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How do vibrating strings and vibrating columns of air create music?

Theory: Standing waves & harmonics

Review: Superposition = what happens when one wave is superimposed  on another wave.  The creates constructive or destructive interference. This can lead to standing waves. Standing waves have nodes and anti-nodes. That’s how a guitar or violin creates music.

Pluck a string. It vibrates at many, many frequencies, all at the same time.

Most of these vibrational modes cancel each other out.

Only the modes which lead to standing waves survive for long.

Here are the first three harmonics of a string, of length L.

Note the relationship between the string’s wavelength (λ, lambda) and its length.

plucked-string-harmonics-and-overtones-giancoli

Enter a caption

“Physics”, Douglas Giancoli, 6th ed., Prentice Hall, 2005

Let’s now see these three in motion.

Note how we interpret these graphs:

(A) If these are strings, they can represent the actual motion of the string

(B) But the same diagrams could represent air molecules vibrating in the pipes of a musical instrument. In that case, these would not show air molecules moving up and down.  Rather, they would show how much each air molecule is being displaced.

fundamental-tone-or-1st-harmonic

1st harmonic (fundamental tone)

1st-overtone-or-2nd-harmonic

1st overtone,
2nd harmonic

.

2nd-overtone-or-3rd-harmonic

2nd overtone,
3rd harmonic

3rd-overtone-or-4th-harmonic

3rd overtone,

4th harmonic

GIFs of harmonics are from Harmonic (Wikipedia)

Lab: Rope and strings harmonics

Get a 10 foot length of rope (Polypropylene Rope)

Affix one end tightly to the bottom of a chair in the “living room.”

Pull the string taut. Since the rope is lying on the ground the entire time, it is basically a 2D object creating a 2D wave.

Create a wave pulse. See if you can get it to reflect.

Change the speed of your pulses. Create a chain of waves moving towards the chair.
Do you observe any reflection?

Slowly change the speed and amplitude of your waves.

Goal: Create a standing wave showing 2nd, 3rd and 4th harmonics.

While viewing the waves, observe the nodes and anti-nodes.
.

Lab: Music with a smartphone app

Use the Android version of Physics Toolbox Sensor Suite  Measuring data with smartphone apps

Lab procedures: Rope and strings harmonics

Get a 10 foot length of rope (Polypropylene Rope)

Affix one end tightly to the bottom of a chair in the “living room.”

Pull the string taut. Since the rope is lying on the ground the entire time, it is a 2D object creating a 2D wave.

Create a wave pulse. See if you can get it to reflect.

Change the speed of your pulses. Create a chain of waves moving towards the chair.
Do you observe any reflection?

Slowly change the speed and amplitude of your waves.

Goal: Create a standing wave showing 2nd, 3rd and 4th harmonics.

Lab procedures: Music with a smartphone app

App: Physics Toolbox Sensor Suite on our phones:

Sound Meter – measures sound intensity.
Tone Detector – measures the frequency of a musical tone.
Oscilloscope – shows us the wave shape and relative amplitude of a musical tone.

  1. Bring a tuning fork near your phone. Read the frequency off of the metal stamp. Analyze the sound intensity, tone, and wave shape, using the app functions noted above. Record your results.
  2. Bring singing wine glasses near your phone. Analyze the sound intensity, tone, and wave shape, using the app functions noted above. Play 2 different notes. Record your results.

Writing the lab report

Lab report

Title, Introduction, Materials, Procedure

Diagrams (show how the rope is connected to a chair –draw standing waves – 1st, 2nd and 3rd harmonics)

Data table

Object Frequency (Hz) Speed of sound (m/s) Wavelength
Tuning fork A
Tuning fork B
Singing glass A
Singing glass B

Data analysis (calculations) section: Use data from the musical notes you analyzed. Assuming that the speed of sound in air is 343 meters/second, calculate the wavelength of two different musical notes. Start with the correct equation, solve for the variable that you need, and plug in numbers only in the last step.

Systematic error: make sure to briefly include this section.

Conclusion: Your observations help you relate to what we learn about waves. You show this by answering these questions:

(1) How many wavelengths is in the 1st harmonic? In the 2nd, 3rd and 4th?

(2) What is a general formula that relates wavelengths to harmonics?

(3) In complete sentences, clearly write what you learned about the tuning fork’s sound intensity, tone, and wave shape.

(4) In complete sentences, clearly write what you learned about the instrument’s sound intensity, tone, and wave shape.

Physics of wind instruments

(11th grade)

Tube Open Both ends Music harmonics and fundamentals

from “Physics”, Douglas Giancoli, 6th ed., Prentice Hall, 2005

Tube closed at one end Music harmonics and fundamentals

from “Physics”, Douglas Giancoli, 6th ed., Prentice Hall, 2005

From Dan Russell

While teaching undergraduate physics at Kettering University for 16 years, I was often frustrated with the depiction of standing sound waves in pipes as it was presented in most elementary physics textbooks… The solution: an animation to visualize particle motion and pressure for longitudinal sound waves….I created the animation below and its accompanying description in an attempt to better explain the behavior of a standing sound wave in a pipe.  Standing Sound Waves (Longitudinal Standing Waves)

The Quality of Sound, and Noise

adapted from Giancoli Physics

When we hear a sound we are aware of its loudness, pitch – and a third aspect called “quality.” For example, when a piano and then a flute play a note of the same loudness and pitch (say, middle C), there is a clear difference in how it sounds – we would never mistake a piano for a flute. This is what is meant by the quality of a sound. For musical instruments, we call this timbre.

The quality (or timbre) of a sound depends on the presence of overtones – their number and their relative amplitudes

Generally, when a note is played on a musical instrument, the fundamental as well as overtones are present simultaneously.

We see here superposition – three wave forms add to create a unique sound.

fundamental and overtone sound

Giancoli Physics, Chapter 12, Pearson Hall

The fundamental and first 2 overtones add together at each point to give a composite waveform. By “waveform” we mean the shape of the wave in space at a given moment.

Quality of Sound overtones

Giancoli Physics, Pearson Education

Interference of sound waves: Beats

text

below, left two waves with slightly different frequencies are travelling to the right. Since the two waves are travelling in the same medium, they travel with the same speed.

The resulting superposition sum wave travels in the same direction and with the same speed as the two component waves. Its local amplitude depends on whether the two individual waves have the same or opposite phase.

The “beat” wave oscillates with the average frequency, and its amplitude envelope varies according to the difference frequency.
– http://www.acs.psu.edu/drussell/Demos/superposition/superposition.html

Anomalous sounds (sound mirages)

Our first article.

How Weather Affects an Outdoor Noise Study by ABD Engineering and Design

This following discussion has helpful images.

Why can a distant sound be heard easier when it travels with the wind? Why does it become weaker if it travels against the wind?

A discussion to consider, from Physics forums, includes this phenomenon:
“Yes. I have a freeway about 10 blocks South of my house. I can hear the traffic very clearly with no wind, or a South wind. If there is even a slight North wind, the traffic noise becomes almost inaudible. If there is a brisk North wind (over 15 MPH), the sound is completely gone.”

https://www.physicsforums.com/threads/does-wind-affect-how-far-sound-can-travel.149392/

Also this “…if the air close to the ground is colder than the air above it then sound waves traveling upwards will be bent downwards. This is called Refraction. These refracted sound waves can act to amplify the sound to someone standing far away.”

http://sciencewows.ie/blog/does-sound-travel-faster-in-warm-or-cold-air/

Sound seems amplified when traveling over water

https://www.school-for-champions.com/science/sound_amplified_over_water.htm#.WoBbQ5M-fVo

Diffraction of sound waves

https://katrinasiron21.wordpress.com/properties-of-sound-waves/diffraction-of-sound-waves/

Temperature inversion and sound waves

http://kxan.com/blog/2015/02/13/why-does-sound-carry-farther-on-cold-calm-mornings/

Also look into: Humans hearing infra sound waves

“Colorado State Climatologist Nolan Doesken says: “When the ground has a thick layer of fresh, fluffy snow, sound waves are readily absorbed at the surface of the snow. However, the snow surface can become smooth and hard as it ages or if there have been strong winds. Then the snow surface will actually help reflect sound waves. Sounds seem clearer and travel farther under these circumstances.””

Related topic: The Hum is a phenomenon, or collection of phenomena, involving widespread reports of a persistent and invasive low-frequency humming, rumbling, or droning noise not audible to all people. Hums have been widely reported by national media in the UK and the United States. The Hum is sometimes prefixed with the name of a locality where the problem has been particularly publicized: e.g., the “Bristol Hum” or the “Taos Hum”. It is unclear whether it is a single phenomenon; different causes have been attributed. ”

Human reactions to infrasound – https://en.wikipedia.org/wiki/Infrasound#Human_reactions

Skyquakes or mystery booms are unexplained reports of a phenomenon that sounds like a cannon or a sonic boom coming from the sky. They have been heard in several locations around the world. – https://en.wikipedia.org/wiki/Skyquake

Apps

app: Wave Interference and Beat Frequency

Massachusetts Arts Curriculum Framework 1999 www.doe.mass.edu

Learning Standards

2016 Massachusetts Science and Technology/Engineering Curriculum Framework

HS-PS4-1. Use mathematical representations to support a claim regarding relationships among the frequency, wavelength, and speed of waves traveling within various media. Examples of situations to consider could include electromagnetic radiation traveling in a vacuum and glass, sound waves traveling through air and water,

SAT subject test in Physics: Waves and optics

General wave properties, such as wave speed, frequency, wavelength, superposition, standing wave diffraction, and Doppler effect.

Massachusetts Arts Curriculum Framework 1999 www.doe.mass.edu

The Arts Disciplines: Music

5.1 Perceive, describe, and respond to basic elements of music, including beat,
tempo, rhythm, meter, pitch, melody, texture, dynamics, harmony, and form.

5.9 Demonstrate knowledge of the basic principles of meter, rhythm, tonality,
intervals, chords, and harmonic progressions in an analysis of music.

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