# Chapter 12 Parts of waves (review) Octaves Stringed Harmonics

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Chapter 12 Parts of waves (review) Octaves Stringed Harmonics How to produce sound Open/Open Harmonics Open/Closed Harmonics Harmonics/Fundamentals

Single wave pulse travels
Transverse waves Single wave pulse travels Transverse waves: Transverse waves: The particles of the disturbed medium move perpendicular to the wave motion particle wave

Longitudinal waves Longitudinal waves: Longitudinal waves:
compression travels Longitudinal waves: The particles of the disturbed medium move parallel to the wave motion Longitudinal waves:

Basic Variables of Wave Motion
Terminology to describe waves Crest: “Highest point” of a wave Wavelength l: Distance from one crest to the next crest. Wavelength l: Distance between two identical points on a wave. Period T: Time between the arrival of two adjacent waves. Frequency f: 1/T, number of crest that pass a given point per unit time

Sound - is a wave (sound wave) - Rarefied and compressed regions Longitudinal wave air molecules move back and forth

Sound Waves Sound waves are longitudinal waves.
They consist of compressed and rarified regions of gas (medium) We can hear (audible) frequencies from about 20 Hz (low) to 20,000 Hz (high). Infrasonic “sound” waves: below ~ 20 Hz Ultrasonic sound waves: above ~ 20,000 Hz The speed of sound in air: c ~ 343 m/s ~ 740 mi/hr ~ 0.2 mi/sec. (dry air, 68F)

Question #1 It is a dark and stormy night.
Lightning strikes in distance. You see the lighting, then, after ten seconds you hear the thunder. How far away did the lighting strike? 1 mile 2 miles 3 miles 4 miles 5 miles

Notes and their fundamental frequency
Octaves: frequency doubles for each tone **Note** An octave will double in frequency each time and this is different than Harmonics

For example: Find middle C (C4) on the chart to the right.
Middle C has a frequency of Hz, one octave higher is C5 which has a frequency of Hz, two octaves higher (C6) has a frequency of Hz. Middle C can also be the fundamental frequency ( Hz) and this is the 1st Harmonic, the 2nd Harmonic (f = Hz) is the same as one octave higher, but the 3rd Harmonic (f = Hz) is NOT the same as two octaves higher.

Creating standing waves:
When two waves are traveling back and forth, under the right conditions (right frequency), we can create standing waves. Standing waves have stationary nodes and antinodes Examples we’ll talk about this chapter: Standing waves on a string. Standing waves in a pipe (open and closed).

String Harmonics frequency L 2f1 3f1 4f1 5f1 6f1
L … Length of string; n … harmonic v … velocity of sound

Standing waves have stationary nodes and anti-nodes
L … Length of string n … harmonic

Fundamental Frequency
String vibrates as a single arc, up and down velocity antinode occurs at center of string This is the fundamental frequency mode Pitch (frequency of vibration) is inversely proportional to string length

Harmonics (Overtones)
In addition, string can vibrate as two half-strings three third-strings etc. These are higher-order frequency modes These modes have higher pitches – overtones

Harmonics in a String In a string, the harmonic pitches are
two times the fundamental frequency (octave) three times the fundamental frequency etc. These integer multiples are called overtones A string harmonics will have all harmonics. i.e. 2nd, 3rd, 4th, 5th, etc.

Producing Sound Thin objects don’t project sound well
Air flows around objects Compression and rarefaction is minimal Surfaces project sound much better Air can’t flow around surfaces easily Compression and rarefaction is substantial Many instruments use surfaces for sound

Open pipe:

Fund. Frequency v… speed of sound v = 343 m/s in air at 20 °C

Open/open tube

A wave reflecting off of the boundary
At an open boundary: the air bounces moving in and out of the boundary. Demo boundary conditions with string and cable motions

Harmonic Vibrations for open-open
In addition, column of air can vibrate as two half-columns three third-columns These higher-order modes are the harmonics Pitches are integer multiples of the fundamental Open/open tube has all integer multiples f,2f,3f,4f,5f OR 1st, 2nd, 3rd, 4th, 5th, etc. (just like a stringed instrument) A open-open tube will have an anti-node at each open end

Fundamental frequency:
open-closed pipe:

Open/closed tube

Wave reflecting off of the closed boundary
At a closed boundary: the wave reflects if it has a high pressure at the wall. The air compresses at the wall and then bounces back.

Harmonic Vibrations for closed-open
These higher-order modes are the harmonics Pitches are odd multiples of the fundamental Closed/open tube only has odd harmonics (e.g., clarinet) f, 3f, 5f, 7f OR 1st , 3rd, 5th, 7th, etc. A closed-open tube will have a node at the closed end

Question #2 You play an open organ pipe with a length of 1m. What is the fundamental frequency? 1 Hz 86 Hz 172 Hz 343 Hz 686 Hz Now you close the pipe at one end. What will the frequency be then?

Standing waves or modes in a column of air
Question #3 Standing waves or modes in a column of air The motions shown are air speeds One of these is a pipe that is closed on one end and the other is open on both ends. Which one is which?

Which one has a lower fundamental frequency? Open/open or open/closed?
Question #4 Which one has a lower fundamental frequency? Open/open or open/closed? closing end of flute! Open/open open/closed

Fundamental Frequency review
Air column vibrates as a single object Pressure antinode occurs at center of open column Velocity antinode occurs at ends of open column Pitch (frequency of vibration) is inversely proportional to column length inversely proportional to air density A closed pipe vibrates as half an open column pressure antinode occurs at sealed end Velocity node occurs at the sealed end frequency is half that of an open pipe