Standing Waves and the Overtone Series

Standing Waves:

Transverse-Stringed Instruments

and

Longitudinal-Wind Instruments

Transverse Standing Waves

Transverse Standing Waves

A standing wave is an interference effect that can occur when two waves overlap.

Transverse Standing Waves

A standing wave is an interference effect that can occur when two waves overlap.

Standing waves can arise with transverse waves, such as those on a guitar string, and also with longitudinal sound waves, such as those in a flute.

Transverse Standing Waves

A standing wave is an interference effect that can occur when two waves overlap.

Standing waves can arise with transverse waves, such as those on a guitar string, and also with longitudinal sound waves, such as those in a flute.

In any case, the principle of linear superposition provides an explanation of the effect, just as it does for diffraction and beats.

Simulation of Standing waves

http://www3.interscience.wiley.com:8100/legacy/college/cutnell/0471151831/concepts/index.htm

Standing wave patterns

The Speed of a Wave on a String

The Speed of a Wave on a String

W

FV

F = Tension in the string.

W = linear mass or mass per length = m/L.

Problem

The G string on a guitar has a fundamental frequency of 196 Hz and a length of 0.62 m. This string is pressed against the proper fret to produce the note C, whose fundamental frequency is 262 Hz. What is the distance L between the fret and the end of the string at the bridge of the guitar?

Longitudinal Standing Waves

Musical instruments in the wind family depend on longitudinal standing waves in producing sound. Since wind instruments (trumpet, flute, clarinet, pipe organ, etc.) are modified tubes or columns of air, it is useful to examine the standing waves that can be set up in such tubes.

Open tube of air

A pictorial representation of longitudinal standing waves on a Slinky (left side) and in a tube of air (right side) that is open at both ends (A, antinode; N, node).

Closed tube of air

A pictorial representation of the longitudinal standing waves on a Slinky (left side) and in a tube of air (right side) that is open only at one end (A, antinode; N, node).

Problem

Sound enters the ear, travels through the auditory canal, and reaches the eardrum. The auditory canal is approximately a tube open at only one end. The other end is closed by the eardrum. A typical length for the auditory canal in an adult is about 2.9 cm. The speed of sound is 343 m/s. What is the fundamental frequency of the canal? (Interestingly, the fundamental frequency is in the frequency range where human hearing is most sensitive.)

Sound Intensity

The sound intensity I is defined as the sound power P that passes perpendicularly through a surface divided by the area A of that surface:

The unit of sound intensity is power per unit area, or W/m2.

Human Ear and Sensitivity

Audible frequency range: 20 Hz – 20,000 Hz

Audible intensity range: 10–12 W/m2 - 10 w/m2

10–12 W/m2 = Threshold of hearing

10 W/m2 = Threshold of pain

The Sensitivity of the Human Ear

16.8 DecibelsThe decibel (dB) is a measurement unit used when comparing two sound intensities.

The intensity level  (expressed in decibels) relative to the threshold of hearing, Io is defined as follows:

TABLE 16.2     Typical Sound Intensities and Intensity Levels Relative to the Threshold of Hearing

  Intensity I (W/m2) Intensity Level (dB)

Threshold of hearing 1.0 × 10-12 0

Rustling leaves 1.0 × 10-11 10

Whisper 1.0 × 10-10 20

Normal conversation (1 meter) 3.2 × 10-6 65

Inside car in city traffic 1.0 × 10-4 80

Car without muffler 1.0 × 10-2 100

Live rock concert 1.0 120

Threshold of pain 10 130