This lecture is about when sound occurs as a standing wave. When we graph a standing
longitudinal wave, where the nodes and antinodes appear depends on which type of graph we're using.
On a displacement graph the nodes are where the compression and rarefaction occur. This is
because those are the places on the standing wave where there's the least displacement.
There's no displacement in the particles in those compressions, and rarefactions of the
standing wave do not move at all, so because those particles aren't moving on a graph that measures
the displacement of the particles on the y axis, those particles that aren't displaced would be
nodes. They just stay at zero throughout the whole time. And you can see that on that top graph,
the displacement graph, the nodes line up with the particles that aren't moving in the animation,
which is also where the compressions and rarefactions in the wave occur. On a graph
of the pressure of the air, the antinodes are where the compression and rarefaction occur,
so even though those particles aren't moving that's where they feel the most or least air
pressure from the other particles. When we're dealing with sound, it's more common to use a
displacement versus distance graph than a pressure versus distance graph, so we'll use displacement
distance graphs for the remainder of this video. You can see that in the standing waves here the
y-axis of the graph is graphing the displacement of the particles, when the particles move to the
left the displacement becomes negative, and when they move to the right it becomes positive, and
there are certain places where the particles are not being displaced at all. These are the nodes
of the graph, so those red lines show you where the particles are not being displaced at all,
and the midpoints of those lines are where they're displaced the most. When sound reflects off of a
surface to create a standing wave, the particles next to the surface it's reflecting off of do not
move in the standing wave. This means that the point at which sound reflects off a surface to
create a standing wave will always be a node on a displacement-distance graph. So you can see
here that the particles on the right side of this animation are being pushed up against a wall that
the air is reflecting off of and moving back to the left, so the particles at the very edge of
that wall are being compressed and then pushing other particles back, but they themselves are not
moving. So because they're not being displaced they always make a node, so the point at which
sound reflects and bounces back on itself will always be the node on a displacement-distance
graph of the longitudinal wave. So that bottom graph is what the displacement distance graph
would look like for this specific standing wave. There's a very popular high school demo
of standing waves and sound that involves a tube submerged in water and a tuning fork. We hold a
tuning fork over the tube submerged in water and hit the fork so that it makes noise. As we raise
the tube over the water we notice a pattern. So I'm going to imagine that we hit the tuning fork,
hold it above the tube, and then observe a pattern in the sound as we lift the tube out of the water
slowly, so just sit back and listen to what occurs when we hit the tuning fork and move the tube.
This pattern exists because the sound is being reflected off the water creating a standing wave
in the tube, at the point of reflection of the standing wave is a node. As we raise the tube,
the point on the wave that the opening is at changes from a node to an antinode and
back because the frequency of the sound doesn't change. It's the same tuning fork. And because
the velocity doesn't change, because it's the same material (air), the wavelength also
doesn't change. So the standing wave isn't going to stretch out at all. So instead,
as we raise the tube, we raise the opening from being at an antinode to a node to an antinode to
a node. Once the sound leaves the opening, it's no longer in a standing wave, it's dissipated in
all directions instead of moving up and down the tube. This means that the point on the standing
wave that the tube is at determines the energy delivered to the air outside the tube. So let's
observe that pattern again, this time where we can see the shape of the standing wave.
I'm going to observe this pattern one more time, this time I'm going to pay attention
to what the air particles inside the tube are doing, remembering that at an antinode,
that's where their displacement is the greatest, and at a node that's where they're not being
displaced at all, they're not oscillating at all. So this is starting off at an antinode,
so when there's an antinode you can see that the air particles are oscillating into and
out of the tube, so they must be delivering a lot of sound energy to the air outside of
them because sound energy is just the oscillation of air particles, so any air particles directly
outside of that tube are getting hit by the air particles inside of the tube,
and then that causes a wave to spread out in all directions, so at the antinode is
where they're moving the most so they're going to deliver the most sound energy to outside.
I've now moved the tube up to where it's at a node in the air,
and here you can see that those top air particles just aren't moving at all, and
so because they're not moving they're not going to oscillate any of the air outside of the tube,
so there's no sound energy delivered to outside of the tube. Remember that
standing wave pattern isn't going to continue once the sound exits the tube,
and so as a result the only thing that determines the energy being delivered to the outside air is
the energy of those particles on the top of the tube, and you can see that those particles are
barely moving and so they're not really going to deliver any sound energy to outside of the tube.
We move it up to another antinode and we can very clearly hear the sound again,
because again this is where the most sound energy is being delivered to outside of the tube.
And this is another node of the standing wave. So here there's no sound energy being delivered
to outside, so that's why that sound pattern emerges as we raise the tube out of the water,
because we can physically hear where the antinodes occur. We can measure the exact
distance between two antinodes. That distance is going to be equal to half the wavelength of
the wave. Because the tuning fork usually has the frequency written on it, we can use the
wavelength and frequency to find the velocity of the sound in the air. So if I listen for
that antinode and then listen for the next one and measure the distance between those two antinodes,
that length is going to be ½ of the full wavelength. As an example problem:
we can imagine that this tuning fork is 256 Hertz and we're holding it above this tube. As we raise
the tube the length between two consecutive points where the sound is loudest is 66 centimeters,
and we want to know the frequency of the sound in the air, so the wavelength 66
centimeters multiplied by two and then divided by a hundred to convert to meters is one point
three two meters. The frequency is 256 Hertz, so the velocity of this sound must be 335 meters per
second. So that's a very straightforward way of calculating the exact velocity of sound waves in
air. Standing waves of sound in a tube behave similarly to transverse standing waves in a
string or other material. For example, in most problems if we have a tube with an open end,
that open end is going to be an antinode. The reason why the open end of a tube is an antinode
isn't because it physically has to be, we only want the open end to be an antinode if we want
the maximum volume from the tube. We usually want the open end to be an antinode to get
the most sound out of it. This isn't always the case but in most problems that we do will assume
that the open end of the tube is an antinode, again not because it physically has to be but
only because we're assuming that the tube has been designed to get the most sound out of that
standing wave. Just like the fixed end of a rope, a closed end of a tube is a place where
sound is reflected, and that's always going to be a node. That is a physical requirement,
that's always going to have to be a node of a displacement graph of the sound standing wave.
That's a place where the particles that the sound is moving through are not themselves oscillating,
so we can get those same harmonic patterns that we got for the string. Using these shapes,
this is what the first harmonic would look like in each shape and these are the relationships between
the wavelength, the length of the material that the wave is in. In the sound example that material
is a tube and the frequency of the wave, just like the string with one open end, if the tube
has one open end we skipped straight from the first harmonic to the third harmonic, so these
are the general patterns that emerge both for standing waves in strings and for standing waves
of sound in tubes. Do that's what you need to know about the properties of sound as a standing wave.