The Physical Principles of Sound
An introductory guide to the physical properties of sound and a basic introduction tohow sound behaves in enclosed spaces.
To help understand other, more practical matters relating to sound it is useful to understand some of the fundamental scientific principles of the subject. This guide provides an introduction to how the complex nature of everyday sounds can be broken down into its simplest elements to help understand the makeup of intricate sounds.
The Physics of Sound
If we take a look at sound in its primitive form, sound is the disturbance of a medium, either gas, liquid or solid. In general we perceieve sound as changes in air pressure which creates waves, soundwaves, that arrive at our ears. For example, when a piano key is struck, the movement of the string disturbs the surrounding medium, air, causing the displacement of molecules within the air. This disturbance has a knock on effect causing adjacent molecules to be disturbed over a certain distance until the initial energy created by the initial displacement has disappeared. The amount of energy eventually decays to zero after being transferred from one molecule to the next, in the process losing an amount through each transferral. This explains how sound cannot be heard after certain distances dependant on the initial energy, where we percieve energy as volume.
Everyday sound is affected in three ways which shapes how it is perceived by a listener.
- The initial character of sound is shaped by the physical properties of the source of the sound. For example a musical instrument's material and shape or the shape of someone's mouth and the character of their vocals chords. The initial sound is also shaped by the nature of the excitation (the manner in which it starts), such as being hit, plucked, breathed into, or shouted, whispered or spoken.
- After it has been initially emitted sound is further affected by the environment in which it travels before it arrives at the listener. For example, a shout in a canyon sounds different than a shout in a bedroom. Sound reflects off certain surfaces to provide echo and is absorbed by other surfaces which dampen the sound.
- Finally, sound is shaped by the listening conditions applied by the listener. Everyone's physiological make-up differs slightly (for example ears come in all kinds of shapes and sizes) which affect how we hear sound. In a similar way the equipment we use to listen to sounds, such as different types of speakers or headphones, also changes the qualities of what we hear.
In air, sound waves have a pressure which alternately deviates from a state of equilibrium. These deviations can be thought of as regions of compression and rarefaction of molecules.
Imagine a set of adjoining springs which is fixed at each end. When the spring at the farthest end is pulled taut and released, this spring will then push the adjoining spring, after which it will be pulled apart back to it's initial state of equilibrium. This push and pull force is then transferred along all of the adjoining springs. The region where a spring is pushed is called a compression, and the region where it is pulled apart is known as a rarefaction. Diagram 1 shows the vibrating system of a tuning fork and the impact on the surrounding molecules.
This method of propagation is expressed as a longitudinal wave (see diagram 2) and provides a useful model for studying how sound travels through air.
Diagram 1. A tuning fork propagates sound waves longitudinally. When struck the prongs expand and contract creating periodic disturbance. This can be seen as periods of compression and rarefaction of molecules, until the prongs return to their natural state.
Sound in materials
Sound travels through solids in a slightly different manner. Imagine again the spring model where the centre most point is pulled from side to side (as opposed to the spring being pushed and pulled). This movement causes a lateral disturbance along both directions of the spring alignment, creating what is known as transverse waves. Propagation of this type is found in the vibrating systems of instruments, such as strings.
Diagram 2 - Although similar in shape to a sine wave, the diagram represents the displacement from equilibrium over time creating a longitudinal sound wave.
The periods of compression in a wave contain the content that is audible to our ears. Within the range of human hearing, sound waves travel at such a speed whereby we perceive there to be a constant tone. This is similar to a light bulb that flickers on and off so fast, that our eyes perceive there to be only constant light.
Speed of sound
Sound pressure in air is a scalar quantity, which means it can be measured at certain points but has no specific direction (as sound spreads from its source). Sound travelling through solids has a speed, or velocity V, in a direction dependent on the elasticity, known as Young's modulus, and the density of the material. This is expressed in the equation:
Vsolid = √ E
where V = speed in metres per second (ms -1 )
E = Youngs' modulus (Nm -2 )
ρ = density (kg/m -3 )
As air contains no elasticity an alternative model to Young's modulus is derived based upon the velocity of sound in air being relative to its temperature and is calculated as:
Velocity in Air = Vair = √ E
where V = speed in metres per second (ms -1 )
ρ = density (kg m -3 )
E = Y
where Y = a constant depending on the gas
P = pressure of the gas (Nm -2 )
Sound travelling through air is proportional to the square root of the absolute temperature. As a result the speed of sound increases by 0.6 metres per second (ms -1) for every unit increase in Celsius in ambient temperature, meaning sound travels faster in warmer environments. Speed of sound in air is calculated as approximately 344 metres per second (ms -1).
It is always advisable when using moving image and sound resources in a live or performance context to take into account the differences in the speeds of sound and light. At a large music concert, for examples, where travels considerably faster than sound it arrives before sound does. An audience member at the back of the audience can experinece this (if they are far enough away), especially when the performance is also being projected on a large screen. They will observe that the sound they hear from the PA arrives later than what they can see on the screen. There will be a noticable delay between what they see and what they hear.
Sound waves are characterised by the principles of harmonic motion and the generic properties of waves. They are graphically represented over two dimensions, plotting amplitude against time (see Diagram 2).
The Sine Wave
A useful tool for analysing sound is the sine wave. It is the fundamental building block of all sounds (discussed later) and is commonly used for testing and analysing audio equipment.
Diagram 3 below shows a rotating wheel and a plot of the constantly moving angle of rotation against time. The plotted graph can be thought of as viewing the wheel along the plane of rotation. If we recap to the analogy of the connected springs, the effect of connecting a rotating wheel to the end of the springs, driving a periodic push/pull motion, can also be seen below. The result is variations in pressure over time which are proportional the angle of the wheel's rotation. The phase angle θ, is derived from the angular velocity multiplied by time t. This is a useful model to describe the theory of simple harmonic motion.
Diagram 3 - The sine of the angle of the wheels rotation is displayed as functions of displacement. A wave cycle is the completion of the peak positive and negative displacement and return to equilibrium. The above diagram shows one cycle of a sine wave. The period of a wave is the time taken to complete one cycle.
It is worth noting that the phase angle is a continually changing variable with time. Phase shift, on the other hand, is the distinction between two separate waves, as shown in diagram 4. When two waveforms have a difference in phase they are often referred to as being out of phase with one another, by the degree of the phase shift. When we hear sounds, the way in which we perceive the position (relative to oneself) of the sound source relies on the difference in time that sound arrives at each of our ears (which is a difference in phase). Quite impressively our brains use this time difference to calculate and interpret the position of a sound source.
Diagram 4- Two sine waves with identical amplitude and frequency have a phase difference of 90°
Properties of sound waves
There are three main properties of sound that describe its audible characteristics. These properties can help us to understand the basis of all sound waves, even when the content of the wave becomes much more complex.
Amplitude is the measurement of sound level displacement above and below the equilibrium pressure. It is measured in the amount of force applied over an area and as such is relative to the energy or intensity of a sound. Amplitude is relative to the perceived volume of sound, although it is not a unit or a meter of volume. Diagram 2 shows the amplitude of a wave plotted against time.
As mentioned above, phase is the angle of displacement at the starting point of the wave, i.e. when t = 0. Phase is usually expressed as theta or θ. Phase is measured in degrees as the offset or onset (before or after) angle from 0°. The top wave in Diagram 2 above begins at point A = 0 on the vertical (y) axis.
- Frequency and Pitch
The frequency of a wave is derived from the amount of cycles (or periods) per unit of time (seconds). A cycle is the full, complete movement of a waveform from 0 amplitide to its full positive and negative displacement and back to 0 amplitude and displacement (the plot on the right hand side of Diagram 3 illustrates one complete cycle of a sine wave). Frequency is a logarithmic scale, measured in Hertz, and derived from the formula:
F = 1/T
Where T = Period = second
Physically, frequency rises with the stiffness of the vibrating system (of the sound source), and falls with its inertial mass.
Frequency is the physical basis for differences in musical pitch whereas pitch is a perceived quality regarding how ‘high' or ‘low' a musical note is.
The model for standard western pitch is known as the Twelve-tone Equal Tempered Scale. This is more commonly known as the twelve semitones from A to G on a piano where the interval between each note is of equal value and has an identical frequency ratio. These twelve semitones make up an octave.
So far, we have only looked at simple waves, like the sine wave, which have a fixed period (period is the time taken to complete one cycle). In sound recording and with everyday sounds waveforms are much more complex, and are more often than not aperiodic waves (with no fixed period). Whereas a tuning fork moves the surrounding air back and forth creating periodic vibration, a cymbal crash contains no fixed period. It creates a short lived oscillation, known as a transient, and is aperiodic. Nearly all sounds we hear are not constant sounds but sounds that decay or end abruptly.
Diagram 5 shows the waveform of middle C played on a grand piano viewed across three zoom levels. In 5a, we can see that there is higher amplitude at the beginning of the waveform which then decays over time (where amplitude is represented horizontally and time is plotted from left to right). Zooming in a level 5b shows that the thick wave is comprised of lots of spikes in the waveform. These spikes are the compressions and rarefactions as a result of the resonating piano string. Zooming in even closer to the beginning of the wave, shown in 5c, we can see the wave in more detail. Regular, but not identical cycles can be seen; there may be similarities in some cycles but the wave is complex and each cycle is unique.
Diagram 5 - Waveforms of a piano string excitation
Harmonic Structure and the Nature of Sound
As the example of the piano note in Diagram 5 above illustrates, real world sounds are much more complex than a single sine wave. These sounds contain a wide range of frequencies that start and stop at different times within the sound, each with their own, unique levels of loudness, harshness, clarity and character (properties collectively known as timbre). At a fundamental level all sounds can be thought of as layers of multiple sine waves. If we accept this principle then sine waves with differing amplitudes, frequency and phase can be combined together to create any sound imaginable.
To illustrate this concept, let's look at a square wave. A square wave (illustrated in Diagram 4d) is useful for analysis. Interestingly (like the sine wave) the square wave is not found in the natural world but is easily attainable through electronics. Although on first glance it may not look like it a square wave is a complex waveform with many elements but it can be understood simply as a fundamental sine wave (f0), with the addition of further sine waves whose frequencies are multiples of the fundamental. These additional waves are known as harmonic frequencies.
A 500Hz square sounds similar to a 500Hz sine but has a different sonic character, known as the timbre. A square wave contains multiple modes of vibration (multiple waves) meaning it is rich in harmonic content. The most dominant wave, the f0, is as expected, 500Hz. By saying the most dominant we refer to the fact that it has the highest amplitude. For a square wave the additional sine waves are odd harmonics of the f0.
Harmonics are multiple integer values of the fundamental frequency f0. When f0 = 500Hz, the first odd harmonics of the series are F3 and F5,which are 1000Hz and 1500Hz respectively.
For a square wave, extra harmonics have an amplitude which is 1 divided by it's number within the harmonic series. Therefore the 3rd harmonic (at 15000 Hz) should be 1/3 of the amplitude of f0, the 4th harmonic at 1/4 and so on.
Diagram 6 - The make up of a square wave
Diagram 6a shows a fundamental wave f0 and the first three odd harmonics that are needed in order to make up a square wave, f3, f5 and f7. The waveform in diagram 6b is the summation of these four waves, which begins to take the form of a square wave. By adding further odd harmonics to the series the crests and peaks become more flattened, which can be seen in diagram 6c which shows a wave with twenty two harmonics added to f0. With the addition of infinite odd harmonics a perfect square wave is formed, which is shown in diagram 6d.
In a musical environment, most sounds are formed with additional waves which are inharmonic . These are known as partial tones. If a note is described as being C, then it has fundamental frequency of C along with partial tones which usually occur very close to the harmonics of the fundamental.
These partial tones affect the quality of the note produced and define the audible character of the resonant body, or the instrument.
Different instruments produce different harmonics and partials at different amplitudes and it is the unique makeup of these which allows us to distinguish the sounds of different musical instruments and sound sources. For example, a trumpet and a piano playing the same note sound very different due to the different harmonic overtones they posses.
From something with such a simple build up of frequencies as a square wave this technique of sine wave addition can be applied mathematically to express far more complex sounds using the principles of Fourier Theory.
Sounds analysed as the build up of sine waves to create more complex waves expressed through a mathematical system called the Fourier Transform. This is based on two principles:
- Any function of time can be represented as a sum of sine waves differing in frequency.
- Each component has a distinct frequency, amplitude and phase.
Using these principles it is then possible to think of a more complex wave, such as a piano string shown in diagram 5, as the sum of many individual sine waves.
Sound outputs a very low energy content. The energy put into generating a sound is generally much greater than that of the sound itself. When measured most musical instruments have approximately 1% (or less) energy efficiency.
Examples of power levels
Bass drum (full volume) 25 Watts
Piano (full volume) 0.4 Watts
Clarinet (full volume) 0.05 Watts
Energy of sound from a sound source is measured using the Inverse Square Law. Following this law, an area twice as far from the source is spread over four times the area and is subsequently one fourth of the intensity. This is explained in Diagram 6 below.
Diagram 7 - The Inverse Square Law
The intensity (I) in the area (A) is derived from the following:
P = I
Where P = Power of the source sound (Watts)
4πr2 = Area of the sphere
As with many mathematical principles, in practice the Inverse Square Law is not always obeyed. This is due to the following reasons.
1. Few systems produce sound equally in all direction. This is certainly true for musical instruments and the human voice.
2. The effect of reflections, where sound reflects off of surfaces during travel (see the section Acoustics below).
3. The effect of absorption, where sound is absorbed by materials such as furniture or people during travel (again, see the section Acoustics below).
Level and Loudness
Sound Intensity level and the decibel scale
The perceived intensity, or loudness, of a sound is depends on what is known as the sound energy density at the position of the listener.
The relationship between sound intensity and perceived sound is measured logarithmically (a method used to scale a large range of values), using a unit called the decibel or db.
The decibel unit is commonly used in consumer and professional audio equipment, be it hi-hi amps or mixing desks, as it allows us to apply a standard measurement for the volume of sounds we listen to.
This db value is measured relative to a 0db reference level, which is typically set at the threshold of human hearing. Therefore the decibel measurement is the ratio of two sounds, one being 0db, and is a relative measure. Sound intensity is calculated using the following equation.
SIL = 10log 10 Intensity source
where Intensity reference = 10 -12 W/m 2
In digital audio systems the maximum available volume level is respresented as 0db, and it is possible to cut and boost the volume by decibel values.
An important point to note when working with decibel measurements is that due to its logarithmical form, doubling the sound intensity at source accounts for a 3.01db increase. Similarly, halving the level of intensity reduces the decibel amount by 3.01db.
Sound Pressure Level
In practice it is difficult to measure sound intensity as two readings are required. Therefore sound pressure level (SPL) (also a logarithmic measure) of a source pressure is measured relatively to a reference level. This is a measure of the pressure levels arriving from a sound source relative to the local ambient pressure level.
Sound pressure, which is measured in unit Pascals (pa), can be measured with a microphone and a dedictaed SPL meter is a useful piece of equipment for measuring SPL values in real-time. SPL meters are useful for measuring musical performance levels and for delivering health and safety standards. They are often used to give accurate readings of peak (maximum) and average levels of a signal.
Examples of sound pressure levels:
(short term exposure)
Loud rock concert
40 – 60db
2 x 10 -3 – 2 x 10 -2 pa
2 x 10 -4 pa
Equation for SPL:
SPL = 20log Pressure source
where Pressure reference = 20μpa
As mentioned above, timbre is a term used to imply the qualities of sounds. For example, descriptors such as harsh, soft, smooth, gentle or piercing all amount to the timbre. As such defining timbre can be quite difficult as it relies on subjective terms which are not always universally agreed upon. Timbre is therefore largely down to our own perception of sound and helps us describe sounds in non-technical terms.
Timbre is that attribute of auditory sensation in terms of which a listener can judge that two sounds similarly presented and having the same loudness and pitch are dissimilar.
American National Standards Institute (1960). USA Standard Acoustical Terminology (Including Mechanical Shock and Vibration) S1.1-1960 (R1976). New York: American National Standards Institute.
Describing sound in terms of timbre is useful in distinguishing the differences between sounds. When one sound is harsher than another, softer than another, warmer, brighter, darker, and so on. It applies to the expressive variations in sounds, and the commonalities within similar instruments. Timbre can be described as quality that differentiates two sounds which are identical in pitch and loudness, such as the example of the trumpet and piano playing the same note.
Acoustics is the study of the behaviour of sound. Sound has three stages, its generation (where is comes from), propagation (its movement in space) and reception (its conditions of arrival at the listener). This section will discuss the basic factors that can shape sound travelling in an enclosed space, and some common methods of treating sound within these spaces.
Sound in enclosed spaces
So far, the concept of wave propagation has been discussed without the consideration of boundaries. As we have already mentioned, sound can be altered by its immediate environment and any physical mediums it arrives at after its initial excitation. To illustrate this let us consider an example of how sound propagates in an enclosed space.
At one end of an empty room a balloon is popped creating a bang that is heard by a listener at the opposite end of the room. Here are three key aspects of how the sound then behaves which explain how it is perceived by the listener.
1. After the excitation of the bang, the listener hears the direct sound after a very short delay. This sound will have travelled the shortest distance possible from the source to the listener and contains the highest intensity.
2. Shortly after the arrival of the direct sound, waves that have been reflected off of one or a small number of surfaces in the room will arrive at the . These are known as early reflections, and these vary in a depending on the positions of the sound source and the listener. Early reflections provide information to the listener that help them perceive the size of the space and the location of the sound source. If there are reflective surfaces far away where sound waves have further to travel to before they reflect and then arrive back at the listener (assuming there is enough energy to allow this extra travel), they are perceived as an echo effect. They are separate from the direct sound and add changes to the timbre of the overall sound.
3. Finally, after the early reflections have arrived, sound from all directions reach the listener.
The effect is this is a dense build up of reflections that arrive later than the initial sound, but smear together in the overall sound. This is called reverberation (or just reverb for short) and adds depth and space to sound. It is what gives a sound the character of being in a room, or when there is alarge reverb effect a space such as a church or a even a tunnel.
The time taken for reverberation to occur is relative to the size of the room as a smaller room has a shorter distance for all reflections to travel. The time between the direct sound and the reverberant refelctions is known as pre-delay, the delay prior to reverberation.
Sound intensity is lost on each impact of reflection and subsequently the reverberation decays over a duration known as the reverberation time.
When a room is excited by an impulse, such as the balloon bang, the surfaces that reflect the waves also help reduce some of the initial energy so that it decays exponentially. In practice this is not always the case and energy can be reflected in cyclic paths. When the distance of a wave's path (i.e. from one wall to another) is an exact multiple of half of the wavelength (where the wavelength is the length of one wave cycle), the energy is actually reflected back to the original position of the impulse. This creates a 'standing wave', and the added energy results in an increase in volume.
Standing waves are common in lower bass frequencies and are undesirable in a environment used for audio work. In studio design non-parallel walls are used to offset the effect of standing waves and bass reducing absorption materials can lessen the audible effects.
Rooms and spaces that are used for audio work can often have a detrimental effect on what we hear. Too many reflective surfaces can often create a harsher and metallic sounding space, where higher frequency sounds are amplified. Similarly too many absorbent surfaces can suck the frequencies from a room creating a dry and flat sounding space. When working with audio a neutral sounding room is desirable where reverberation, background noise and standing waves are as minimal as possible. As a result it can be necessary to apply treatment to a room, and depending on the use and the desired result this will mean different types of treatment.
There are three main types of acoustic treatment that can alter the character of a sound. Acoustic treatment is commonly done by the addition or subtraction of surfaces which have strong acoustic properties. Three key properties are reflection, absorption and diffusion.
- Reflective treatments
Surfaces that refect sound waves are made from materials such as stone, glass or wood panelling. Most reflective surfaces accenuate higher frequencies due to the nature of the waves (low frequency waves travel through materials much easier than high frequency waves), therefore the addition of reflective surfaces can make a room sound brighter.
- Absorbant treatment
Reverberation and echo are often undesirable in a room used for sound analysis and can even increase the volume of certain frequency ranges at different positions in the room. To eliminate these effects absorbent surfaces that absorb waves can be applied to deaden the sound. Low frequency waves tend to congregate in corners of rooms and can often cause strong audible differences in rooms with symmetrical surfaces through standing waves, as we have seen. Absorbent materials can also be installed to counter these effects.
- Diffusive treatment
Diffusion is the spread of frequencies caused by a medium. The use of diffusion is often used in the acoustic treatment of rooms and spaces to stop sound waves from grouping together, which can create inconsistencies from one point in a room to another. Diffusion helps to reduce standing waves, and can make a small room appear audibly larger through creating a sense of openness. It can be used to stop early reflections from room boundaries merging with the sound from the initial source without absorbing the frequencies, hence there being no loss in energy and overall frequency content.
Most professional listening rooms are designed and built by experienced professionals. For everyone else it is a case of judging for yourself if the space you are using to monitor needs improving, and this is not an easy task to undertake. Sometimes, applying treatment to a room may not be possible and in these cases it is important to optimise the equipment and layout of the space to the natural sound of the room (for further information see the Advice document: Preparing your workstation).
Here are two basic exercises to help you begin to understand the effect your room may be having on what you hear.
1. Listen to a recognisable audio file first through headphones and then through your monitoring system at a reasonable level.
This could be a favourite song you that you are very familiar or a piece of music you have been working on and therefore understand well. Firstly likely to be struck by differences caused by the change of equipment, between your headphones and speakers. You may hear imperfections in the room that change the sound when listening through speakers. Perhaps the piece sounds ‘muddy' or lacking in clarity, it may be the result of too many reflective surfaces and not enough absorption. You may notice an increase in frequency content, for example too much or too little bass. This could be attenuated by your systems' equaliser.
2. Apply a ‘sine sweep' through your monitoring system.
This is done by amplifying a sine wave test tone at a fixed amplitude at different frequencies within the audible threshold. By literally sweeping the tone across the frequency range or by changing the frequency to set tones across the spectrum (the standard frequency bands being 32, 64, 125, 250, 500, 1k, 2k, 4k, 8k, 16k), you may find some frequencies appear to be louder than others. If so, then this is likely to be the result of standing waves and low frequency attenuation in the shape of the room. Many digital audio software platforms have a test tone feature built in. It is recommended that you refer to the appropriate manual for directions on how to use this.
Decibel - db
A logarithmical unit which measures the intensity or level of a signal.
The act of movement from a state of equilibrium.
The act of causing the initial displacement of an objects material.
A quick succession of reflected sounds which occurs between two parallel surfaces and is normally stimulated by a transient sound.
n.b. Flutter is also the result of mechanical error when working with analogue tape.
A mathematical term for the ratio of values expressed by the base 10 or function e.
The manner in which an acoustic wave is propagated, as characterized by the particle motion in the wave (shear, Lamb, surface or longitudinal).RW
The physical action of the spreading and movement of waves
Speed which is defined as the distance travelled per unit of time.
The vibration of an object or medium at a specific frequency.
A continuous single-frequency periodic waveform whose amplitude varies as the sine of the linear function of time. Sometimes referred to as a sinusoidal wave.
Simple Harmonic Motion
A back and forth periodic motion, which is neither driven nor damped, which repeats about a central equilibrium point.
Angus, J. Howard, D. Acoustics and Psychoacoustics. Focul Press, Third Edition, 2006.
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