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 1^st
2. NANA’s frequency is 3x that of NAN’s, hence called the 3^rd 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 1^st
2. NANAN’s frequency is 2x that of NAN’s, hence called the 2^nd 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).

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(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).

    

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.