphin or bat) serves as both the source and the detector of the sound. The sound bounces off of a surface and is reflected back to the animal. How long it takes for the sound to return, and the change in frequency of the sound, can be used to determine the position of objects in the environment and the speed at which they are moving.

The Doppler effect

The Doppler effect can be visualized by considering the sound waves in front of a moving object as being compressed, while the sound waves behind the object are stretched out

To which properties of a sound wave do amplitude and frequency correspond?

The amplitude of a wave is related to its sound level (volume). The frequency of a wave is related to its pitch.

Equilibrium position:

The equilibrium position is the point with zero displacement in an oscillating system

Fundamental frequency

The lowest frequency (longest wavelength) of a standing wave that can be supported in a given length of string is known as the fundamental frequency (first harmonic) The frequency of the standing wave given by n = 2 is known as the first overtone or second harmonic. This standing wave has one-half the wavelength and twice the frequency of the first harmonic. The frequency of the standing wave given by n = 3 is known as the second overtone or third harmonic, as shown in Figure 7.6c. All the possible frequencies that the string can support form its harmonic series.

How does applying a force at the natural frequency of a system change the system?

The object will resonate because the force frequency equals the natural (resonant) frequency. The amplitude of the oscillation will increase

Traveling wave:

Traveling waves have nodes and antinodes that move with wave propagation

What phenomena can be detected or treated using ultrasound?

Ultrasound can be used for prenatal screening or to diagnose gallstones, breast and thyroid masses, and blood clots. It can be used for needle guidance in a biopsy, for dental cleaning, and for treating deep tissue injury, kidney stones, certain small tumors, cataracts, among many other applications.

Ultrasound

Until this point we've focused on sound in the audible range; however, in medicine we can also use sound waves to visualize organs, anatomy, and pathology. This imaging modality can be used for prenatal screening, or to diagnose gallstones and breast or thyroid masses, or for needle guidance in a biopsy. Ultrasound uses high frequency sound waves outside the range of human hearing to compare the relative densities of tissues in the body. An ultrasound machine consists of a transmitter that generates a pressure gradient, which also functions as a receiver that processes the reflected sound Because the speed of the wave and travel time is known, the machine can generate a graphical representation of borders and edges within the body by calculating the traversed distance. Note that ultrasound ultimately relies on reflection; thus, an interface between two objects is necessary to visualize anything. ---- The transmitter (sender) generates a wave, which reflects off of an object and returns to the transmitter (which also functions as a receiver).

Wave speed:

Wave speed is the rate at which a wave transmits the energy or matter it is carrying. Wave speed is the product of frequency and wavelength.

Doppler effect

We've all witnessed the Doppler effect: an ambulance or fire truck with its sirens blaring is quickly approaching from the other lane, and as it passes, one can hear a distinct drop in the pitch of the siren. This phenomenon affecting frequency is called the Doppler effect, which describes the difference between the actual frequency of a sound and its perceived frequency when the source of the sound and the sound's detector are moving relative to one another. If the source and detector are moving toward each other, the perceived frequency, f′, is greater than the actual frequency, f. If the source and detector are moving away from each other, the perceived frequency is less than the actual frequency f' = perceived frequency f= actual emitted frequency v=speed of the sound in the medium vD= speed of the detector vs= speed of the source

Phase difference

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 would have lamba/2 or 180 degrees

If two objects are traveling toward each other, how does the apparent frequency differ from the original frequency? What if two objects are traveling away from each other? What if one object is following the other?

When two objects are traveling toward each other the apparent frequency is higher than the original signs: toward = top + on top half - on bottom half ---- When two objects are traveling away from each other, the apparent frequency is lower than the original frequency signs: away = bottom - on top half + on bottom half ---- When one object follows the other, the apparent frequency could be higher, lower, or equal to the original frequency depending on the relative speeds of the detector and the source either halves + or both -

Open boundaries

allow maximal oscillation and correspond to antinodes The open end of a pipe and the free end of a flag are both open boundaries.

Standing waves

standing waves are produced by the constructive and destructive interference of a traveling wave and its reflected wave. More broadly, we can say that a standing wave will form whenever two waves of the same frequency traveling in opposite directions interfere with one another as they travel through the same medium. Standing waves appear to be standing still—that is, not propagating—because the interference of the wave and its reflected wave produce a resultant that fluctuates only in amplitude. As the waves move in opposite directions, they interfere to produce a new wave pattern characterized by alternating points of maximum displacement (amplitude) and points of no displacement. The points in a standing wave with no fluctuation in displacement are called nodes. The points with maximum fluctuation are called antinodes.

For each of the following diagrams, label the type of pipe or string it represents, a node and antinode, and the relevant equation relating λ and L:

open pipe with 2 nodes: frequency for open pipes = 2L/n ---- closed pipe with one node and antinode frequency for closed pipes= 4L/n ---- string with 2 antinodes and one node frequency for strings: 2L/n

Fundamental pitch and multiple overtones

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.

Nodes

points in the wave that remain at rest (where amplitude is constantly zero)

Antinodes

points midway between the nodes fluctuate with maximum amplitude

Period (T)

the inverse of frequency (cycles per second) seconds per cycle T=1/f

Not every frequency of traveling wave will result in standing wave formation.

the length of the medium dictates the wavelengths (and by extension the frequencies) of traveling waves that can establish standing waves Objects that support standing waves have boundaries at both ends

Frequency (f)

the number of wavelengths passing a fixed point per second measured in hertz (hz) or cycles per second (cps)

Loudness/volume of sound

the way in which we perceive its intensity Perception of loudness is subjective, and depends not only on brain function, but also physical factors such as obstruction of the ear canal, stiffening of the ossicles, or damage to cochlear hair cells by exposure to loud noises or with age

Transverse waves

those in which the direction of particle oscillation is perpendicular to propagation (movement) of the wave 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.

Closed boundaries

those that do not allow oscillation and that correspond to nodes The closed end of a pipe and the secured ends of a string are both considered closed boundaries.

Propagation speed (v)

v=fλ

Equilibrium position -displacement -amplitude

waves oscillate about a central point called the equilibrium position displacement (x): describes how far a particular point on the wave is from the equilibrium position, expressed as a vector quantity. amplitude (A): the maximum magnitude of displacement in a wave (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).)

Constructive Interference

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

Destructive 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

Principle of superposition

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

Calculating new sound level when changed by some factor

where If/Ii is the ratio of the final intensity to the initial intensity.

Harmonic: Equation that relates the wavelength of a standing wave and the length of the string that supports it

where n is a positive nonzero integer (n = 1, 2, 3, and so on) called the harmonic. The harmonic corresponds to the number of half-wavelengths supported by the string

Hurtz (Hz)

1/s

Open pipes

Pipes that are open at both ends are called open pipes If the end of the pipe is open, it will support an antinode. An open pipe, being open at both ends, has antinodes at both ends. If a standing wave is set up such that there is only one node between the two antinodes at the ends, the length of the pipe corresponds to one-half the wavelength of this standing wave This is analogous to a string except that the ends are both antinodes instead of nodes. The analogy continues throughout: the second harmonic (first overtone) has a wavelength equal to the length of the pipe, The third harmonic (second overtone) has a wavelength equal to two-thirds the length of the pipe, an open pipe can contain any multiple of half-wavelengths; the number of half-wavelengths corresponds to the harmonic of the wave. The relationship between the wavelength λ of a standing wave and the length L of an open pipe that supports it is wavelength = 2L/n and the possible frequencies of the harmonic series are f=nv/2L just like a string --- picture = first second and third harmonics of an open pipe nodes are where they cross and we have antinodes at the end

Sound intensity (intensity equation)

Sound intensity, on the other hand, is objectively measurable. Intensity is the average rate of energy transfer per area across a surface that is perpendicular to the wave. In other words, intensity is the power transported per unit area. SI unit: W/m^2

Damping (attenuation)

Sound is not transmitted undiminished. Even aer the decrease in intensity associated with distance, real world measurements of sound will be lower than those expected from calculations. This is a result of damping, or attenuation. Oscillations are a form of repeated linear motion, so sound is subject to the same nonconservative forces as any other system, including friction, air resistance, and viscous drag. The presence of a nonconservative force causes the system to decrease in amplitude during each oscillation. Because amplitude, intensity, and sound level (loudness) are related, there is a corresponding gradual loss of sound. Note that damping does not have an effect on the frequency of the wave, so the pitch will not change. This phenomenon, along with reflection, explains why it is more difficult to hear in a confined or cluttered space than in an empty room: friction from the surfaces of the objects in the room actually decreases the sound waves' amplitudes. Over small distances, attenuation is usually negligible.

How is sound produced and transmitted?

Sound is produced by mechanical vibrations. These are usually generated by solid objects like bells or vocal cords, but occasionally can be generated by fluids. Sound is propagated as longitudinal waves in matter, so it cannot propagate in a vacuum.

Beat frequency

Sound volume can also vary periodically due to interference effects. When two sounds of slightly different frequencies are produced in proximity, as when tuning a pair of instruments next to one another, volume will vary at a rate based on the difference between the two pitches being produced. The frequency of this periodic increase in volume can be calculated by the equation: Where f1 and f2 represent the two frequencies that are close in pitch, and fbeat represents the resulting beat frequency.

Ultrasonic

Sound waves with frequencies above 20,000 Hz are called ultrasonic waves. Both dog whistles, which emit frequencies between 20 and 22 kHz, and medical ultrasound machines, which emit frequencies in excess of 2 GHz, are examples of ultrasonic waves.

Infrasonic waves

Sound waves with frequencies below 20 Hz are called infrasonic waves,

Standing wave:

Standing waves have defined nodes and antinodes that do not move with wave propagation.

Closed pipe

closed at one end (and open at the other) are called closed pipes If it is closed, it will support a node. In the case of a closed pipe, the closed end will correspond to a node, and the open end will correspond to an antinode. The first harmonic in a closed pipe consists of only the node at the closed end and the antinode at the open end In a sinusoidal wave, the distance from a node to the following antinode is one-quarter of a wavelength. Indeed, unlike strings or open pipes, the harmonic in a closed pipe is equal to the number of quarter-wavelengths supported by the pipe. Because the closed end must always have a node and the open end must always have an antinode, there can only be odd harmonics. This is because an even number of quarter-wavelengths would be an integer number of half- wavelengths—which would necessarily have either two nodes or two antinodes at the ends. The first harmonic has a wavelength that is four times the length of the closed pipe. The third harmonic (first overtone) has a wavelength that is four-thirds the length of the closed pipe The fih harmonic (second overtone) has a wavelength that is four-fihs the length of the closed pipe ---- The equation that relates the wavelength λ of a standing wave and the length L of a closed pipe that supports it is: λ = 2L/n where n can only be an odd integers (n = 1, 3, 5, and so on). --- The frequency of the standing wave in a closed pipe is f=nv/4L where v is the speed

damping/attenuation

decrease in amplitude of a wave caused by an applied or nonconservative force

Wavelength (lambda)

distance from one maximum (crest) of the wave to the next

Partially constructive and partially deconstructive

if waves are not perfectly in phase or out of phase 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. 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.

Sound

longitudinal wave transmitted by the oscillation of particles in a deformable medium As such, sound can travel through solids, liquids, and gases, but cannot travel through a vacuum.

Sinusoidal waves

may be transverse or longitudinal, individual particles oscillate back and forth with a displacement that follows a sinusoidal pattern

Angular frequency

measured in radians per second and is often used in consideration of simple harmonic motion in springs and pendula

Noise

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.

Longitudinal waves

one in which the particles of the wave oscillate parallel to the direction of propagation; the wave particles are oscillating in the direction of energy transfer sound waves are the classic example of longitudinal waves