Vibrations and Waves
There are only two methods of transferring energy and thus information from place to place in the universe: particles and waves. This chapter begins the study of waves.
We begin with the motion of a pendulum. We could use any periodic motion, but a pendulum is common and easy to understand. Imagine a pen attached to the bottom of the bob so that it can write on a piece of long paper. Pull the paper perpendicular to the swing plane of the pendulum and the pen will draw a sine wave on the paper. The sine wave is the shape of a transverse wave, so the motion of a pendulum is a type of wave motion.
Some terms:
Transverse – medium moves in a direction perpendicular to the direction of the wave.
Longitudinal – medium moves in the same direction as the direction of the wave.
Equilibrium – place where the medium would come to rest without the wave traveling through it. In the case of the pendulum, equilibrium is when the bob is directly under the place of support.
Crest – place of maximum displacement from equilibrium
Trough – same as the crest but in the opposite direction
Amplitude (A) – distance from equilibrium to a crest or a trough
Wavelength (l) – distance between consecutive, identical places on the wave – distance a wave travels during one vibration
Cycle – one complete motion of the wave
Period (T) – time for one complete cycle – time to generate one complete wave – time for wave to travel one wavelength
Frequency (f) – number of complete cycles of periodic motion in a period of time – measured in cycles/sec or hertz
We did experiments on pendulum motion to see what made the period change. We found that only the length and the acceleration of gravity affects the period.
Period µ Length
Period µ 1/(acc. Of gravity)
We found that the period did NOT depend on the mass of the bob or the amplitude of the swing (essential for making pendulum clocks).
Relationships: v = fl
F = 1/T
Interference is a characteristic of waves. When more than one wave occupies the same place at the same time, the result is a point-by-point addition of the waves (superposition) to produce the resultant wave. If the resultant amplitude is diminished, we call that destructive interference and if the resultant’s amplitude is enhanced, we call this constructive interference. If two identical waves are interfering, the places where a crest meets a crest results in twice the amplitude of either of the original waves (total constructive interference). If a crest meets a trough, zero displacement results (total destructive interference).
Standing waves result when identical waves (same l and A) moving in opposite directions interfere. The wave still oscillates but doesn’t appear to travel in the medium. Nodes are places of minimum displacement, while anti-nodes are places of maximum displacement.
The Doppler Effect happens for all waves. Due to the motion of the source (or the observer), waves appear to stack up in the direction of motion and we measure too short of a wavelength. Waves appear to stretch out in the direction opposite the motion and we measure too long a wavelength. In sound pitch corresponds to frequency. So a race car coming at us will be pitched higher (shorter wavelength – higher frequency), while the car moving away will be pitched lower (longer wavelength – lower frequency). By noting the direction of the wavelength changed (longer or shorter), we can measure the direction of travel, and the size of the shift gives us the speed. For light these same concepts are called blueshift (approaching) or redshift (receding).
In three dimensions the “stacking up” of the waves caused by the Doppler Effect produces a bow wave, a conical shaped wave coming from the leading edge of the object. When the object moves at the local speed of sound the stacking of the waves puts all of the sound energy along a shock front that travels with the object. The sound energy is delivered all at once as it passes an observer and a sonic boom is the result.
Terms:
Subsonic – speed is less than the speed of sound
Supersonic – speed is greater than the speed of sound
Mach number – ratio of the speed of the object to the speed of sound