Category: 10 Superposition

159 Fundamental and Harmonics

Can you differentiate between the sounds of sinusoidal, saw-tooth and square waves? Only the sinusoidal wave is monotonic. It truly consists of one single pitch. The saw-tooth and square waves are actually the superposition of many sinusoidal waves of different frequencies. So they contains higher harmonics on top of the fundamental. While they all sound like the same pitch to us, they also sound different because of the different (amplitudes) of the harmonics they contain.

The sound produced by the guitar (string) and the recorder (open pipe) are also not monotonic. It takes a bit of imagination to realize that the standing waves formed on the string and the air in the pipe actually consists of many frequencies at the same time. But that’s what allows the sound produced by different instruments to have different timbres.

140 Standing Wave Odd Harmonics

This video shows the standing waves that can be formed on a string that is fixed at one end or loose at the other end.

Any standing wave that forms on this string must have a node at the fixed end, and antinode at the loose end.

The simplest standing wave fulfills these conditions is the NA. (Node-Antinode) The next standing wave that can be formed is the NANA, followed by

  1. NA,
  2. NANA,
  3. NANANA,
  4. NANANANA,
  5. NANANANANA, and so on.

Notice that each ANA corresponds to 2 quarter-wavelength segment. This means

  1. NA packs 1 quarter-wavelength along the length of the string,
  2. NANA packs 3 quarter-wavelengths along the length of the string,
  3. NANANA packs 5 quarter-wavelengths along the length of the string,
  4. NANANANA packs 7 quarter-wavelengths along the length of the string,
  5. NANANANANA packs 9 quarter-wavelengths along the length of the string, and so on.

Which means that

  1. NA’s wavelength is called the fundamental wavelength,
  2. NANA’s wavelength is 3x as short as that of NAN’s,
  3. NANANA’s wavelength is 5x as short as that of NAN’s,
  4. NANANANA’s wavelength is 7x as short as that of NAN’s,
  5. NANANANANA’s wavelength is 9x as short as that of NAN’s, and so on.

Which means that

  1. NA’s frequency is called the fundamental frequency, or 1st
  2. NANA’s frequency is 3x that of NAN’s, hence called the 3rd harmonic,
  3. NANANA’s frequency is 5x that of NAN’s, hence called the 5th harmonic,
  4. NANANANA’s frequency is 7x that of NAN’s, hence called the 7th harmonic,
  5. NANANANANA’s frequency is 9x that of NAN’s, hence called the 9th harmonic, and so on.

139 Standing Wave Harmonics

This video shows the standing waves that can be formed on a string fixed at both ends.

Since the string is fixed at both ends, any standing wave that forms on the string must have nodes at both ends.

The simplest standing wave that has nodes at both ends is the NAN. (Node-Antinode-Node) The next standing wave that can be formed is the NANAN, followed by

  1. NAN,
  2. NANAN,
  3. NANANAN,
  4. NANANANAN,
  5. NANANANANAN, and so on.

Notice that each NAN corresponds to one half-wavelength segment. This means

  1. NAN packs 1 half-wavelength along the length of the string,
  2. NANAN packs 2 half-wavelengths along the length of the string,
  3. NANANAN packs 3 half-wavelengths along the length of the string,
  4. NANANANAN packs 4 half-wavelengths along the length of the string,
  5. NANANANANAN packs 5 half-wavelengths along the length of the string, and so on.

Which means that

  1. NAN’s wavelength is called the fundamental wavelength,
  2. NANAN’s wavelength is 2x as short as that of NAN’s,
  3. NANANAN’s wavelength is 3x as short as that of NAN’s,
  4. NANANANAN’s wavelength is 4x as short as that of NAN’s,
  5. NANANANANAN’s wavelength is 5x as short as that of NAN’s, and so on.

Which means that

  1. NAN’s frequency is called the fundamental frequency, or 1st
  2. NANAN’s frequency is 2x that of NAN’s, hence called the 2nd harmonic,
  3. NANANAN’s frequency is 3x that of NAN’s, hence called the 3rd harmonic,
  4. NANANANAN’s frequency is 4x that of NAN’s, hence called the 4th harmonic,
  5. NANANANANAN’s frequency is 5x that of NAN’s, hence called the 5th harmonic, and so on.

138 Wave Reflection

A wave always undergoes reflection when it hits the end of the road. If it is a hard reflection (like a fixed end), the reflection come with a 180° phase change (so the pulse returns on the other side of the slinky). If it is a soft reflection (like a loose end), the reflection comes with no phase change (so the pulse returns on the same side of the slinky).

(Beyond H2 syllabus)

Actually, when a wave encounters a discontinuity in the medium (aka medium boundary), only part of it is reflected, and the remaining part is transmitted. The fraction that is reflected depends on the degree of discontinuity. The more abrupt the change in medium, the higher the fraction that is reflected (and lower the fraction that is transmitted).

1 hard    2 soft

Compared to the slinky, the fixed end represents an infinitely heavy slinky, and the loose end represents an infinitely light slinky. For the wave traveling down the slinky, both the fixed end and the loose end represent the most drastic change in medium possible. That’s why 100% of the pulse was reflected and 0% was transmitted. If the medium change is not so abrupt, we will see some of the wave being reflected, and some being transmitted. As illustrated by the below.

3 reflect1     4 reflect2

Standing Wave on Free String — xmdemo 091

The string is fixed at one end but free on the other end. Standing waves formed on such a string must have a node at the fixed end, and an antinode at the free end.

As such, the fundamental (1st harmonic) that can be formed corresponds to one quarter-segment. The next standing wave that can be formed corresponds to three quarter-segments (3rd harmonic), followed by five quarter-segments (5th harmonic) and so on.

Even harmonics are not possible because half-segments will require nodes to be formed at both ends, which is not possible for such a string with one free end. This is very similar to the situation in open pipes, where only odd harmonics can be formed.

Water Standing Wave — xmdemo 075

Water waves travelling towards along the water container are reflected off the walls when they reach either ends. Incident and reflected waves thus superpose and interfere with one another. A standing wave of significant amplitude can be formed only when the water waves have certain “correct” wavelengths.

Since water waves reflect off a wall without incurring any phase change, displacement antinodes are formed at the boundaries. What was shown in the video were the 1st, 2nd and 3rd harmonics.

067 Standing Sound Wave: Ruben’s Tube Inversion

This video shows that as the gas throttle is increased, the positions of tallest and shortest flames are swapped.

What’s happening?

First of all, the height of the flames must depend on the rate of propane flow, which depends on the excess pressure PTP0, which is pressure difference between the tube pressure PT and atmospheric pressure P0. The larger the excess pressure, the faster the rate at which propane is pumped out of the perforations, and thus the taller the flames.

At low throttle, PT is practically equal to P0. The excess pressure at the pressure nodes is a low constant, resulting in short flames. The excess pressure at the pressure antinodes however can reach as high as Ps, where Pis the amplitude of the standing wave, resulting in tall flames.

At high throttle, PT >> P0. The excess pressure at the pressure nodes is a constant value of PTP0, resulting in taller flames than before. At the pressure antinodes, the excess pressures reach as high as PTP0 + Ps and as low as PTP0Ps. Even though the average excess pressure is also PTP0 (same as at the pressure nodes), the average flow rate is actually lower thanks to the non-linear relationship between flow rate and excess pressure. Hence shorter flames are found at pressure antinodes.

For some helpful animated diagrams, watch this video.

066 Standing Sound Wave: Ruben’s Tube

Sound waves reflect repeated off the two ends of the tube and interfere with one another. At certain frequencies, standing sound waves are formed, resulting in alternating pressure nodes and antinodes along the tube. This must be the basis for the formation of alternating tall and short flames along the tube.

To figure out the wavelength of the sound wave, we don’t really have to know whether the positions of tallest (or shortest) flame correspond to pressure nodes, or antinodes. We just need to know that the distance between two tallest (or shortest) flames must correspond to half a wavelength of the sound wave.

If you’re interested in knowing which is which, you should watch the next demo.