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Introduction to sound review

Review the key terms and skills for sound waves, including how to identify the nodes and antinodes for standing waves in tubes.

Pojęcia kluczowe

Term	Meaning
Open tube	Tube that is open at both ends. Also called an open pipe.
Closed tube	Tube that is open at one end, and closed at the other. Also called a closed pipe.
Resonance	Another term for standing sound waves.

Standing sound waves open and closed tubes

Sound waves are longitudinal waves in a medium such as air. The molecules in the medium vibrate back and forth from their equilibrium position. This leads to the molecules being compressed in some parts of the wave, which results in variations in pressure in a predictable pattern. For example, when a musician blows into a tube such as a flute, the sound produced comes from waves that travel along the length of the tube.

Just like other waves, the energy of sound waves increases with the wave amplitude. The loudness or volume of the sound also increases with wave amplitude.

In open and closed tubes, sounds waves can exist as standing waves as long as there is at least one node.

Open tubes

In an open tube, the medium (ex. air) at the open ends vibrates horizontally parallel to the tube length. This means the standing wave has displacement antinodes at the ends of the tube for all harmonics, and a node in the middle for the fundamental.

The simplest standing wave in an open tube is the fundamental, which has

2

antinodes and

1

node. Thus, there is half of a wavelength between the antinodes. For an open tube with length

L

, the wavelength

λ

of the standing wave that corresponds with the fundamental frequency is:

\begin{aligned} L & = \frac{λ}{2} \\ λ & = 2 L \end{aligned}

Where the fundamental frequency is:

\begin{aligned} f_{1} & = \frac{v}{λ} \\ f_{1} & = \frac{v}{2 L} \end{aligned}

Standing waves with any integer multiple of the fundamental frequency can fit in an open tube.

Another way to think about standing waves in open tubes is by instead considering how the air pressure varies along the length of the tube. For open tubes, the air pressure at the ends equalizes with the atmosphere. Thus, the pressure stays constant at the open ends and they are pressure nodes.

The open end of a tube is a pressure node as well as a displacement antinode. This is admittedly confusing. The pressure at the open end of a tube has a fixed constant value (i.e node) equal to one atmosphere (assuming your tube is exposed to the air near Earth's surface). But the displacement of the air molecules at the open end of the tube has a maximally varying value (i.e. antinode) since the air is free to move at the open end. In other words, if you're drawing a diagram of the pressure in the tube, the open ends will be nodes. If you're drawing a diagram of the displacement of the air in the tube, the open ends will be antinodes.

Closed tubes

The air molecules are not free to vibrate back and forth parallel to the tube, so the displacement standing wave has is a node at the closed end. The open end of the tube is always an antinode since the air molecules can vibrate horizontally parallel to the length of the tube.

The simplest standing wave case in a closed tube has

1

antinode and

1

node. Thus, there is one quarter of a wavelength between the antinodes.For a closed tube with length

L

, the standing wave that corresponds with the fundamental frequency is:

\begin{aligned} L & = λ / 4 \\ λ & = 4 L \end{aligned}

The fundamental frequency is:

\begin{aligned} f_{1} & = \frac{v}{λ} \\ f_{1} & = \frac{v}{4 L} \end{aligned}

For closed tubes, we can have only odd-numbered harmonics. That’s because closed tubes by definition have a node at one end and antinode at the other, so there’s no way for even-numbered frequencies to be present.

We can also think about standing waves in closed tubes in terms of the air pressure along the length of the tube. For closed tubes, the air pressure at the closed end varies since it does not need to equalize with the atmosphere. The pressure at the open end is constant, so the pressure wave has a node at the open end and an antinode at the closed end.

Często spotykane błędy i nieporozumienia

Sometimes people forget that sound waves require a medium. Sound waves can’t travel through a vacuum (empty space) because there are no air molecules to vibrate and cause pressure variations.

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