Chapter 15: Waves and Sound
Example: pulse on a string
speed of pulse = wave speed = v
depends upon tension T and inertia (mass per length )
v
T


T
mL
motion of “pulse”
actual motion of string
Phys 250 Ch15 p1
Reflections at a boundary
fixed end = “hard” boundary
Pulse is inverted
Reflections at a boundary
free end = “soft” boundary
Pulse is not inverted
Phys 250 Ch15 p2
Reflections at an interface
light string to heavy string = “hard” boundary
faster medium to slower medium
heavy string to light string = “soft” boundary
slower medium to faster medium
Principle of Superposition: When Waves Collide!
When pulses pass the same point,
add the two displacements
Phys 250 Ch15 p3
Periodic Waves
a.k.a. Harmonic Waves, Sine Waves ...
Important characteristics of periodic waves
wave speed v: the speed of the wave, which depends upon the medium only.
wavelength : : (greek lambda) the distance over which the wave repeats, it is also the
distance between crests or troughs.
frequency f : the number of waves which pass a given point per second. The period of the
wave is related to the frequency by T = 1/f.
Wavelength, speed and frequency are related by:
v=f
Amplitude A: the maximum displacement from equilibrium. The amplitude does not affect
the wave speed, frequency, wavelength, etc.
A
 Elastic Potential Energy and Kinetic Energy associated with wave depend upon
Amplitude.
Energy per time (power) carried by a wave is proportional to the square of the
amplitude.
“Loudness” depends upon amplitude.
Phys 250 Ch15 p4
Types of waves
Transverse Waves: “disturbance” is perpendicular to wave velocity, such as for
waves on a string. (disturbance is a shear stress, only occurs in solids!)
Longitudinal Waves: “disturbance” is parallel to wave velocity, such as the
compression waves on the slinky.
Water surface waves: mixture of longitudinal and transverse
Phys 250 Ch15 p5
Standing Waves
vibrations in fixed patterns
effectively produced by the superposition of two traveling waves
y = y0sin(x/t/T)
constructive interference: waves add
destructive interference: waves cancel
 = 2L
 = 2L
node
antinode
Phys 250 Ch15 p6
antinode
 = 2L
 = 2L
characteri stic wavelengt hs
2L
n  1,2,3
n
longest wa velength  lowest frequency
 = 2L

= fundamenta l frequency
f1 
v


v
2L
f1 
1
T
2L m L
 = 2L
 = 2L
harmonics
fn 
v
n
n
v
 nf1
2L
 = 2L
Resonance
When a system is subjected to a periodic force with a frequency equal to one of its
natural frequencies, energy is rapidly transferred to the system.
examples:
musical instruments with fundamental or overtones
mechanical vibrations
Phys 250 Ch15 p7
Example: What is the wave speed of a guitar string whose fundamental frequency is 330
Hz if the length of string free to vibrate is 0.651m? What is the tension in the string of the
string’s linear mass density is 0.441 g/m3?
Phys 250 Ch15 p8
Sound Waves:
pressure waves
Intensity decreases with square of distance (for spherical waves)
speed of sound in air v(T) = (331.5+.6*T) T in celsius
Human hearing: 20Hz to 20,000 Hz
subsonic: frequencies below 20 Hz
ultra sonic: frequencies above 20,000 Hz
Intensity  Loudness
frequency  pitch
Example: A loudspeaker on a tall pole standing in a field generates a high frequency sound at
an intensity of 1E-5 W/m2 for someone directly below the speaker whose ears are 8 m from the
speaker. How loud is the sound when the person is 24 m from the speaker?
Phys 250 Ch15 p9
Sound Levels
Human hearing can detect a wide range of intensities
Standard Scale: decibels (dB), a logarithmic scale
LI (in dB)  10 log 10
I
I0
I0 = 1E-12 W/m2 , “Barely audible”
human hearing is not uniform, most sensitive from 2000-5000 Hz
20 dB increase in L means 100x in intensity, 3dB change means 2x
0 dB is barely audible, not 0 Intensity!
Pain at about 120 dB
Phys 250 Ch15 p10
Doppler Effect
Motion of source or detector can affect measured frequency
Stationary Source, Observer
Moving Source, Stationary Observer
f '
f
(1  vs v)
moving source
Stationary Source, Moving Observer
f '  f  (1  vO v)
Phys 250 Ch15 p11
moving observer
Example: A police car emits a 250 Hz tone when sitting still. What frequency does a
stationary observer hear if the car sounds it horn while approaching at a speed of 27 m/s?
What frequency is hear if the horn is sounded as the car is leaving at 27 m/s?
What sound is heard if the police car is stationary, but the observer is approaching/receding
at 27 m/s?
Phys 250 Ch15 p12
Beat Frequency
effect of superimposing two “close” frequencies
f beat | f1 - f 2 |
Phys 250 Ch15 p13
Standing Waves II pipe open at one end
node
 = 4L
5 = 4L
 = 4L
7 = 4L
node
antinode
antinode
4L
; n  1, 3, 5
n
v
v
1
T
f1  
f1 
 4L
4L m L

harmonics (only odd harmonics)
v
v
fn 
n
 nf1 n  1, 3, 5
n
4L
Phys 250 Ch15 p14