82C - Sound

Sound is a disturbance of mechanical energy that propagates through matter as a longitudinal wave. Sound is characterized by the properties of sound waves, which are frequency, wavelength, period, amplitude, and speed. Humans perceive sound by the sense of hearing. By sound, we commonly mean the vibrations that travel through air and can be heard by humans. However, scientists and engineers use a wider definition of sound that includes low and high frequency vibrations in air that cannot be heard by humans, and vibrations that travel through all forms of matter, gases, liquids and solids.

The matter that supports the sound is called the medium. Sound propagates as waves of alternating pressure, causing local regions of compression and rarefaction. Particles in the medium are displaced by the wave and oscillate. The scientific study of sound is called acoustics. Noise is often used to refer to an unwanted sound. In science and engineering, noise is an undesirable component that obscures a wanted signal.

The speed at which sound travels depends on the medium through which the waves are passing, and is often quoted as a fundamental property of the material. In general, the speed of sound is proportional to the square root of the ratio of the elastic modulus (stiffness) of the medium and its density. Those physical properties and the speed of sound change with ambient conditions. For example, the speed of sound in air and other gases depends on temperature. In air, the speed of sound is approximately 344 ms-1, in water 1500 ms-1 and in a bar of steel is 5000 ms-1.

82C1 - Wave Interference

Here you will learn to:

Identify constructive and destructive interference, and

Use the Superposition Principle to determine wave patterns arising from interference.


What happens when two waves meet moving through the same medium? Will the two waves reflect off each other on meeting (much like two billiard balls would) or will the waves pass through each other? These questions involving the meeting of two or more waves in the same medium will help us to understand wave interference.

Two wave pulses travelling in opposite directions
interfere with one another.

Wave interference (superposition) is the phenomenon that occurs when two waves meet while traveling through the same medium. The interference of waves causes the medium to take on a shape that results from the net effect of adding the two waves together. Let's consider two pulses traveling in different directions along the same medium. As the pulses move towards each other, there will eventually be a moment in time when they are completely overlapped. At that moment, the resulting shape of the medium would be an upward displaced pulse with a total amplitude equal to the sum of the two amplitudes at each point along the wave. The animation on the left shows the combined wave as the two pulses overlap.

This type of interference is sometimes called constructive interference. Constructive interference is a type of interference that occurs at any location along the medium where the two interfering waves have a displacement in the same direction. In this case, both waves have an upward displacement; consequently, the medium has an upward displacement that is greater than the displacement of the two interfering pulses. Generally constructive interference is observed at any location where the two interfering waves are displaced in the same direction. This can also include two waves being displaced in the downwards or negative direction.

Destructive interference is a type of interference that occurs at any location along the medium where the two interfering waves have a displacement in the opposite direction. For instance, when a sine pulse with a maximum displacement of +1 unit meets a sine pulse with a maximum displacement of -1 unit, destructive interference occurs. The result is that the two pulses completely destroy each other when they are completely overlapped. At the instant of complete overlap, there is no resulting displacement of the particles of the medium. This cancelling out is not a permanent condition. As soon as the waves pass each other they are restored to their original amplitude.

Superposition Principle

The superposition principle can be applied to waves that pass each other in two dimensions. The resultatant waveform is the sum of the amplitudes which would have been produced by the individual waves separately. For example, two waves traveling towards each other will pass right through each other without any distortion on the other side, but while they overlap, the resultant wave will be the sum of their amplitudes at any point along the wave.

Two waves 360° out of phase (a) constructively interfere to
produce a wave with a larger amplitude. Waves 180° out of
phase (b) destructively interfere to produce a wave with a
smaller amplitude.

The phenomenon of interference between waves is based on this idea. When two or more waves traverse the same space, the net amplitude at each point is the sum of the amplitudes of the individual waves. In some cases, such as in noise-cancelling headphones, the summed variation has a smaller amplitude than the component variations; this is called destructive interference. In other cases, the summed variation will have a bigger amplitude than any of the components individually; this is called constructive interference.

Waves travelling with their crests and troughs not aligned are said to be out of phase. Waves travelling with their crests and troughs aligned are said to be in phase because each wavelength is perfectly aligned. Angles are used to describe by how much waves are in or out of phase. One whole wavelength is 360° and half a wavelength is 180°. Radians are also sometimes used. The diagram on the right shows some phase relationships. Maximum constructive interference occurs when waves are whole number multiples of a wavelength or multiples of 360° out of phase. Maximum destructive interference occurs when waves are multiples of half of a wavelength or 180° out of phase.

The interference pattern produced from
the diffraction of light passing through
two narrow slits.

Light waves can also interfere constructively and destructively. Constructive interference shows up as bright areas and destructive interference shows up as dark areas. This phenomenon was demonstrated famously by Thomas Young in the early 1800s when he passed light through two narrows slits and observed the interference pattern produced (bottom right). This was an important experiment at the time because it pretty much convinced everyone that light was a wave (and it isn't as it turns out).

Check out this simulation showing the interference of light.

Stuff to Do

TutorialTutorial 82C1 - Interference
TutorialTutorial 82C1 - Interference - Answers
MovieDan Fullerton - Wave Interference


82C2 - Sound

In this section you will learn to:

Describe how humans perceive sound in the ear,

Describe some properties of sound such as pitch and loudness,

Describe how sound is produced by a musical instrument such as a violin, and

Calculate the frequency, wavelength and velocity of standing waves in strings.

What is Sound?

Sound is a sequence of longitudinal pressure waves that propagate through compressible media such as air or water. Sound that is perceptible by humans has frequencies from about 20 Hz to 20,000 Hz. In air at standard temperature and pressure, the corresponding wavelengths of sound waves range from 17 m to 17 mm. During propagation, waves can be reflected, refracted, or attenuated by the medium. The behavior of sound propagation is generally affected by three things:

A relationship between density and pressure. This relationship, affected by temperature, determines the speed of sound within the medium.

  • The propagation is also affected by the motion of the medium itself. For example, sound moving through wind. Independent of the motion of sound through the medium, if the medium is moving, the sound is further transported.
  • The viscosity of the medium also affects the motion of sound waves. It determines the rate at which sound is attenuated. For many media, such as air or water, attenuation due to viscosity is negligible.
  • When sound is moving through a medium that does not have constant physical properties, it may be refracted (either dispersed or focused).

The mechanical vibrations that can be interpreted as sound are able to travel through all forms of matter: gases, liquids, solids, and plasmas. The matter that supports the sound is called the medium. Sound cannot travel through a vacuum so without an atmosphere on Earth none of us would be able to hear anything.

How Do We Hear Sound?

The anatomy of the human ear showing the inner, middle and outer parts.

Hearing is the ability to perceive sound by detecting vibrations, changes in the pressure of the surrounding medium through time, through an organ such as the ear. Sound may be heard through solid, liquid, or gaseous matter. It is one of the traditional five senses. In humans and other vertebrates, hearing is performed primarily by the auditory system. Sound waves produce vibrations in the ear which are turned into nerve impulses that are perceived by the brain (primarily in the temporal lobe). Like touch, audition requires sensitivity to the movement of molecules in the world outside the organism.

There are three main components of the human ear: the outer ear, the middle ear and the inner ear. The outer ear includes the visible part of the ear (or the pinna), the auditory canal and the eardrum. The eardrum is made of an airtight flap of skin. Sound waves arriving at the eardrum cause it to vibrate with the same waveform as the sound wave. The eardrum simplifies incoming air pressure waves to a single change with a certain amplitude. This allows for the differentiation of sound. The middle ear consists of a small air filled chamber that is located behind the eardrum. Within this chamber are the three smallest bones in the body, known collectively as the ossicles. They aid in the transmission and amplification of the vibrations from the ear drum to the inner ear. The inner ear contains the cochlea, which is a spiral shaped, fluid filled tube that is considered the organ of auditory transduction. It is divided lengthwise by the basilar membrane, a structure that vibrates when waves from the middle ear propagate through the cochlear fluid–membrane system. The basilar membrane is tonotopic, so that each frequency has a characteristic place of resonance along it. Characteristic frequencies are high at the basal entrance to the cochlea, and low at the apex. Basilar membrane motion causes the movement of the hair cells, specialized auditory receptors located within the basilar membrane. The space–time pattern of vibrations in the basilar membrane is converted to a spatial–temporal pattern of firings on the auditory nerve, which transmits information about the sound to the brain where it is interpreted.

Characteristics of Sound

The frequency of sound is related to a characteristic that we call pitch. High frequency sounds have a high pitch and low frequency sounds have a low pitch. In western music, the scale is defined by some very precise relationships between the frequency of notes in it. For example, a note one octave higher than another will have a frequency that is double that of the lower note.

The amplitude of a sound wave is directly related to its loudness. High volume sound waves have large amplitudes and low volume sound waves have low amplitudes. Turning up the volume of a sound system increases or amplifies the amplitude of the sound wave so it sounds louder.

Standing Waves

A guitar string has a number of frequencies at which it will naturally vibrate. These natural frequencies are known as the harmonics of the guitar string. The natural frequency at which an object vibrates at depends upon the tension of the string, the linear density of the string and the length of the string. Each of these natural frequencies or harmonics is associated with a standing wave pattern. Standing waves are produced when waves reflect from fixed ends of the string and interfere. At certain frequencies the interference pattern produces a standing wave which appears to not be moving.

The first three harmonics or standing waves produced in a vibrating

The diagram depicts the standing wave patterns for the lowest three harmonics or frequencies of a guitar string. The wavelength of the standing wave for any given harmonic is related to the length of the string (and vice versa). If the length of a guitar string is known, the wavelength associated with each of the harmonic frequencies can be found. Thus, the length-wavelength relationships and the wave equation can be combined to perform calculations predicting the length of string required to produce a given natural frequency. And conversely, calculations can be performed to predict the natural frequencies produced by a known length of string. Each of these calculations requires knowledge of the speed of a wave in a string. The diagram depicts the relationships between the key variables in such calculations. These relationships will be used to assist in the solution to problems involving standing waves in musical instruments.

Sound in Musical Instruments

The sound waves made by a musical instrument such as a guitar occur because the string vibrates and that vibration is transferred to the molecules of air around it. We hear it if the sound wave in air makes it to our ears.

Let's examine what happens in a guitar string to produce a sound. When the string is plucked it oscillates. Waves travel up and down the string and are reflected from each fixed end. These waves interfere with each other and and the standing waves produced occur at unique frequencies called harmonics. The actual pitch we hear as an A, B flat or C is determined by the standing wave produced at the first or lowest frequency harmonic (sometimes called the fundamental frequency). A note from a violin that we hear as an A is actually the string vibrating with a fundamental frequency of 440 Hz. The quality of the sound heard in the violin is determined by the way in which other standing waves or harmonics with higher frequencies interfere to produce one complex waveform in the string that we hear. Making electronic sounds that sound real is both a a complex art and science in that the pattern of harmonics in the real instrument are analysed and then reproduced to produce an authentic sound as close to the real thing as possible.

Stuff to Do

TutorialTutorial 82C2 - Sound
TutorialTutorial 82C2 - Sound - Answers
MovieBozeman Science - Sound
MovieDan Fullerton - Standing Waves