Chapter 7: Waves and Sound Flashcards
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.
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. In this case, the longitudinal wave created by the person moving the piston back and forth causes air molecules to oscillate through cycles of compression and rarefaction (decompression) along the direction of motion of the wave.
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.
Describing waves
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 propagation speed (ν) of a wave:
ν = fλ
Period (T)
If frequency defines the number of cycles per second, then its inverse—period (T)—is the number of seconds per cycle:
T= 1/F
Angular frequency (ω)
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:
Equilibrium position, displacement, amplitude
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 its amplitude (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).
Sound
Sound is a 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. The speed of sound is given by the equation
V= square root B/ ρ
where B is the bulk modulus, a measure of the medium’s resistance to compression (B
increases from gas to liquid to solid), and ρ is the density of the medium. The speed of sound 343 m/s.
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.
f’=f (v +_ vd)/ (v +_ vs)
where f′ is the perceived frequency, f is the actual emitted frequency, ν is the speed of sound in the medium, νD is the speed of the detector, and νS is the speed of the source.
Top sign for “toward”
Bottom sign for “away”
Intensity and loudness
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. The SI units of intensity are therefore watts per square meter (N/m^2) . Intensity is calculated using the equation
I=P/A
where P is the power and A is the area. Rearranging this equation, we could consider that the power delivered across a surface, such as the tympanic membrane (eardrum), is equal to the product of the intensity I and the surface area A, assuming the intensity is uniformly
distributed.
Sound level
This is a huge range, which would be unmanageable to express on a linear
scale. To make this range easier to work with, we use a logarithmic scale, called the sound level (β), measured in decibels (dB):
β= 10 log I/ Io
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
Fbeat= l f1-f2 l
where f1 and f2 represent the two frequencies that are close in pitch, and fbeat represents the resulting beat frequency
Standing nodes
The points in a standing wave with no fluctuation in displacement are called nodes.
The points with maximum fluctuation are called antinodes.
Nodes are places of No Displacement
Closed boundaries are 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. Open boundaries are those that allow maximal oscillation and correspond to antinodes. The open end of a pipe and the free end of a flag are both open boundaries
Strings
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. This pattern suggests that the length L of a string must be equal to some multiple of half-wavelengths ( and so
on).
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. From therelationship that where ν is the wave speed, the possible frequencies are:
λ= 2L/n
Frequency pf a standing wave
Frequency = nv/2L
