Unit 7.1: Waves and Harmonic Motion
When an object or material is displace and then returned to its original starting position over and over again, it is
considered to have simple harmonic motion. This means that the object or material is vibrating or moving consistently
in the same path of motion for several cycles. A spring with a mass hanging from it will bounce up and down
consistently over and over again as the spring is stretched and then relaxed. A pendulum will swing back and forth
following the same path of motion as the mass hanging from it swings to and fro.
A wave is a disturbance in the molecules such that it carries through matter or space. There are two types of wave
formations that occur: Transverse and Longitudinal. Transverse waves have vertical movement in the wave as it moves
forward while longitudinal waves have horizontal compressions that push the wave forward.
Transverse Wave
Longitudinal Wave
There are several aspects of a wave that can be looked at in regards to both wave formations. Crests are the high points
on transverse waves and high pressure of compression regions in longitudinal waves. Troughs are the low points on
transverse waves and low pressure or compression regions in longitudinal waves. The wavelength ( ) of a wave is the
distance from once spot on a wave to the same spot on the next adjacent wave. For transverse waves this could be
from crest to crest or trough to trough while for a longitudinal wave it is from compression to the next compression.
The amplitude of a wave is the size of the wave from top to bottom. For a transverse wave it is the distance from the
bottom of the trough to the top of the crest while for the longitudinal wave it is the amount or density of the
compression. All wave formations also have a period (T) of oscillation which is the time it takes for one wave to
complete a full movement or cycle. The inverse of period is frequency (f) which describes how often a wave occurs. It
represents the number of times a wave occurs each second. Again these two have the relationship of inverses of each
other.
We can also measure the speed of a wave using the basic speed equation but by changing the variables in terms of wave
motion.
Examples:
1.
2.
3.
A merry-go-round makes 12 rotations
in 15 s. What is the merry-go-rounds
period and frequency?
rot  12
t  15s
Find T & f
What is wave speed of a water wave
that has a wavelength of 3 m and a
frequency of 5.2 Hz?
  3m
What is the wavelength of a wave
that is moving at 97 m/s and makes
one complete cycle in 8.2 s?
v  97 m / s
T  8.2 s
Find 
f  5.2 Hz
Find v
15
T  time / rotation  12
 1.25s
f 
1
1

 .8Hz
T 1.25
v  f  3(5.2)  15.6m / s
v  f  97   ( 81.2 )    97(8.2)  795.4m / s
Waves Interference
If two waves collide they create forms of interference. However the interference only disrupts at the moment the
waves collide causing what is called superposition. When a crest and a trough collide, they cancel (destructive
interference) to the degree of two waves as shown in the first and third wave collisions below. However, when two of
the same type of formation collide (like two crests in the second wave collision), they combined (constructive
interference) to make a larger wave.
Wave Collision 1
Wave Collision 2
Wave Collision 3
When waves encounter boundaries (areas of greater or lesser density) the wave reflects some of its energy back. If a
wave hits a soft boundary (a medium of lesser density) then it will reflect a small portion of the energy back with the
same vertical orientation as the rest of the wave that transmits through the new medium. However if the wave hits a
hard boundary (a medium of greater density) then it will reflect most of the energy back but opposite the waves original
vertical orientation while only a little bit gets transmitted through the new medium.
When a wave reflects between two hard boundaries over and over again, the waves interfere with one another as
discussed previously. However, if the wave frequency has a wavelength that is of any 1:2 proportions to the distance
between the boundaries, the wave will reflect continuously and
constructively interfere in the same places and destructively interfere
in the same places. When this happens it is called a standing wave.
Where the places of constant destructive interference are called nodes
and places of constant constructive interference are call anti-nodes.
Standing waves are the main contributor to the generation of musical
instruments. Different notes on each instrument are created by
different wave frequencies and varying lengths. For example, a pipe
organ has a pipe cut to specific lengths for each not that can be played.
When a standing wave is produced in the tubes of a pipe organ it
causes the metal to resonate and produce the high quality sound we
hear with our ears. In the graphic above we have several different
notes being produced in the same length of tube by changing the
wavelength of the wave resonating within it.