Physics of Sound & Music:

Day 06

The Harmonic Series

Homework!Try them all!

Bending Sound with Sound

Array of speakers

Finely tuned phases

Acoustic “bottle” can• Guide sound• Trap particles• Cloak objects• Levitate small objects

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Bending Sound with Sound

Array of speakers

Finely tuned phases

Acoustic “bottle” can• Guide sound• Trap particles• Cloak objects• Levitate small objects

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WarmUp Response: Nodes

From the discussion of standingwaves, what is a "node"? Are nodes something that only standing waves can have, or can a regular traveling wave have nodes too?~67% → Nodes: locations of minimal (zero) motion

~25% → Nodes: locations of minimal (zero) amplitude

~17% → Regular traveling waves can have nodes

~75% → Only standing waves can have nodes

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WarmUp Response: Nodes

“A node is a location along a standing wave at which there is no motion of the medium. Only standing waves can have nodes because a node is the lack of motion in said wave. All other waves are constantly moving, so a node cannot exist.”

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Standing Waves

• A special case of interference occurs when identical waves are moving in opposite directions.

• As the waves move, they go in and out of phase with each other from moment to moment, alternately creating constructive and destructive interference.

• The resulting movement iscalled a standing wave because it doesn't travel.

• Notation:– Nodes: No movement– Antinodes: Max movement– Loops: Section between nodes

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End Reflections

• Critical to the formation of standing waves and music is the fact that when a wave reaches the end of a string (or the end of a tube) it is reflected.

• Depending on whether the end of the string is fixed or free, two different things happen.

• If the end is free, the wave is reflected "as is".• If the end is fixed, the wave is inverted as it is

reflected.• Thus we can create standing waves with only one

source of waves, because the other end will reflect!

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WarmUp Response: Harmonic

You read about the harmonic series. What makes two frequencies "harmonic" in this sense? Meaning, what is the special relationship between the frequencies of the waves in the harmonic series?~17% → The frequencies are related by a whole-

number ratio.

~25% → One frequency is a multiple of the other

~33% → Both relate to the same fundamental

~25% → Fuzzy or incorrect

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WarmUp Response: Harmonic

“Two frequencies are harmonic when the the nodes of the standing waves match up. the relationship between harmonic frequencies is that the frequency is doubled while the wavelength is halved.”

“To make two frequencies harmonic they would have to have frequencies that are multiples of each other (the lower frequency being the fundamental frequency),”

“harmonic frequencies have a specific ratio between them, such as 1:2, 2:3, 3:4, 4:5.”

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Frequencies on a Fixed String

• If we consider what frequencies/wavelengths will create standing waves on a string we have a few rules:–Both ends of the string must be nodes (fixed!)–All loops must be the same size (and amplitude).

• Following these rules, we make an interesting discovery:–All the frequencies that create standing waves are

multiples of the lowest one!

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Say you had a guitar string that was 1 meter long. What is the wavelength of the standing wave that has five loops in it?

A) 0.1 meters

B) 0.2 meters

C) 0.4 meters

D) 0.5 meters

E) 0.8 meters

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Harmonics of a Fixed String

• Calculations like what you just did lead to a formula for the frequency of standing waves on a fixed string:

where v is the speed of sound on that string.• The lowest frequency that forms a standing wave is

called the fundamental or 1st harmonic.• If the higher frequencies that form standing waves

are all multiples of the fundamental, they are given harmonic numbers as well.

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2

vf N

L

The Harmonic Series!

• Confusingly theterminology of overtones is onenumber off fromthe terminology of harmonics.

• Often, "overtone" is used to mean any otherfrequency components of a sound (above the fundamental), even if it isn't harmonic (e.g., gongs)

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Difficulties/Interests

“The thing that i found most confusing was how you can have harmonics that are not octaves. I could see how having a wavelength half the size of the original creates a harmonic, but how do you get a wave that produces harmonics like 3rds and 7ths.”

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The Sound of Harmonics

• What does this sequence of frequencies sound like?• This sequence of frequencies happens to be very

pleasing to our ears, creating the foundation of most musical scales, the world around.

• However, given thatmost modern americanmusic is based on theequal-tempered scale,not all of these notesfit well on the musicalstaff.

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Difficulties/Interests

“Why do the frequencies in the overtone series sounds pleasing to the human ears compared to those that are out of "tune"?”

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Pythagoras Again

• Among Pythagoras' discoveries about music was the idea that string length helps determine pitch.

• He used a monochord to explore the sounds of a stretched string.

• From this he found that strings with lengths of 6, 8, 9, 12 yielded three of the intervals that greeks considered “consonant”.

• This is perhaps the reason that the early Greek lyre had 4 strings.

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Reading:

Tuesday (9/9) → 3.3 – 3.5

Thursday (9/11) → Mini-Exam on Ch. 1-3

Notes:

Homework #3 due Friday:Ch. 2: Q: 7, 9, 15, 17 + Ch. 3: P: 1 + Theory

WarmUp due Monday by 10 PM

Homework #4 (due next Thursday at the Quiz):Ch. 3: Q: 2, 4, 6, 7, 8, 9, 11, 12, 14   P: 3, 4

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