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Chapter 6
Waves and Sound
2
Waves — types and properties
A wave is a traveling disturbance consisting of coordinate vibrations that transmit energy with not net movement of matter. The disturbance is frequently called an
oscillation or vibration. The substance through which the wave travels
is called the medium.
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Waves — types and properties, cont’d
There are two main wave types: Transverse waves have oscillations that are
perpendicular (transverse) to the direction the wave travels.
Examples include waves on a rope, electromagnetic waves and some seismic waves.
Longitudinal waves have oscillations that are along the direction the wave travels.
Examples include sound and some seismic waves.
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Waves — types and properties, cont’d
This figure illustrates the two main types of waves.
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Waves — types and properties, cont’d
Consider a string of length l and mass m. The speed at which a wave travels on the
string when it is under a tension T is
is the mass per unit length:
.T
v
.m
l
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Waves — types and properties, cont’d
From this we see that the speed: Increases as the tension increases.
The string has a greater restoring force that attempts to straighten it out.
Is faster for smaller strings. The string has less mass that has to be moved by
the restoring force. Is independent of the length.
The speed depends on the mass per length, not on just the length.
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ExampleExample 6.1
A student stretches a Slinky out on the floor to a length of 2 meters. The force needed to keep the Slinky stretched in measured and found to be 1.2 newtons. The Slinky’s mass is 0.3 kilograms. What is the speed of any wave sent down the Slinky by the student?
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ANSWER:
The problem gives us:
The linear mass density is
ExampleExample 6.1
2 m
1.2 N
0.3 kg
l
F
m
0.3 kg0.15 kg/m
2 m
m
l
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ANSWER:
The wave speed is then
ExampleExample 6.1
1.2 N2.8 m/s.
0.15 kg/m
Fv
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Waves — types and properties, cont’d
The speed of a sound wave when the air is at a temperature T is
The temperature must be in Kelvin.
20.1 .v T
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ExampleExample 6.2
What is the speed of sound in air at room temperature (20ºC = 68ºF)?
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ANSWER:
The problem gives us:
We need to convert this temperature from celsius to kelvin:
The sound speed is then
ExampleExample 6.2
20ºT C
20 273 293 KT
20.1 293 344 m/sv
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DISCUSSION:
The factor of 20.1 depends on the properties of air.
For other gases: Helium:
Carbon dioxide:
ExampleExample 6.2
58.8 .v T
15.7 .v T
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Waves — types and properties, cont’d
The amplitude of a wave is the maximum displacement of the wave from the equilibrium position. It is just the distance equal to the height of a
peak or the depth of a valley.
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Waves — types and properties, cont’d
The wavelength is the distance between successive “like” points on a wave. “Like” points might be peaks, valleys, etc.
The wavelength is denoted by the Greek letter lambda: .
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Waves — types and properties, cont’d
Here is an illustration of changing the wavelength and/or amplitude. Lower amplitude implies
smaller height/depth. Shorter wavelength
implies more complete waves “fit” in a given distance.
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Waves — types and properties, cont’d
The frequency of a wave indicates the number of cycles of a wave that pass a given point per unit time. It is the number of oscillations per second.
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Waves — types and properties, cont’d
We use different terminology for the “peaks” and “valleys” of a longitudinal wave. A compression is where the medium is squeezed
together. A expansion is where the medium is spread apart.
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Waves — types and properties, cont’d
The wavelength and frequency are related to the wave speed according to:
v is the wave’s speed, f is the wave’s frequency, and is the wave’s wavelength.
.v f
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ExampleExample 6.3
Before a concert, musicians in an orchestra tune their instruments to the note A, which has a frequency of 440 Hz. What is the wavelength of this sound in air at room temperature?
The speed of sound at this temperature is 344 m/s.
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ANSWER:
The problem gives us:
The relation between frequency, wavelength and wave speed is
The wavelength is then
ExampleExample 6.3
440 Hz
344 m/s
f
v
v f
344 m/s0.78 m
440 Hz
v
f
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Waves — types and properties, cont’d
A complex wave is any continuous wave that does not have a sinusoidal shape.
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Aspects of wave propagation
There are two approaches to represent a wave. A wavefront is a circle representing the
location of a wave peak. A ray is an arrow
representing the direction that a wave segment is traveling.
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Aspects of wave propagation, cont’d
A reflection is when a wave abruptly changes direction.
A wave is reflected whenever it reaches a boundary of its medium or encounters an abrupt change in the properties of its medium.
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Aspects of wave propagation, cont’d
We can use either model to examine reflection from a flat mirror.
The point behind the mirror from which the reflection appears to originate is called the image.
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Aspects of wave propagation, cont’d
For a curved surface, the reflections can be focused to a point. Examples include satellite dishes, radar
receivers, etc.
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Aspects of wave propagation, cont’d
The Doppler effect is an apparent change in a wave’s wavelength due to the relative motion between the source and receiver.
Consider a source emitting waves and moving to the right. The crests appear
closer together in the direction the source moves.
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Aspects of wave propagation, cont’d
The crests appear farther apart in the direction opposite to the source’s motion.
These changes cause the frequency to sound different since the wave travels at the same speed relative to the medium.
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Aspects of wave propagation, cont’d
A shock wave occurs whenever the speed of the source is greater than the wave speed.
The medium cannot respond fast enough to propagate the wave. The crests essentially “pile up” in front of the
source. This build-up causes a large-amplitude pulse.
This produces a sonic boom for supersonic aircraft.
A fast boat creates a bow wave.
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Aspects of wave propagation, cont’d
Watching the waves expand from the source over time, we can construct a leading edge for the shock wave. The angle of this leading-
edge was an important discovery for jet-planes.
It led to delta-wing configurations.
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Aspects of wave propagation, cont’d
Diffraction results whenever a wave has to travel past a barrier or obstruction.
As the wave travels through the opening, the outgoing waves bend.
The amount of diffraction depends on the wavelength and the size of the obstruction.
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Aspects of wave propagation, cont’d
When the opening is much larger than the wavelength, there is little diffraction.
The amount of diffraction increases as the wavelength becomes more similar to the size of the opening.
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Aspects of wave propagation, cont’d
Diffraction explains why you can hear a sound through a door even if you’re behind a wall. The sound’s wavelength is
much longer than the size of the door, so the sound wave bends around the wall.
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Aspects of wave propagation, cont’d
Interference occurs whenever two or more waves overlap.
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Aspects of wave propagation, cont’d
When the waves interfere to create a larger amplitude, we call it constructive interference.
When the waves interfere to reduce the amplitude, it is called destructive interference.
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Sound
The speed of sound in a substance depends on: the mass of its constituent atoms, and the strength of the forces between the atoms.
The speed of sound is large when: the atoms have small mass — they’re easier
to move, and/or the forces between the atoms are larger — an
atom pushes harder on its neighbor.
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Sound, cont’d
Typically we represent a sound wave as a transverse wave (even though it is not). A region of compression is drawn as a crest. A region of expansion is drawn as a trough.
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Sound, cont’d
A waveform of a sound wave is a graph of the air-pressure fluctuations causes by the sound wave versus time. A pure tone is a sound
with a sinusoidal waveform.
A complex wave is a sound that is not pure.
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Sound, cont’d
Noise is sound that has a random waveform. It does not have a definite wavelength or
period. Sound with frequencies below our audible
range is called infrasound. Below about 20 Hz.
Sound with frequencies above our audible range is called ultrasound. Above about 20,000 Hz.
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Production of sound
Sound can be produced by: Causing a body to vibrate:
e.g., plucking a string. Varying an air flow:
e.g., buzzing your lips. Abrupt changes in an object’s temperature:
e.g., a lightning flash creates thunder. By creating a shock wave:
e.g., flying faster than the speed of sound.
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Production of sound
A piano produces sound by: The player presses a key so that the hammer strikes
the wire. The wire vibrates
and transmits this vibration to the soundboard.
The soundboard then radiates the sound to the room.
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Production of sound
A flute produces sound by: The player blows across the opening to create a
varying airflow. The airflow reaches
the end of the flute and radiates to the room.
Or the player opens a note-hole to release part of the wave.
The tube is then effectively shortened.
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Propagation of sound
Imagine a person talking in the middle of an empty arena.
The amplitude of the sound wave decreases as one moves farther away from the person.
1amplitude .
d
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Propagation of sound, cont’d
In a room, we have to deal with the multiple reflections off the walls and other objects. It gets a lot more complicated to determine the
amplitude at an arbitrary place in the room. The process of repeated reflections of sound
in an enclosure is called reverberation.
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Propagation of sound, cont’d
A hand clap in an open field is a simple pulse since there is no echo.
But in a gym, there are multiple echoes which tends to make the sound fade away gradually.
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Propagation of sound, cont’d
This effect can be useful or detrimental. Speech might be easy to understand in a field
but can become muddled with too much reverberation.
Reverberation is useful in music to help blend sounds.
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Perception of sound
We have to be careful when we discuss sound.
There are physical properties we can measure.
But our ears do not just measure these physical properties.
We have to deal with the perception of the sound.
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Perception of sound, cont’d
Pitch is the perception of highness or lowness of a sound. The pitch depends primarily on the frequency
of the sound. It also depends on the duration.
A very short sound might sound like a click even if it has a definite frequency.
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Perception of sound, cont’d
Loudness is the perception of whether a sound is easy to hear or painful to hear. It depends primarily on the amplitude of the
sound. It also depends on whether the sound is
played with other sounds (before, after, concurrently, etc).
It even depends on the frequency. Our ears are more sensitive to higher frequencies
and less sensitive to lower frequencies.
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Perception of sound, cont’d
Our eardrums respond to sound pressure level. A louder sound creates a larger compression,
i.e., higher pressure, than a quiet sound. We typically call the sound pressure level just
the sound level. It is measured in decibels (dB).
0 dB corresponds to inaudible. Normal conversation is about 50 dB. ~120 dB starts causing pain.
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Perception of sound, cont’d
The sound level of the quietest sound is called the threshold of hearing.
The sound level at which we start experiencing pain is called the threshold of pain.
The minimum increase in sound level that is noticeable is about 1 dB.
For a sound to be judged as “twice as loud,” the original sound must be increased by 10 dB.
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Perception of sound, cont’d
To make a sound “twice as loud,” you would need ten equal sources.
Two sounds with equal sound levels cause an increase of 3 dB over a single such sound.
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Perception of sound, cont’d
Tone quality is a measure by which two sounds of the same frequency and amplitude sound different. A sax sounds different from a trumpet playing
the same note because the two instruments have different tone qualities.
We typically refer to tone quality as timbre or tone color.
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Perception of sound, cont’d
Any complex waveform is equivalent to a combination of two or more sinusoidal waveforms with definite amplitudes and specific frequencies.
Theses component waveforms are called harmonics.
The frequencies of the harmonics are whole-numbered multiples of the complex waveform’s frequency.
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Perception of sound, cont’d
We can construct the waveform on the left by adding the three waveforms (harmonics) on the right.
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Perception of sound, cont’d
The specific tone quality of a sound depends on: the number of harmonics that are present, and the relative amplitudes of these harmonics.
A spectrum analyzer displays a complex waveform in terms of the constituent harmonics.
