The Physical Principles of Sound
An introductory guide to the physical properties of sound and a basic introduction to the acoustics of enclosed spaces.
To aid the understanding of any technical matters relating to sound, as often the case with any discipline, it is crucial to understand the fundamental scientific principles of the subject and how they are commonly interpreted.
This guide offers an introduction to the basic physics of sound including the build up of sound waves and their properties, the speed of sound, how it is shaped in acoustical environments, and how treatment can be applied to listening rooms. This document does not, on the whole, provide advice, but presents information for reference with which to aid an overall understanding of sound in a practical working environment.
The Physics of Sound
At its most stripped back level, sound is the mechanical disturbance of a medium, either gas, liquid or solid. For example, when a piano key is struck, the movement of the string disturbs the surrounding medium, air, causing the displacement of molecules. 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 energy decays to zero after being transferred from one molecule to the next, in the process losing an amount through each transferral.
In a more practical context sound can be described as the transmission of pressure, from an initial sound source to a listener through the air.
Sound has three stages which affect how it is perceived by a listener.
- The initial character is shaped by the properties of the sound source (i.e. an instruments' material and shape) and its excitation, such as being hit, plucked or breathed into.
- The environment and the mediums which sound travels through to the listener. For example, a shout heard in a canyon sounds different than if heard in a small room, or through a pair of speakers. See section Elementary Acoustics)
- The listening conditions applied by the listener. The individuals subjective perception of sound as well as their physical condition (i.e. the shape of the outer ears, or someone's age), affects how sound is perceived by the listener (this is discussed in the section, Sound and the Ear)
Sound pressure is transmitted in air as a wave like motion. In air, sound waves have a pressure which alternately deviates from a state of equilibrium. These deviations are 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 is an accurate model for how sound travels through air.
A tuning fork propagates sound waves longitudinally. Upon excitement 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 travels through solids in a different manner. Imagine the spring model, as discussed, where the centremost spring is pulled from side to side, as opposed to pushed and pulled. This lateral movement causes lateral disturbance along both directions of the spring alignment, known as transverse waves. Propagation of this type is found in the vibrating systems of instruments, such as strings.
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. 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, as light travels considerably faster than sound and arrives before sound does to an audience which is at a distance from the two sources.
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
The most common tool for understanding sound due to its spectral purity is the sine wave. It is the fundamental building block of all sounds (discussed later) and is used extensively for testing and analysis of audio equipment.
Diagram 3, below, shows a rotating wheel where the constantly angle of rotation is plotted against time. This 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 to the sine of the angle of the wheels rotation. The phase angle θ, is derived from the angular velocity multiplied by time t. i.e. the speed of rotation measured in radians per second multiplied by time t. This model describes the theory of simple harmonic motion.
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 and can be constant, 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). The brain uses this time difference to calculate and interpret the position of the sound source.
Two sine waves with identical amplitude and frequency have a phase difference of 180°
There are three main properties of a sine wave 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. As a result, amplitude is also 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 expressed as Theta or θ. Phase is measured in degrees as the offset or onset (before or after) angle from 0°. The 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). 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 (discuss?).
Frequency is the physical basis for differences in musical pitch.
How is pitch different from frequency? 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.
AUDIO EXAMPLE HERE
Periodic and Aperiodic Waves
So far, only periodic waves have been considered. 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, a transient, and is aperiodic. In fact, nearly all sound we hear is not constant sounds but sounds that decay or end abruptly.
Diagram 5 shows the waveform of middle C played on a grand piano viewed at three levels of resolution. From 5a, we can see that the amplitude of the note starts at high level at the time of excitation and then decays over time. Zooming into the beginning of the impulse, diagram 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 up close to the beginning of the wave in 5c, is proof that this wave is aperiodic. There may be similarities in some cycles but the wave is complex and each cycle is unique.
Waveforms of a piano string excitation
Harmonic Structure and the Nature of Sound
Musical and environmental 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, with differing levels of loudness, harshness, clarity and character (known as Timbre, discussed further). At a fundamental level, all sounds can be thought of as the build up of multiple sine waves. Therefore, in theory, sine waves with differing amplitudes, frequency and phase can be added together to create any sound imaginable.
To clarify this concept, the deconstruction of a square wave shall be considered. A square wave (illustrated in Diagram 4d) is another wave used for analysis, not found in the natural world but attainable through electronics. Technically it is a very complex waveform with many elements but it can be understood simply as a fundamental sine wave, or f0, with the lowest frequency present, and the summation of a further set of sine waves of the appropriate amplitude, frequency, and phase.
A 500Hz square sounds similar to a 500Hz sine but has a different character, or timbre. A square wave contains multiple modes of vibration meaning it is rich in harmonic content (in theory it is infinite, but in practice this is impossible to recreate). The most dominant wave, the f0, is as expected, 500Hz. 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 2nd and 3rd harmonics are 1000Hz and 1500Hz and so on.
For a square wave, these extra harmonics have an amplitude which is 1 divided by it's number within the series of harmonics. 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.
The making of a square wave
Diagram 6a shows the fundamental wave f0 and the first three odd harmonics are needed 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 harmonics and partials at different amplitudes and it is the makeup of these which allows us to distinguish the sounds of different musical instruments and sound sources.
From something with such a simple build up of frequencies as a square this technique of sine wave addition can be applied mathematically to express far more complex sounds using the principles of Fourier Theory.
The collage of sine waves to create more complex waves can be expressed through a mathematical system called the Fourier Transform. This is based on two principles.
1. Any function of time can be represented as a sum of sine waves differing in frequency.
2. Each component has a distinct frequency, amplitude and phase.
Using these principles it is then possible to think of the complex wave of a piano string 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.
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 scientific theories, in practice the Inverse Square Law is not always obeyed. This is due to the following reasons.
1. Few systems emit sound equally in all direction. This is certainly true for musical instruments and the human voice.
2. The effect of reflections (see the section Elementary Acoustics below).
3. The effect of absorption (see the section Elementary Acoustics below).
Level and Loudness
Sound Intensity level and the decibel scale
The perceived intensity of a sound is dependant on the sound energy density at the position of the listener.
The relationship between sound intensity and perceived sound is measured logarithmically (which is a method used to scale a large range of values), using a unit called the decibel or db.
The decibel is a unit 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
It is common in the majority of audio systems that the maximum volume level is 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 reading are required. The sound pressure level SPL is also a logarithmic measure of a source pressure relative to a reference level. It 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. SPL meters are a practical tool which display readings of a connected microphone 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.
(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 Pressuresource
where Pressurereference = 20μpa
It is important, especially in the context of sound as a musical form to understand the meaning of Timbre. In fact, everyone is already aware of it, and uses it on a daily basis, but Timbre helps us understand the science of sound as well as music of sound.
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.
Essentially, timbre is the descriptive element of sound quality. It is the distinction that 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.
A useful analogy is trying to apply colours as descriptive words of music, a cello sounds brown or a guitar sounds yellow. Another method is to try and describe a colour without actually using the name of the colour.
Timbre is not an acoustical attribute but is a perceptual and subjective sensation that is processed in the mind. It is how we perceive and understand sound in a non-scientific form.
Acoustics is the scientific study of sound behaviour. Sound has three stages, generation, propagation and reception. This section will discuss the basic factors that can shape sound travelling in an enclosed space during propagation to a listener, 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 mentioned at the beginning of this document, sound can be altered by its immediate environment and any further physical mediums it passes through after initial excitation. In order to understand this better, let us consider an example of how sound propagates in an enclosed space.
At one end of an empty room a balloon is popped which is heard by a listener at the opposite end of the room. There are three key aspects to how the sound 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 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 (bounced) off one or more surfaces will be heard. These are known as early reflections, and would vary in the same room depending on the positions of the source and the listener. Early reflections provide information to listener that is used to perceive the size of the space and the location of the source. If reflections are long, i.e. have further to travel, they are perceived as an echo effect. They are separate from the direct sound and as such can add changes to the timbre of the overall sound.
3. Finally, after the early reflections have arrived, sound from many other reflection paths in all directions reach the listener.
As there are so many possible paths the effect is of a dense build up of reflections, which smears the overall sound together. This effect is called reverberation and is a coveted feature in music sound as it adds depth and space to sound.
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 the reflections to travel. This time between the direct sound and the reverberation is commonly 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, the surfaces reflect the waves and at each reflection absorb the 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 waves path (i.e. from one wall to another) is an exact multiple of half of the wavelength (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.
For example, if a wave is transmitted towards a metal surfcae which is one wave cycle away in distance, the reflected energy will be added to the energy from the source signal resulting in the waves being added together. This summation is heard as an increase in volume of the wave and is called resonance.
These standing waves (or resonant modes) occur in the audible lower frequencies and are undesirable in a environment used for audio work. Non-parallel walls are used to offset the effect of standing waves and bass reducing absorption materials can lessen the audible effects in a room.
Rooms and spaces 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 frequencies are amplified. Similarly too many absorbent surfaces suck the frequencies from the room create a dry and empty sounding room. When monitoring audio a neutral sounding room is desired which has as little effect as possible on the signal path between the monitor source and the listener. 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 a sound's characteristics in a physical environment. Treatment is commonly done by the addition or subtracting of surfaces which have strong acoustic properties. The three main properties are reflection, absorption and diffusion.
Surfaces which are reflective can be made from materials such as stone 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.
Reverberation and echo are often undesirable in a room used for sound analysis as they add extra frequency content which can vary at different positions in the room. To eliminate these effects absorbent surfaces that absorb (audibly deaden) waves can be applied. 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. Adequately absorbent materials can be installed to counter these effects.
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, which can create inconsistencies from one point in a room to another. Diffusion helps to reduce standing waves and flutter, 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 monitoring spaces are designed and built by experienced professionals. For everyone else it is a case of measuring for yourself if the space you are using to monitor needs improving. 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 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. You may instantly hear imperfections in the room that colour the sound. If 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 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.
A lot of DAWs have a test tone feature built in to the system, often as a plug-in. It is recommended that you refer to the appropriate manual for directions on how to use this.
There are a number of companies who supply acoustic treatment materials which are simple to install. If you require structural modifications to a building for acoustic improvements then it is advisable to consult a qualified acoustician.
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.
An impulse of sound
The act of communication through wave movement
Angus, J. Howard, D. Acoustics and Psychoacoustics. Focul Press, Third Edition, 2006.