•WAVE MOTIONS
AND SOUND
Vibrations are
common in many
elastic materials, and
you can see and hear
the results of many
in your
surroundings. Other
vibrations in your
surroundings, such
as those involved in
heat, electricity, and
light, are invisible to
the senses.
• Forces and Elastic Materials.
• Forces and Vibrations.
– A vibration is a repeating motion that moves back and
forth.
A mass on a frictionless surface is at rest at an
equilibrium position (A) when undisturbed. When the
spring is stretched (B) or compressed (D), then
released (C), the mass vibrates back and forth because
restoring forces pull opposite to and proportional to
the displacement.
• Describing Vibrations.
– A vibrating mass is described by measuring several
variables.
• The extent of displacement from the equilibrium
position.
• A cycle is the movement from some point, to another
point and back again.
• A period (T) is the time required for one complete
cycle.
• Frequency (f) is the number of cycles per second.
– Frequency is measured in Hertz (Hz)
– The period is the time for one cycle and the
frequency is the cycles per second, the relationship
is: T=1/f
» F = 1/T
A vibrating mass
attached to a spring is
displaced from rest, or
equilibrium, position,
and then released. The
maximum displacement
is called the amplitude
of the vibration. A cycle is one complete vibration.
The period is the time required for one complete
cycle. The frequency is a count of how many
cycles it completes in 1s.
A graph of simple harmonic motion is described by a
sinusoidal curve.
• Waves.
• Kinds of Waves.
– Longitudinal Wave
• A wave that travels in a back and forth movement
– Transverse Wave
• A wave that disturbs particles in a perpendicular motion in the
direction of the wave.
(A) Longitudinal waves are created in a spring when
the free end is moved back and forth parallel to the
spring. (B) Transverse waves are created in a spring
when the free end is moved up and down.
• Waves in Air.
– Condensation
• When a longitudinal wave forces particles closer together.
• This results in a pulse of increased density and pressure.
– Rarefaction
• A zone of reduced density and pressure.
– Sound
• A vibrating object produces condensation and rarefactions that
expand from the source.
• The vibrations can be interpreted as sound by the human ear if
the frequency of the waves is between 20 and 20,000 Hz.
When you open one door into this room, the other
door closes. Why does this happen? The answer is that
the first door creates a pulse of compression that
moves through the air like a sound wave. The pulse of
compression pushes on the second door, closing it.
(A) Swinging the door inward
produces pulses of increased
density and pressure called
condensations. Pulling the door
outward produces pulses of
decreased density and pressure
called rarefactions. (B) In a
condensation, the average
distance between gas
molecules is momentarily
decreased as the pulse passes.
In a rarefaction, the average
distance is momentarily
increased.
A vibrating tuning fork produces a series of
condensations and rarefactions that move away from
the tuning fork. The pulses of increased and decreased
pressure reach your ear, vibrating the eardrum. The ear
sends nerve signals to the brain about the vibrations,
and the brain interprets the signals as sounds.
Compare the (A) back-and-forth vibrations of a tuning
fork with (B) the resulting condensations and
rarefactions that move through the air and (C) the
resulting increases and decreases of air pressure on a
surface that intercepts the condensations and
rarefactions
• Hearing Waves in Air.
– Infrasonic
• Longitudinal waves with frequencies below 20 Hz
– Ultrasonic
• Longitudinal waves with frequencies greater that 20,000 Hz
– Since humans can only hear waves in the 20 – 20,000 Hz
range, they hear neither infrasonic nor ultrasonic waves.
– Waves move the eardrum in and out with the same
frequency as the wave, which the brain interprets as
sound.
• Wave Terms.
• Wave Crest
– The maximum disturbance a wave will create from the resting
position
• Wave trough
– Maximum displacement a wave will create in the opposite
direction from the resting position.
• Amplitude
– The magnitude of the displacement to either the crest or the trough.
• Period
– The time required for a wave to repeat itself
– This is the time that is required to move through one full wave
cycle.
Here are some terms associated with periodic waves. The
wavelength is the distance from a part of one wave to the
same part in the next wave, such as from one crest to the
next. The amplitude is the displacement from the rest
position. The period is the time required for a wave to
repeat itself, that is the time for one complete wavelength
to move past a given location.
• Wavelength
– The distance from one crest of a wave to the crest of the
next wave.
– Given the Greek symbol lambda ()
• Wave Equation
– The wave equation tells us that the relationship between
the velocity of sound waves and the frequency is:
• v=f
• Sound Waves.
• Introduction
– The movement of sound waves requires a medium
through which the waves can travel.
– The nature of the medium determines the velocity of the
sound through the medium
• This is due to the fact that the waves are propagated through
molecular interactions and is determined by:
• Inertia of the molecules
• Strength of the interactions between molecules
(A) Spherical waves move outward from a sounding
source much as a rapidly expanding balloon. This twodimensional sketch shows the repeating condensation as
spherical wave fronts. (B) Some distance from the source,
a spherical wave front is considered a linear, or plane,
wave front.
(A) Since sound travels faster in warmer air, a wave front
becomes bent, or refracted, toward the earth's surface
when the air is cooler near the surface. (B) When the air is
warmer near the surface, a wave front is refracted upward,
away from the surface.
This closed-circuit TV control room is acoustically
treated by covering the walls with sound-absorbing
baffles.
• Velocity in Air.
– As the gas molecules that make up the air increase in temperature,
the velocity of sound waves increases due to increased kinetic
energy (energy of motion).
– This increase is 0.60 m/s for each degree Celsius increase in
temperature.
– At sea level, in dry air the velocity of sound is 331.0 m/s.
– The velocity of sound at different temperatures can be calculated
from the following equation:
0.60m / s
Vtp (m / s)  VO  ( O
)(Tp )
C
• For feet per second simply substitute in 2.0 ft/s for 0.60 m/s.
(A) At room temperature,
sound travels at 343 m/s. In
0.10 s, sound would travel
34 m. Since the sound must
travel to a surface and back
in order for you to hear the
echo, the distance to the
surface is one-half the total
distance. (B) Sonar
measures a depth by
measuring the elapsed time
between an ultrasonic
sound pulse and the echo.
The depth is one-half the
round trip.
• Refraction and Reflection
– Sound waves are reflected or refracted from a boundary,
which means a change in the medium through which they
are being transmitted.
• Interference.
– Constructive interference
• Reflected waves that are in phase with the incoming waves
undergo constructive interference.
– Destructive interference
• Waves that are out of phase undergo destructive interference.
(A) Constructive interference occurs when two equal, inphase waves meet. (B) Destructive interference occurs
when two equal, out-of-phase waves meet. In both cases,
the wave displacements are superimposed when they
meet, but they then pass through one another and return to
their original amplitudes.
Two waves of equal amplitude but slightly different
frequencies interfere destructively and constructively. The
result is an alternation of loudness called a beat.
• Energy and Sound.
• Loudness.
– The energy of a sound wave is called the wave intensity
and is measured in Watts per square meter.
– The intensity of wound is expressed on the decibel scale,
which relates to changes in loudness as perceived by the
human ear.
The intensity, or energy, of a sound wave is the rate of
energy transferred to an area perpendicular to the waves.
Intensity is measured in watts per square meter, W/m2.
• Resonance.
– All elastic objects have natural frequencies of vibration
that are determined by the materials they are made of and
their shapes.
– When energy is transferred at the natural frequencies,
there is a dramatic increase of amplitude called
resonance.
– The natural frequencies are also called resonant
frequencies.
When the frequency of an applied force, including the
force of a sound wave, matches the natural frequency of
an object, energy is transferred very efficiently. The
condition is called resonance.
Different sounds that you hear include (A) noise, (B) pure
tones, and (C) musical notes.
• Sources of Sounds.
• Vibrating Strings.
– Standing Waves
• When reflected waves interfere with incoming waves
• Created by a patter on nodes and antinodes
– Nodes
• Places of destructive interference, which show no disturbance
– Antinodes
• Loops of constructive interference which take place where
crests and troughs produce a disturbance that rapidly alternates
upward and downward.
– Fundamental Frequency
• The longest wave that can make a standing wave on a string has
a wavelength that is twice the length of the string
• This longest wavelength has the lowest frequency and is called
the fundamental frequency.
• The fundamental frequency determines the pitch of the basic
musical note being sounded and is called the first harmonic.
An incoming wave on a
chord with a fixed end
(A) meets a reflected
wave (B) with the same
amplitude and
frequency, producing a
standing wave (C).
Note that a standing
wave of one
wavelength has three
nodes and two
antinodes.
A stretched string
of a given length
has a number of
possible resonant
frequencies. The
lowest frequency is
the fundamental,
f1; the next higher
frequencies, or
overtones, shown
are f2 and f3.
A combination of the fundamental and overtone
frequencies produces a composite waveform with
characteristic sound quality.
– Overtones
• Any whole number of halves of the wavelength will permit a
standing wave to form.
• The frequencies of these wavelengths are called overtones or
harmonics.
– The relationship for finding the fundamental frequency
and the overtones when the string length and velocity are
known is:
nv
fn 
2L
• Vibrating Air Columns.
– Closed Tube.
• The air in a closed tube can be made to vibrate by a reed,
turbulence, or some other way.
• In a closed tube the longest wave that can make a standing
wave has a wavelength that is four times the length of the
column.
• The relationship between the fundamental frequency, the
overtones, the length of the column and the velocity are given
by the equation:
nv
fn 
4L
Standing sine wave patterns of air vibrating in a closed
tube. Note the node at the closed end and the antinode at
the open end. Only odd multiples of the fundamental are
therefore possible.
– Open Tube.
• An open tube can have an antinode at both ends since the air is
free to vibrate at both places.
• The longest wavelength that can fit into the open tube is
therefore the distance between the 2 antinodes or ½ 
• The relationship between frequency, velocity, and length for an
open column is give by:
nv
fn 
2L
Standing waves in these open tubes have an antinode at
the open end, where air is free to vibrate.
Standing sine wave patterns of air vibrating in an open
tube. Note that both ends have anitnodes. Any whole
number of multiples of the fundamental are therefore
possible.
• Sounds from Moving Sources.
– A moving source of sound or a moving observer
experiences an apparent shift of frequency called the
Doppler Effect.
– If the source is moving as fast or faster than the speed of
sound, the sound waves pile up into a shock wave called
a sonic boom.
– A sonic boom sound very much like the pressure wave
from an explosion
(A) Spherical sound waves
from a stationary source
spread out evenly in all
directions. (B) If the source
is moving, an observer at
position P will experience
more wave crests per second
than an observer at position
P'. The observer at P
interprets this as a higher
pitch, and the phenomenon
is called the Doppler effect.