Acoustics/Print version

Acoustics is the science that studies sound, in particular its production, transmission, and effects. Sound can often be

considered as something pleasant; music is an example. In that case a main application is room acoustics, since the purpose

of room acoustical design and optimisation is to make a room sound as good as possible. But some noises can also be

unpleasant and make people feel uncomfortable. In fact, noise reduction is actually a main challenge, in particular in the

industry of transportations, since people are becoming increasingly demanding. Furthermore, ultrasounds also have applications

in detection, such as sonar systems or non-destructive material testing. The articles in this wikibook describe the

fundamentals of acoustics and some of the major applications.

Fundamentals of Acoustics

Introduction

Sound is an oscillation of pressure transmitted through a gas, liquid, or solid in the form of a traveling wave, and can be generated by any localized pressure variation in a medium. An easy way to understand how sound propagates is to consider that space can be divided into thin layers. The vibration (the successive compression and relaxation) of these layers, at a certain velocity, enables the sound to propagate, hence producing a wave. The speed of sound depends on the compressibility and density of the medium.

In this chapter, we will only consider the propagation of sound waves in an area without any acoustic source, in a homogeneous fluid.

Equation of waves

Sound waves consist in the propagation of a scalar quantity, acoustic over-pressure. The propagation of sound waves in a stationary medium (e.g. still air or water) is governed by the following equation (see wave equation):

$\nabla ^2 p - \frac{1}{{c_0 ^2 }}\frac{{\partial ^2 p}}{{\partial t^2 }} = 0$

This equation is obtained using the conservation equations (mass, momentum and energy) and the thermodynamic equations of state of an ideal gas (or of an ideally compressible solid or liquid), supposing that the pressure variations are small, and neglecting viscosity and thermal conduction, which would give other terms, accounting for sound attenuation.

In the propagation equation of sound waves, $c_0$ is the propagation velocity of the sound wave (which has nothing to do with the vibration velocity of the air layers). This propagation velocity has the following expression:

$c_0 = \frac{1}{{\sqrt {\rho _0 \chi _s } }}$

where $\rho _0$ is the density and $\chi _S$ is the compressibility coefficient of the propagation medium.

Helmholtz equation

Since the velocity field $\underline v$ for acoustic waves is irrotational we can define an acoustic potential $\Phi$ by:

$\underline v = \text{grad }\Phi$

Using the propagation equation of the previous paragraph, it is easy to obtain the new equation:

$\nabla ^2 \Phi - \frac{1}{{c_0 ^2 }}\frac{{\partial ^2 \Phi }}{{\partial t^2 }} = 0$

Applying the Fourier Transform, we get the widely used Helmholtz equation:

$\nabla ^2 \hat \Phi + k^2 \hat \Phi = 0$

where $k$ is the wave number associated with $\Phi$. Using this equation is often the easiest way to solve acoustical problems.

Acoustic intensity and decibel

The acoustic intensity represents the acoustic energy flux associated with the wave propagation:

$\underline i (t) = p\underline v$

We can then define the average intensity:

$\underline I = < \underline i >$

However, acoustic intensity does not give a good idea of the sound level, since the sensitivity of our ears is logarithmic. Therefore we define decibels, either using acoustic over-pressure or acoustic average intensity:

$p^{dB} = 20\log \left(\frac{p}{{p_\mathrm{ref} }}\right)$  ; $L_I = 10\log \left(\frac{I}{{I_\mathrm{ref} }}\right)$

where $p_\mathrm{ref} = 2.10^{ - 5} Pa$ for air, or $p_\mathrm{ref} = 10^{ - 6} Pa$ for any other media, and $I_\mathrm{ref} = 10^{ - 12}$ W/m².

Solving the wave equation

Plane waves

If we study the propagation of a sound wave, far from the acoustic source, it can be considered as a plane 1D wave. If the direction of propagation is along the x axis, the solution is:

$\Phi (x,t) = f\left(t - \frac{x}{{c_0 }}\right) + g\left(t + \frac{x}{{c_0 }}\right)$

where f and g can be any function. f describes the wave motion toward increasing x, whereas g describes the motion toward decreasing x.

The momentum equation provides a relation between $p$ and $\underline v$ which leads to the expression of the specific impedance, defined as follows:

$\frac{p}{v} = Z = \pm \rho _0 c_0$

And still in the case of a plane wave, we get the following expression for the acoustic intensity:

$\underline i = \pm \frac{{p^2 }}{{\rho _0 c_0 }}\underline {e_x }$

Spherical waves

More generally, the waves propagate in any direction and are spherical waves. In these cases, the solution for the acoustic potential $\Phi$ is:

$\Phi (r,t) = \frac{1}{r}f\left(t - \frac{r}{{c_0 }}\right) + \frac{1}{r}g\left(t + \frac{r}{{c_0 }}\right)$

The fact that the potential decreases linearly while the distance to the source rises is just a consequence of the conservation of energy. For spherical waves, we can also easily calculate the specific impedance as well as the acoustic intensity.

Boundary conditions

Concerning the boundary conditions which are used for solving the wave equation, we can distinguish two situations. If the medium is not absorptive, the boundary conditions are established using the usual equations for mechanics. But in the situation of an absorptive material, it is simpler to use the concept of acoustic impedance.

Non-absorptive material

In that case, we get explicit boundary conditions either on stresses and on velocities at the interface. These conditions depend on whether the media are solids, inviscid or viscous fluids.

Absorptive material

Here, we use the acoustic impedance as the boundary condition. This impedance, which is often given by experimental measurements depends on the material, the fluid and the frequency of the sound wave.

Fundamentals of Room Acoustics

Introduction

Three theories are used to understand room acoustics :

1. The modal theory
2. The geometric theory
3. The theory of Sabine

The modal theory

This theory comes from the homogeneous Helmoltz equation $\nabla ^2 \hat \Phi + k^2 \hat \Phi = 0$. Considering a simple geometry of a parallelepiped (L1,L2,L3), the solution of this problem is with separated variables :

$P(x,y,z)=X(x)Y(y)Z(z)$

Hence each function X, Y and Z has this form :

$X(x) = Ae^{ - ikx} + Be^{ikx}$

With the boundary condition $\frac{{\partial P}} {{\partial x}} = 0$, for $x=0$ and $x=L1$ (idem in the other directions), the expression of pressure is :

$P\left( {x,y,z} \right) = C\cos \left( {\frac{{m\pi x}} {{L1}}} \right)\cos \left( {\frac{{n\pi y}} {{L2}}} \right)\cos \left( {\frac{{p\pi z}} {{L3}}} \right)$

$k^2 = \left( {\frac{{m\pi }}{{L1}}} \right)^2 + \left( {\frac{{n\pi }}{{L2}}} \right)^2 + \left( {\frac{{p\pi }}{{L3}}} \right)^2$

where $m$,$n$,$p$ are whole numbers

It is a three-dimensional stationary wave. Acoustic modes appear with their modal frequencies and their modal forms. With a non-homogeneous problem, a problem with an acoustic source $Q$ in $r_0$, the final pressure in $r$ is the sum of the contribution of all the modes described above.

The modal density $\frac{{dN}}{{df}}$ is the number of modal frequencies contained in a range of 1Hz. It depends on the frequency $f$, the volume of the room $V$ and the speed of sound $c_0$ :

$\frac{{dN}}{{df}} \simeq \frac{{4\pi V}}{{c_0^3 }}f^2$

The modal density depends on the square frequency, so it increase rapidly with the frequency. At a certain level of frequency, the modes are not distinguished and the modal theory is no longer relevant.

The geometry theory

For rooms of high volume or with a complex geometry, the theory of acoustical geometry is critical and can be applied. The waves are modelised with rays carrying acoustical energy. This energy decrease with the reflection of the rays on the walls of the room. The reason of this phenomenon is the absorption of the walls.

The problem is this theory needs a very high power of calculation and that is why the theory of Sabine is often chosen because it is easier.

The theory of Sabine

Description of the theory

This theory uses the hypothesis of the diffuse field, the acoustical field is homogeneous and isotropic. In order to obtain this field, the room has to be sufficiently reverberant and the frequencies have to be high enough to avoid the effects of predominating modes.

The variation of the acoustical energy E in the room can be written as :

$\frac{{dE}}{{dt}} = W_s - W_{abs}$

Where $W_s$ and $W_{abs}$ are respectively the power generated by the acoustical source and the power absorbed by the walls.

The power absorbed is related to the voluminal energy in the room e :

$W_{abs} = \frac{{ec_0 }}{4}a$

Where a is the equivalent absorption area defined by the sum of the product of the absorption coefficient and the area of each material in the room :

$a = \sum\limits_i {\alpha _i S_i }$

The final equation is : $V\frac{{de}}{{dt}} = W_s - \frac{{ec_0 }}{4}a$

The level of stationary energy is : $e_{sat} = 4\frac{{W_{abs} }}{{ac_0 }}$

Reverberation time

With this theory described, the reverberation time can be defined. It is the time for the level of energy to decrease of 60 dB. It depends on the volume of the room V and the equivalent absorption area a :

$T_{60} = \frac{{0.16V}}{a}$ Sabine formula

This reverberation time is the fundamental parameter in room acoustics and depends trough the equivalent absorption area and the absorption coefficients on the frequency. It is used for several measurement :

• Measurement of an absorption coefficient of a material
• Measurement of the power of a source
• Measurement of the transmission of a wall

Fundamentals of Psychoacoustics

Due to the famous principle enounced by Gustav Theodor Fechner, the sensation of perception doesn’t follow a linear law, but a logarithmic one. The perception of the intensity of light, or the sensation of weight, follow this law, as well. This observation legitimates the use of logarithmic scales in the field of acoustics. A 80dB (10-4 W/m²) sound seems to be twice as loud as a 70 dB (10-5 W/m²) sound, although there is a factor 10 between the two acoustic powers. This is quite a naïve law, but it led to a new way of thinking acoustics, by trying to describe the auditive sensations. That’s the aim of psychoacoustics. By now, as the neurophysiologic mechanisms of human hearing haven’t been successfully modelled, the only way of dealing with psychoacoustics is by finding metrics that best describe the different aspects of sound.

Perception of sound

The study of sound perception is limited by the complexity of the human ear mechanisms. The figure below represents the domain of perception and the thresholds of pain and listening. The pain threshold is not frequency-dependent (around 120 dB in the audible bandwidth). At the opposite side, the listening threshold, as all the equal loudness curves, is frequency-dependent.

Phons and sones

Phons

Two sounds of equal intensity do not have the same loudness, because of the frequency sensibility of the human ear. A 80 dB sound at 100 Hz is not as loud as a 80 dB sound at 3 kHz. A new unit, the phon, is used to describe the loudness of a harmonic sound. X phons means “as loud as X dB at 1000 Hz”. Another tool is used : the equal loudness curves, a.k.a. Fletcher curves.

Sones

Another scale currently used is the sone, based upon the rule of thumb for loudness. This rule states that the sound must be increased in intensity by a factor 10 to be perceived as twice as loud. In decibel (or phon) scale, it corresponds to a 10 dB (or phons) increase. The sone scale’s purpose is to translate those scales into a linear one.

$\log (S) = 0,03(L_{ph} - 40)$

Where S is the sone level, and $L_{ph}$ the phon level. The conversion table is as follows:

Phons Sones
100 64
90 32
80 16
70 8
60 4
50 2
40 1

Metrics

We will now present five psychoacoustics parameters to provide a way to predict the subjective human sensation.

dB A

The measurement of noise perception with the sone or phon scale is not easy. A widely used measurement method is a weighting of the sound pressure level, according to frequency repartition. For each frequency of the density spectrum, a level correction is made. Different kinds of weightings (dB A, dB B, dB C) exist in order to approximate the human ear at different sound intensities, but the most commonly used is the dB A filter. Its curve is made to match the ear equal loudness curve for 40 phons, and as a consequence it’s a good approximation of the phon scale.

Example : for a harmonic 40 dB sound, at 200 Hz, the correction is -10 dB, so this sound is 30 dB A.

Loudness

It measures the sound strength. Loudness can be measured in sone, and is a dominant metric in psychoacoustics.

Tonality

As the human ear is very sensible to the pure harmonic sounds, this metric is a very important one. It measures the number of pure tones in the noise spectrum. A broadwidth sound has a very low tonality, for example.

Roughness

It describes the human perception of temporal variations of sounds. This metric is measured in asper.

Sharpness

Sharpness is linked to the spectral characteristics of the sound. A high-frequency signal has a high value of sharpness. This metric is measured in acum.

Blocking effect

A sinusoidal sound can be masked by a white noise in a narrowing bandwidth. A white noise is a random signal with a flat power spectral density. In other words, the signal's power spectral density has equal power in any band, at any centre frequency, having a given bandwidth. If the intensity of the white noise is high enough, the sinusoidal sound will not be heard. For example, in a noisy environment (in the street, in a workshop), a great effort has to be made in order to distinguish someone’s talking.

Sound Speed

The speed of sound c (from Latin celeritas, "velocity") varies depending on the medium through which the sound waves pass. It is usually quoted in describing properties of substances (e.g. see the article on sodium). In conventional use and in scientific literature sound velocity v is the same as sound speed c. Sound velocity c or velocity of sound should not be confused with sound particle velocity v, which is the velocity of the individual particles.

More commonly the term refers to the speed of sound in air. The speed varies depending on atmospheric conditions; the most important factor is the temperature. The humidity has very little effect on the speed of sound, while the static sound pressure (air pressure) has none. Sound travels slower with an increased altitude (elevation if you are on solid earth), primarily as a result of temperature and humidity changes. An approximate speed (in metres per second) can be calculated from:

$c_{\mathrm{air}} = (331{.}5 + (0{.}6 \cdot \theta)) \ \mathrm{m/s}\,$

where $\theta\,$ (theta) is the temperature in degrees Celsius.

Details

A more accurate expression for the speed of sound is

$c = \sqrt {\kappa \cdot R\cdot T}$

where

• R (287.05 J/(kg·K) for air) is the gas constant for air: the universal gas constant $R$, which units of J/(mol·K), is divided by the molar mass of air, as is common practice in aerodynamics)
• κ (kappa) is the adiabatic index (1.402 for air), sometimes noted γ
• T is the absolute temperature in kelvins.

In the standard atmosphere :

T0 is 273.15 K (= 0 °C = 32 °F), giving a value of 331.5 m/s (= 1087.6 ft/s = 1193 km/h = 741.5 mph = 643.9 knots).
T20 is 293.15 K (= 20 °C = 68 °F), giving a value of 343.4 m/s (= 1126.6 ft/s = 1236 km/h = 768.2 mph = 667.1 knots).
T25 is 298.15 K (= 25 °C = 77 °F), giving a value of 346.3 m/s (= 1136.2 ft/s = 1246 km/h = 774.7 mph = 672.7 knots).

In fact, assuming an ideal gas, the speed of sound c depends on temperature only, not on the pressure. Air is almost an ideal gas. The temperature of the air varies with altitude, giving the following variations in the speed of sound using the standard atmosphere - actual conditions may vary. Any qualification of the speed of sound being "at sea level" is also irrelevant. Speed of sound varies with altitude (height) only because of the changing temperature!

 Altitude Temperature m/s km/h mph knots Sea level (?) 15 °C (59 °F) 340 1225 761 661 11,000 m–20,000 m (Cruising altitude of commercial jets, and first supersonic flight) -57 °C (-70 °F) 295 1062 660 573 29,000 m (Flight of X-43A) -48 °C (-53 °F) 301 1083 673 585

In a Non-Dispersive Medium – Sound speed is independent of frequency, therefore the speed of energy transport and sound propagation are the same. For audio sound range air is a non-dispersive medium. We should also note that air contains CO2 which is a dispersive medium and it introduces dispersion to air at ultrasound frequencies (28kHz).
In a Dispersive Medium – Sound speed is a function of frequency. The spatial and temporal distribution of a propagating disturbance will continually change. Each frequency component propagates at each its own phase speed, while the energy of the disturbance propagates at the group velocity. Water is an example of a dispersive medium.

In general, the speed of sound c is given by

$c = \sqrt{\frac{C}{\rho}}$

where

C is a coefficient of stiffness
$\rho$ is the density

Thus the speed of sound increases with the stiffness of the material, and decreases with the density.

In a fluid the only non-zero stiffness is to volumetric deformation (a fluid does not sustain shear forces).

Hence the speed of sound in a fluid is given by

$c = \sqrt {\frac{K}{\rho}}$

where

K is the adiabatic bulk modulus

For a gas, K is approximately given by

$K=\kappa \cdot p$

where

κ is the adiabatic index, sometimes called γ.
p is the pressure.

Thus, for a gas the speed of sound can be calculated using:

$c = \sqrt {{\kappa \cdot p}\over\rho}$

which using the ideal gas law is identical to:

$c = \sqrt {\kappa \cdot R\cdot T}$

(Newton famously considered the speed of sound before most of the development of thermodynamics and so incorrectly used isothermal calculations instead of adiabatic. His result was missing the factor of κ but was otherwise correct.)

In a solid, there is a non-zero stiffness both for volumetric and shear deformations. Hence, in a solid it is possible to generate sound waves with different velocities dependent on the deformation mode.

In a solid rod (with thickness much smaller than the wavelength) the speed of sound is given by:

$c = \sqrt{\frac{E}{\rho}}$

where

E is Young's modulus
$\rho$ (rho) is density

Thus in steel the speed of sound is approximately 5100 m/s.

In a solid with lateral dimensions much larger than the wavelength, the sound velocity is higher. It is found by replacing Young's modulus with the plane wave modulus, which can be expressed in terms of the Young's modulus and Poisson's ratio as:

$M = E \frac{1-\nu}{1-\nu-2\nu^2}$

For air, see density of air.

The speed of sound in water is of interest to those mapping the ocean floor. In saltwater, sound travels at about 1500 m/s and in freshwater 1435 m/s. These speeds vary due to pressure, depth, temperature, salinity and other factors.

For general equations of state, if classical mechanics is used, the speed of sound $c$ is given by

$c^2=\frac{\partial p}{\partial\rho}$

where differentiation is taken with respect to adiabatic change.

If relativistic effects are important, the speed of sound $S$ is given by:

$S^2=c^2 \left. \frac{\partial p}{\partial e} \right|_{\rm adiabatic}$

(Note that $e= \rho (c^2+e^C) \,$ is the relativisic internal energy density).

This formula differs from the classical case in that $\rho$ has been replaced by $e/c^2 \,$.

Speed of sound in air

Impact of temperature
θ in °C c in m/s ρ in kg/m³ Z in N·s/m³
−10 325.4 1.341 436.5
−5 328.5 1.316 432.4
0 331.5 1.293 428.3
+5 334.5 1.269 424.5
+10 337.5 1.247 420.7
+15 340.5 1.225 417.0
+20 343.4 1.204 413.5
+25 346.3 1.184 410.0
+30 349.2 1.164 406.6

Mach number is the ratio of the object's speed to the speed of sound in air (medium).

Sound in solids

In solids, the velocity of sound depends on density of the material, not its temperature. Solid materials, such as steel, conduct sound much faster than air.

Experimental methods

In air a range of different methods exist for the measurement of sound.

Single-shot timing methods

The simplest concept is the measurement made using two microphones and a fast recording device such as a digital storage scope. This method uses the following idea.

If a sound source and two microphones are arranged in a straight line, with the sound source at one end, then the following can be measured:

1. The distance between the microphones (x)
2. The time delay between the signal reaching the different microphones (t)

Then v = x/t

An older method is to create a sound at one end of a field with an object that can be seen to move when it creates the sound. When the observer sees the sound-creating device act they start a stopwatch and when the observer hears the sound they stop their stopwatch. Again using v = x/t you can calculate the speed of sound. A separation of at least 200 m between the two experimental parties is required for good results with this method.

Other methods

In these methods the time measurement has been replaced by a measurement of the inverse of time (frequency).

Kundt's tube is an example of an experiment which can be used to measure the speed of sound in a small volume, it has the advantage of being able to measure the speed of sound in any gas. This method uses a powder to make the nodes and antinodes visible to the human eye. This is an example of a compact experimental setup.

A tuning fork can be held near the mouth of a long pipe which is dipping into a barrel of water, in this system it is the case that the pipe can be brought to resonance if the length of the air column in the pipe is equal to ( {1+2n}/λ ) where n is an integer. As the antinodal point for the pipe at the open end is slightly outside the mouth of the pipe it is best to find two or more points of resonance and then measure half a wavelength between these.

Here it is the case that v = fλ