The Perception of Sound – Part 2

This is the second part of a small series on the topic of sound. In the first part we dealt with ‘The Perception of Sound’.  Here we now try to clarify physical facts of sound propagation in gases. In particular, we derive the one-dimensional wave equation from a simple model.

As discussed in Part 1 of this series,  a sound source causes the air surrounding it to oscillate in small vibrations. In the process, small rapid pressure fluctuations take place in the transmitting air or the gas or the fluid.  Due to the compressibility of the air, these are transmitted into the environment and thus brought to the human ear.

Apart from environmental influences such as reflections from walls or attenuation with distance, sound signals sound the same at every observation point.  They have the same frequency composition.  Their signal shape does not change when they propagate in gases.  This type of propagation is called non-dispersive.

The non-dispersive propagation of sound in gases is by no means self-evident.  In the case of structure-borne sound, for example, the propagation of bending waves on bars and plates is dispersive and the frequency composition of the signals does change.  The non-dispersive propagation of sound in gases and thus the preservation of the signal shape is an essential prerequisite for our verbal communication.

Gas Equations
The physical state of a given gas mass M is described by the volume V it occupies, its density ρ, the pressure p in it, and its temperature T.  The density ρ = M/V appears as a redundant quantity resulting from the volume.

Heating of a gas mass in a container with a constant volume leads to an increase in pressure in the container.  If the volume of a gas mass in a piston is rapidly reduced by compression, for example in a bicycle pump, the temperature in the gas increases. If the volume of the same mass of gas is increased, the pressure or temperature decreases.

These proportional relationships are familiar to us from everyday life and technology. They are summarised in the so-called Boyle-Mariotte equation for ideal gases.            It reads:

pGVG=MMmolRTG

(1) 

Here

Mmol

is a material constant, the so-called molar mass.  It denotes the molecular weight in grams per mole of the respective substance.  For example, the molecular weight of nitrogen is:

Mmol(N2)=28g

and the molecular weight of oxygen:

Mmol(O2)=32g

Air consists of about 20% oxygen and about 80% nitrogen, so the molecular weight of air is:

Mmol(air)=28.8g

The constant R is the general proportionality constant for ideal gases.  The assumption of ideal gases is experimentally well founded for airborne sound in the audible frequency range.  It is:

R = 8.314 Joule/K

Kelvin K is the  unit of measurement of absolute temperature. 0ºC corresponds to about 273K.

When observing very large masses and volumes, such as those of interest in sound fields, the description of state equivalent to:

p=RMmolρT

(2)

by pressure, density and temperature is appropriate.

Sound events in a gaseous continuum consist of very small, temporally and spatially distributed changes in the static variables pressure, density and temperature. It is

ptotal=p0+p(x,t) ρtotal=ρ0+ρ(x,t)   and Ttotal=T0+T(x,t)

wherin

p0,ρ0 and T0

represent the static variables pressure, density and temperature in the medium without exposure to sound. The sound field variables p, ρ and T , which are superimposed on the static variables and depend on location x and time t, are referred to as sound pressure, sound density and sound temperature.

The sound field sizes are minutely small compared to the static sizes.  As already described, the effective sound pressure value is just 2 N/m² for a sound reinforcement with dangerously loud 100dB.

The air pressure on the other hand has a value of about 100,000 N/m². Now, the static variables

p0,ρ0 and T0

as well as the total variables

ptotal,ρtotal and Ttotal

must satisfy the Boyle-Mariotte equation, but not the sound field variables p, ρ and T alone.  If we insert the total quantities into the Boyle-Mariotte equation, we obtain for the relative quantities p , ρ  and as well as the total variables ptotal, ρtotal and Ttotal must satisfy the Boyle-Mariotte equation, but not the sound field variables p, ρ and T alone. If we insert the total quantities into the Boyle-Mariotte equation, we obtain for the relative quantities

pp0,ρρ0, and TT0

the linear equation:

pp0=ρρ0+TT0

(3)

 The adiabatic equation of state

With the exception of very low frequencies, sound fields are subject to very rapid temporal changes. Because heat conduction processes are, in contrast, very slow, sound processes take place in the gas without the involvement of heat conduction.  Changes of state in a gas without involvement of heat conduction are called adiabatic. The relationship between pressure and density changes in a gas under the assumption that no heat is absorbed during the compression process is described by the adiabatic equation of state

ptotalp0=(ρtotalρ0)k

 (4)

where

k=cpcv

denotes the ratio of the specific heat of the gas at constant pressure, Cp and at constant volume CV. For the diatomic gases of almost exclusive interest in acoustics, k = 1.4. For monatomic gases, k = 1.67.

Because the pressure fluctuations around the operating point

(p0,ρ0)

generated by sound are very small, the adiabatic equation of state (4) can be linearized to

pp0=kρρ0

(5)

Solving for the density gives

ρ=pc2 with c2=kp0ρ0

(6)

From the equations (3) and (6) we can see that sound pressure, sound density and sound temperature have the same temporal and spatial signal shape.  They only differ by a scaling constant.  In most cases, sound pressure is used to describe the sound field because it is easily accessible for measurement with microphones.

Calculation of sound speed

The constant c in equation (6) has a special physical meaning.  It describes the speed of propagation of sound waves in gas – the speed of sound.  The wave speed will be considered in more detail in the following section.

To calculate c, we apply the Boyle-Mariotte equation (2) to the static variables. We get

c=kRMmolT0

(7)

The speed of sound thus depends only on the medium and the absolute temperature, but not on the static pressure and density.  For air we get the known value of c = 341 m/s.

Oscillations and Waves

A sound source causes the air surrounding it to oscillate slightly. Oscillations are temporally periodic processes that repeat after a period of time T.

A typical example of an oscillation is a punctiform mass m, which moves with constant speed on a circle with radius a.  The circle is traversed once in the time T – the period duration. Because of the constant speed, the path x(t) of the mass point is obviously T – periodic.

The simplest T periodic functions are

sin(2πT) and cos(2πT)

They can be used to describe the circular path as well as oscillations in general.  We call

ω:=2πT=2πf

the angular frequency (f frequency of oscillation).

Another typical example of an oscillation is a mass m (elastic pendulum) suspended from an elastic spring.  If the spring is tensioned and the mass is deflected from its rest position by the length a, the restoring force F is proportional to the deflection according to the Hookes law

F  = −sa,

where s is a spring constant.

We speak of a wave when a temporally periodic process – an oscillation – is also locally or spatially periodic.

Sound waves are spherical waves.  Spatially we can imagine a spherical balloon.  In a narrow spherical shell around the balloon surface, the pressure increases periodically and decreases to the normal pressure that prevails at rest. For an idealized wave propagating in only one dimension, we first think of a sound field that only depends on a single spatial coordinate, for example an air-filled pipe with a rigid wall, in which the sound field is virtually confined and thus forced into one direction of propagation – the pipe axis.

We get a simple and plausible explanation for the wave transport processes, if we  think of this one-dimensional column of air in the tube divided into disc-shaped segments, each alternately as a mass and as a spring. In this way, a so-called chain conductor is created as a model for the air column.  When the outermost mass segment of the air column is set in motion and deflected in the direction of the tube axis, the adjacent spring segment is compressed. The spring again exerts a force on the next mass segment. Because masses are inert, the gas mass does not react immediately, but only after a delay with a deflection.

Fig. 1: Air column divided into segments, alternately representing a mass and a spring

In doing so, it again tensions the next gas spring and is itself braked by this.

The whole thing is repeated along the chain. There will come up a delay of transmission of deflection from mass to mass.  The perturbation of the resting state impressed on one side spreads along the waveguide at a certain travelling speed.

The model idea can be transferred to the spherical wave by imagining that there are infinitely many small chain conductors between the sound source and the surface of the sphere.

The travel speed of the disturbance pattern is the wave speed – the speed of sound c.  The wave speed must be carefully distinguished from the velocity of the local gas particles oscillating around their resting position while the sound wave passes over them, so to speak.

The velocity of local gas elements is denoted by v in contrast to the propagation speed c of the wave. If  ξ(x, t) is the deflection of the gas particles of the chain conductor at the place x at time t, there applies

v(x,t)=ξt

(8)

The Continuity Equation

The compression within a mass segment – within a so-called gas parcel – with moving end faces can be derived from the fact that the mass present between the two ends remains unchanged.  So the elastic deformation results in a change in its density.

One can show that the sound density ρ results directly – that is, linearly – from the local derivative ∂ξ of the particle deflection ξ, such that the following applies

ρ=ρ0ξ(x)x

(9)

This equation is called the continuity equation. The local derivation of the particle deflection is also called elongation or dilatation.

 

The Inertia Law of Acoustics

For a gas spring element of length ∆x with side face S of the chain conductor, the product of the side face S and the sound pressure p is the spring force of the gas spring. According to Hooke’s law, this is proportional to the deflection

Sp=s[ξ(x+Δx)ξ(x)]

The proportionality factor s – the spring constant – is given by

s:=Sρ0c2Δx

where

E:=ρ0c2

is a material constant of the gas, the so-called modulus of elasticity, which is related to the propagation speed of the sound wave.

According to the law of inertia of Newton, the acceleration ∂²ξ/∂t² of an element of volume caused by the spring force is the smaller the greater the mass m of the element.
For the mass we set m = volume times density = ∆xSρ0. Let the length of the segment ∆x become infinitesimal small and obtain the so-called law of inertia of acoustics

ρ02ξt2=px

(10)

The Wave Equation

From the continuity equation (9) and the law of inertia of acoustics (10) we can derive the one-dimensional wave equation. In equation (9) we derive twice by time and in equation (10) by place. From this follows

2px2=2ρt2

By means of ρ = p/c2 according to equation (6) we replace the sound density by the sound pressure and obtain

2px2=1c22pt2

(11)

Equation (11) is called the wave equation.  The wave equation is a partial differential equation of second order.  All sound events satisfy it.  In particular, it is fulfilled by sound and oscillation events with a harmonic, cosine time response, the pure tones

p(x,t)=p0 cos(ω(t±x/c))

(12)

Summary

Sound consists of very small changes in pressure, density and temperature in gases that propagate in waves in the medium at wave speed c.
Heat conduction does not occur with these very rapid changes due to sound. The three state variables therefore satisfy both the Boyle-Mariotte equation and the adiabatic equation of state. The one-dimensional wave equation can be derived from the model of chain conductors, which consist alternately of mass and spring segments.