Because both masses suspended on a spring and waves at
the beach exhibit periodic motion, we can use much of the same vocabulary
and mathematical tools to describe both. However, there is a significant
difference: waves are extended in space, while a mass on a spring
just oscillates back and forth in one place.
A familiar and concrete example of wave motion is the
“wave” spectators create at sporting events by standing up and sitting
down at appropriate intervals. Each person stands up just as that
person’s neighbor stands up, transmitting a form of energy all the
way around the stadium. There are two things worth noting about
how this works:
Waves are transmitted through a medium: The
energy and the “wave” are both created by the successive action
of people standing up and down. If there were no people in the stadium,
no wave could exist and no energy could be transmitted. We call
the people at the stadium, the water at the beach, the air molecules
transmitting sound, etc., the medium through which these waves are
medium itself is not propagated: For the “wave” to work,
each person in the stadium only needs to stand up and sit back down.
The “wave” travels around the stadium, but the people do not.
Think of waves as a means of transmitting energy over
a distance. One object can transmit energy to another
object without either object, or anything in between them, being
permanently displaced. For instance, if a friend shouts to you across
a room, the sound of your friend’s voice is carried as a wave of
agitated air particles. However, no air particle has to travel the
distance between your friend and your ear for you to hear the shout.
The air is a medium, and it serves to propagate sound energy without
itself having to move. Waves are so widespread and important because
they transmit energy through matter without permanently displacing
the matter through which they move.
Crests, Troughs, and Wavelength
Waves travel in crests
although, for reasons we will discuss shortly, we call them compressions
dealing with longitudinal waves
. The terms crest
used in physics just as you would use them to refer to waves on
the sea: the crest of a wave is where the wave is at its maximum
positive displacement from the equilibrium position, and the trough
is where it is at its maximum negative displacement. Therefore,
the displacement at the crest is the wave’s amplitude, while the
displacement at the trough is the negative amplitude. There is one
crest and one trough in every cycle of a wave. The wavelength
, of a traveling wave is the distance between
two successive crests or two successive troughs.
The period of oscillation, T,
is simply the time between the arrival of successive wave crests or
wave troughs at a given point. In one period, then, the crests or
troughs travel exactly one wavelength. Therefore, if we are given
the period and wavelength, or the frequency and wavelength, of a
particular wave, we can calculate the wave speed, v:
attaches a stretched string to a mass that oscillates up and down
once every half second, sending waves out across the string. He
notices that each time the mass reaches the maximum positive displacement
of its oscillation, the last wave crest has just reached a bead
attached to the string 1.25 m away. What are the frequency, wavelength,
and speed of the waves?
The oscillation of the mass on the spring determines the
oscillation of the string, so the period and frequency of the mass’s
oscillation are the same as those of the string. The period of oscillation
of the string is T = 0.5 s,
since the string oscillates up and down once every half second.
The frequency is just the reciprocal of the period: f = 1/T = 2 Hz.
The maximum positive displacement of the mass’s oscillation
signifies a wave crest. Since each crest is 1.25
apart, the wavelength,
, is 1.25
Determining wave speed:
Given the frequency and the wavelength, we can also calculate
the wave speed:
Imagine placing a floating cork in the sea so that it
bobs up and down in the waves. The up-and-down oscillation of the
cork is just like that of a mass suspended from a spring: it oscillates
with a particular frequency and amplitude.
Now imagine extending this experiment by placing a second
cork in the water a small distance away from the first cork. The
corks would both oscillate with the same frequency and amplitude,
but they would have different phases: that is, they
would each reach the highest points of their respective motions
at different times. If, however, you separated the two corks by
an integer multiple of the wavelength—that is, if the two corks
arrived at their maximum and minimum displacements at the same time—they
would oscillate up and down in perfect synchrony. They would both
have the same frequency and the same phase.