Ch. 14  Sound


14.1 Producing a Sound Wave


Sound is produced by the vibration of something, and carried by a medium (solid, liquid, or gaseous).




14.2 Characteristics of Sound Waves


Sound is a longitudinal wave. The elements of the wave travel back & forth along the direction that the wave travels (as opposed to a transverse wave).


Characteristics based on Human Perception:


Infrasonic :      Frequency < 20 Hz

Audible:           Frequency 20 Hz  20 kHz

Ultrasonic:       Frequency > 20 kHz


Applications: ultrasonic imaging: fraction reflected is related to density  patient given an anti-reflection coating! (read the details of this section)


14.3 Speed of Sound


Liquids:   where  is the mass density and B is bulk modulus: . The bulk modulus is the response of a volume of the liquid to a change in pressure, and acts sort of like a 3D “spring constant”.


Solid Rods:  where Y is Young’s modulus of the solids  “stress over strain” (see Ch. 9).




Because atoms in solids are in closer contact than in liquids, and those in liquids are in closer contact than those in gases, it is generally true that  




Example: Problem #1

Suppose that you hear a clap of thunder 16.2 s after seeing the associated lightning stroke. The speed of sound waves in air is 343 m/s and the speed of light in air is 3.00 x 108 m/s. How far are you from the lightning stroke?



14.4 Energy and Intensity of Sound Waves


Intensity I: Energy flowing per unit time through a unit area A oriented perpendicular to the wave:


Human Ear Sensitivity (average person):






Decibels: Intensity level or decibel level:


Threshold of Hearing = 0 dB

Threshold of Pain = 120 dB


Relative Intensities: an increase of 10 dB in intensity corresponds to a factor of 10 in intensity. A 60 dB sound is 100 times as intense as a 40 dB sound.


Example: Problem #10:

The area of a typical eardrum is about 5.0 x 10 -5 m2. Calculate the sound power (the energy per second) incident on an eardrum at (a) the threshold of hearing and (b) the threshold of pain.


Example: Problem #12:

The intensity level of an orchestra is 85 dB. A single violin achieves a level of 70 dB. How does the intensity of the sound of the full orchestra compare with that of the violin’s sound?


14.5 Spherical and Plane Waves


Sound emitted by a “small” spherical source travels equally in all directions.



Because the area A that the sound crosses increases with the square of the distance r from the source:


For a given wave front, the distance r it has traveled in time  is







Very far from the source, the curvature of the spherical wave fronts is so small that we will refer to them as plane waves.

Example: Problem #16:

An outside loudspeaker (considered a small source) emits sound waves with a power output of 100 W. (a) Find the intensity 10.0 m from the source. (b) Find the intensity level, in decibels, at this distance. (c) At what distance would you experience the sound at the threshold of pain, 120 dB



14.6 The Doppler Effect


In general, a Doppler effect is experienced whenever a source of sound and an observer are moving toward or away from one another:


No Motion: Observer hears the “stationary” frequency  

Toward: Observer hears higher frequency

Away: Observer hears lower frequency

Case A  Stationary source , observer in motion with speed  toward the source

In time  the observer covers a distance . Due to motion, crosses a number of additional wave fronts than if stationary. That “extra” number will be , and so the “extra” number crossed peer second will be . So the observed frequency will be  




Because  we find that: .


SIGN CONVENTION: take  to be positive when observer approaches source, negative when moving away.


RULE OF THUMB: When observer is in motion toward source, the observer hears higher frequency. When their motion is away from the source, they hear a lower frequency.


Case B  Stationary observer  and source moving toward observer with speed  


Source emits a wave front, then in time , the period of the wave front production, travels toward it a distance . The observer will detect a different wavelength shortened by this amount: , So the frequency heard is  

RULE OF THUMB: When source is in motion toward observer, the observer hears higher frequency. When their motion is away from the source, they hear a lower frequency.

Case C
 Both source and observer in motion:     


Generalized RULE OF THUMB: When there is a net motion of source and observer toward one another, the observer hears a higher frequency. When there is a net motion of source and observer away from one another, the observer hears a lower frequency.




Example 14.6 The Noisy Siren

An ambulance travels down a highway at a speed of 75.0 mi/hr, its siren emitting a sound at a frequency of 400 Hz. What frequency is heard by a passenger in a car traveling at 55.0 mi/hr in the opposite direction as the car (a) approaches? (b) moves away from the ambulance?





Let  and convert to SI units , so that  and .










Sound waves propagate radially with speed  cover a distance  in time . In same time, source travels distance . From geometry , where the .


When  the wave fronts pile up faster than the air can respond due to its speed of sound, and a shock front is produced. At supersonic speeds, aircraft will produce sonic booms.


Example: Problem #28

A supersonic jet traveling at Mach 3 at an altitude of 20 000 m is directly overhead at time t = 0, as in Figure P14.28. (a) How long will it be before the ground observer encounters the shock wave? (b) Where will the plane be when it is finally heard? (Assume an average value of 330 m/s for the speed of sound in air.)






14.7 Interference of Sound Waves



If the sound waves from a source can be split and made to take 2 different paths to the observer, the crests & troughs from the 2 parts might not arrive “in phase”. If they do not, the crests from one part will “fill in” the troughs of another part.


Constructive interference: path difference is zero or an integer multiple of the wavelength:


Destructive interference:



The same problem occurs if two speakers are driven by the same signal  the reason that speaker wires are color-coded.


Interference occurs with all wave phenomena, water waves & electromagnetic waves.


Example: Problem #31


The ship in this figure travels along a straight line parallel to the shore and 600 m from the shore. The ship’s radio receives simultaneous signals of the same frequency from antennas at points A and B. The signals interfere constructively at point C, which is equidistant from A and B. The signal goes through the first minimum at point D. Determine the wavelength of the radio waves.











14.8 Standing Waves


A string that is forced to vibrate will respond in a manner that depends on how it is attached at the “other” end. A fixed attachment will cause a reflected wave to be propagated back with an inverted amplitude.


If the reflected wave crosses another incident wave, their amplitudes will partly cancel.

 In the case where the reflection is “perfect” and the amplitude of the reflected pulse equals that of the incident pulse, ea point of perfect cancellation can occur


A string fixed at both end will have nodes where there is no movement. Under the right conditions, nodes will occur between the ends as well. The locations of greatest amplitude are antinodes.



Distance between nodes:  



These “standing” waves occur ant “natural frequencies” determined by the nature of the string, its tension, and its length.


String of length L fixed at both ends:

(b)  plucking string in middle produces standing wave with


This is called the “fundamental frequency” of the string and is

   first harmonic

(c)  next natural frequency:       second harmonic


(d)  next:     third harmonic




Stringed Instruments: plucking, striking, bowing sets strings into various harmonics


Note that the frequencies can be altered by changing L, F or μ.

Can you think of examples?




Example: Problem # 38

A cello A string vibrates in its fundamental mode with a frequency of 220 vibrations/s. The vibrating segment is 70.0 cm long and has a mass of 1.20 g. (a) Find the tension in the string. (b) Determine the frequency of the string when it vibrates in three segments.


14.9 Forced Vibrations and Resonance (read this section)


Applying a force in a periodic manner  forced vibrations


If the forcing period matches a resonant frequency of the system very large amplitudes result.


Example: a swing  it has a natural frequency  that of the equivalent pendulum. Pushing someone on a swing with a frequency of the “pendulum” will produce large “amplitude” swings!


Tacoma Narrows Bridge Failure  Nov. 7, 1940


Often discussed as an example of forced resonance, there may be more to it than this. Visit this site.


This is the famous little movie of the bridge.



14.10 Standing Waves in Air Columns


Organ pipes, flutes, and the trusty old recorder (that many young students learn to “play”) are examples of standing waves inside a column of air.


Closed ends are Nodes

Open ends are Antinodes

Open at both ends: all harmonics present:  

Closed at one end: only odd harmonics!  


Read the Applying Physics examples on page 447!! Wonderful practical examples of the physics of everyday things!!


Example: Problem #43

The windpipe of a typical whooping crane is about 5.0 feet long. What is the lowest resonant frequency of this pipe assuming it is a pipe closed at one end? Assume a temperature of 37°C.

Example: Problem # 44

The overall length of a piccolo is 32.0 cm. The resonating air column vibrates as in a pipe open at both ends. (a) Find the frequency of the lowest note a piccolo can play, assuming the speed of sound in air is 340 m/s. (b) Opening holes in the side effectively shortens the length of the resonant column. If the highest note a piccolo can sound is 4 000 Hz, find the distance between adjacent antinodes for this mode of vibration.


14.11 Beats


Beats occur when sound waves of slightly different frequencies interfere:




Example: Problem #50

The G string on a violin has a fundamental frequency of 196 Hz. It is 30.0 cm long and has a mass of 0.500 g. While this string is sounding, a nearby violinist effectively shortens (by sliding her finger down the string) the G string on her identical violin until a beat frequency of 2.00 Hz is heard between the two strings. When this occurs, what is the effective length of her string?



14.12 Quality of Sound (read)



14.13 The ear (read)