1. Standing Waves and the Overtone Series
Transverse-Stringed Instruments
and
Longitudinal-Wind Instruments

2. Transverse Standing Waves

3. Transverse Standing Waves
A standing wave is an interference effect that can occur when two waves overlap.

4. 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.

5. 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.

6. Simulation of Standing waves

7. Standing wave patterns

8. The Speed of a Wave on a String

9. The Speed of a Wave on a String
F = Tension in the string.
W = linear mass or mass per length = m/L.

10. 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?

11. 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.

12. 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).

13. 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).

14. 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.)

15. 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.

16. Human Ear and Sensitivity
Audible frequency range: 20 Hz – 20,000 Hz
Audible intensity range: 10–12 W/m w/m2
10–12 W/m2 = Threshold of hearing
10 W/m2 = Threshold of pain

17. The Sensitivity of the Human Ear

18. 16.8 Decibels
The decibel (dB) is a measurement unit used when comparing two sound intensities.
The intensity level b  (expressed in decibels) relative to the threshold of hearing, Io is defined as follows:

19. Normal conversation (1 meter) 3.2 × 10-6 65
TABLE 16.2     Typical Sound Intensities and Intensity Levels Relative to the Threshold of Hearing
Intensity I (W/m2)
Intensity Level b (dB)
Threshold of hearing
1.0 × 10-12
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
130