# How are standing waves a result of constructive and destructive interferences?

For constructive I can understand. But destructive I can't.

I can not picture the shape of two pulses or waves maybe that form the resulting standing wave. The places where waves are canceled just look so perfect and so like a normal wave.

• This is a weird question to me. I'd more say that standing waves are examples of "normal" waves, and other waveforms result from constructive and destructive interference of standing waves. – Jerry Schirmer Jul 30 '15 at 19:34
• But my book says antinode is place of maximum constructive interference and nodes the opposite. – most venerable sir Jul 30 '15 at 19:35

Take a look at this Desmos animation. Either animate it by clicking the play button next ot the time (t) variable, or drag the slider around to watch the behavior of the waves.

Watch carefully what the standing wave (the black trace) looks like when the waves constructively interfere (I.E. at times when they both look identical) and when the waves destructively interfere (I.E. at times when they looked like mirror images of each other flipped over the y-axis). As you can see, the standing wave is simply the addition of the two traveling waves.

If you have access to a class room setting, find a long, small diameter, stiff spring (e.g., 1" dia, 10' long) and attach one end to the handle of an immovable object. Take the other end of the spring and step back several steps so the spring is stretched. Now, hold the end tightly with one hand, and use your other hand to displace the spring several inches, approximately 1' from the end. Let go of the displacement and note the pulse wave travelling down the spring.

When the pulse hits the immovable object, it will reflect, but the reflection will be inverted. This is characteristic of waves when they reflect off of fixed boundaries (see https://phet.colorado.edu/en/simulation/wave-on-a-string for an excellent example of this).

For a standing wave to exist, you will shake your end of the spring up and down at a constant frequency, sending a continuous train of pulses down the spring. The reflections will be inverted and will interfere with the pulses that you are sending. However, only very PARTICULAR frequencies will produce the standing wave pattern.

The frequency that produces the longest wavelength (the lowest frequency) standing wave, with a node on each end and an antinode in the middle, is known as the fundamental frequency. Twice that frequency produces a standing wave that is one full wavelength long, with nodes at each end and one in the middle, and two antinodes between the nodes, and is known as the 2nd harmonic. Three times the fundamental frequency produces the third harmonic, and so on. Note that for a string or spring, physics requires a node on each end.

For a pipe, you have a different situation. Physics requires a node on the closed end and an antinode on the open end. If you place the longest wave that you can in the pipe (closed on one end), you will find that the pipe will accommodate 1/4 of a wavelength. This means that the wavelength of the fundamental frequency is 4 times the length of the pipe, and different length pipes have different fundamental frequencies. In addition, the next harmonic for the pipe closed on one end will fit 3/4 of a wavelength, then 5/4 of a wavelength, etc., again due to the requirement of a node at the closed end and an antinode at the open end.

For a pipe open on both ends, an antinode must be on each end, and the harmonic series is much akin to the harmonic series that you would see on a string or spring.

Of course, if you want to see what is really happening, try the PhET simulation. Even better, set the spring up in a class room, shake it at one of its harmonic frequencies, and have a student film the action in high speed (I recently did this). When the video is slowed way down, you can clearly see the spring alternate between its "flat" state and its "maximum antinode" state as the pulses and their reflections are interfering with one another.