This lecture is going to be about phases of a wave. These two waves have the exact
same wavelength, period, frequency, amplitude, and velocity, but they still
look different from each other. We need one more vocab word to describe the
physical difference between two waves. So the reason why we do physics is to
describe and understand the world, and if there's a part of the world that we
don't have vocabulary for, like the difference between these two waves, we
need to invent a new vocabulary term to describe it. Before we talk about this
difference we need to make an observation about the shape of a wave.
Any wave graph can be understood as the graph of the vertical height of the
radius line of a circle on the y axis and the angle of the vector on the x
axis. So as this circle goes around, you'll notice that if we track just the
vertical height of that radius line as it goes around a circle that produces a
wave so all waves actually share this property where they can be understood as
the vertical height of that radius line on the y axis and the degree of the
circle on the x axis so each of these graphs can show the y axis height of the
radius line for every angle around the circle so I'm just going to make a graph
of the vertical height of the radius line as it goes around the circle and
the x axis again we'll have those angles on it in both degrees and radians
what I'm trying to get across here is that certain angles can be associated
with certain parts of a wave and the change in an angle by a certain amount
corresponds to the change in a wave by a certain amount the specific angle that
we start from in the circle is kind of arbitrary we could also make wave graphs
that look like this just starting from different angles around the circle and
that still produces the same pattern so the specific angle that a wave is at at
different points isn't really a constant but there's something that is constant
in the shape of a wave so the thing that's not arbitrary about the
relationship between the angle and the wave is what fraction of the wave a
change in angle produces for example if we graph the full wave using that circle
no matter which part of the wave we're looking at a ninety degree or PI over
two radians in the circle results in a change of one quarter of the wave a
hundred and eighty degrees goes with half the wave 270 goes with
three-quarters of the wave and 360 goes with the full wave we can now understand
specific fractions of a wave as being equal to specific angles over the full
wave we can use this new way of recording the fraction of a wave to
describe how two waves that are otherwise completely alike are at
different points in their motion this is called the phase difference between two
waves this is the difference between which part of the waves full period or
wavelength each wave is on expressed as the angle difference between the two
waves at the same point on their wavelength or frequency are said to be
in phase so there's no phase difference between two waves that are perfectly
alike so as an example I can see that the bottom wave is exactly the same as
the top wave it's just shifted by a quarter of its length so that means that
the phase difference is going to be one quarter times 360
degrees or two pi so the phase difference is equal to ninety degrees or
PI over two so I would describe the bottom wave as having a phase difference
of 90 degrees or PI over two from the top wave here I can see that the wave is
shifted by this much which is 5/8 of the wave using that bottom equation for
phase difference 5/8 times 360 or two pi gets me a phase difference of 225
degrees or 5 PI over four this is a displacement time graph but the exact
same rule still works for finding the phase difference I just need to compare
the difference in the waves and I can see that this is a difference in 7/8 of
the wave so to find the phase difference I just multiply 7/8 by 360 or 2 pi and
that gets me a phase difference of 315 degrees or 7 PI over 4 simple harmonic
motion can also be described with phase differences because it can be graphed
with the shape of a wave for example in this picture that bottom portion of
simple harmonic motion is 3/8 along its full path compared to the original point
at that one maximum point of compression so what's happened here is that that
spring that oscillates has been released from rest moves through the equilibrium
position and continues to move between the equilibrium position and the other
maximum so at that point it's gone along 3/8 of its full oscillation because it
has to come all the way back to that other max for it to be a full
oscillation so because this is 3/8 of its full oscillation we would say that
the phase difference is 3/8 times 360 degrees which is equal to a 135 degree
phase difference or a 3 PI over 4 phase difference if it's going backwards that
means that it's already achieved that maximum and is moving back in the
opposite direction so that means that it has now covered 5/8 of its total motion
so the phase difference there would be 225 degrees or 5 PI over
for just one last note a negative phase difference just means a shift to the
left on the graph rather than to the right so here this graph has been
shifted by one fourth of its period to the left so that means the fraction of
the wave that's been shifted is negative 1/4 so the phase difference is negative
90 degrees or negative PI over 2