Departamento de

Engenharia Mecânica

Sound intensity and power

Professor Phil Joseph

IMPORTANCE OF SOUND INTENSITY AND SOUND

POWER MEASUREMENT

Sound pressure is the quantity usually used to quantify

sound fields. However, it is often satisfactory as an measure

of source because the pressure propagates as a wave

which, due to multi-path interference, may lead to

fluctuations with observer position.

Sound pressure, unlike measures of sound energy, are not

conserved.

Performance of noise control systems often specified in

terms of energy, e.g., transmission loss, absorption

coefficient.

Instantaneous sound intensity I(t) is the rate of acoustic energy flowing

through unit area in unit time (Wm-2). If, in a point in space, the

acoustic pressure p(t) produces at the same point a particle velocity

u(t), the rate at which work is done on the fluid per unit area I(t) at time

t is given by

I ut p t t

INSTANTANEOUS INTENSITY

Note that I is a vector quantity in the direction of the particle velocity u

‘PROOF’

The work done on the fluid by Force F

acting over a distance d in the

direction of the force is Fd

The work done per unit area A in unit

time T, i.e., the sound intensity, is

given by

uxI pTA

F

d

EXAMPLES FOR WHICH SOUND INTENSITY AND

MEAN SQUARE PRESSURE ARE SIMPLY RELATED

1. Plane progressive waves

2. Far field of a source in free field

3. Hemi-diffuse field

However, in general there is no simple relation between intensity and pressure

ENERGY CONSERVATION

yxt

Iyx

x

Iyx

t

E yx

Net rate of change of energy = Rate of energy in – Rate of energy out

In 3 - dimensions

0.

t

t

EI

x

I

x

I

x

I zyx

I.

RELATIONSHI[P BETWEEN SOUND IINTENSITY

AND SOURCE SOUND POWER

0.

I

t

E

Applying Gauss's divergence theorem

SV

dSdV nA.A ˆ.

SV

dSdVt

EW nI. ˆ

to

gives

GENERAL PROPERTIES OF SOUND INTENSITY FIELDS

Sound intensity (sometimes called sound power flux density) is a vector

quantity acting in the direction of the particle velocity vector u(t).

The instantaneous sound intensity I(t) is in general, rapidly oscillating. A

non-zero time averaged intensity represents a net overall flow of

energy and is called active intensity. A value is referred to as

reactive intensity and is characteristic of strong near fields comprising

strong circulations of energy, which do not propagate to the far field.

SO

UR

CE

Active intensity

Reactive intensity

I

I

INTERFERENCE BETWEEN SOUND INTENSITY FIELDS

Interfering monopoles

tttptp 2121 uuI

Intensity at the microphones is

Sum of intensities

generated by each

source individually.

Intensities are therefore

not generally additive.

ttpttpttpttp 12212211 uuuu

This term represents

correlations. It is zero for

statistically unrelated

source strengths, i.e., if the

sources are incoherent,

SOUND INTENSITY FIELD FOR INTERFERING (COHEENT)

MONOPOLES

Note presence of active ‘sinks’.

Thus, suppressing a portion of an

extended radiator may increase

total power radiation

PRINCIPLES OF SOUND INTENSITY MEASUREMENT

In general sound intensity can only be determined by the measurement

of acoustic pressure and particle velocity simultaneously. This has only

been possible fairly recently with the advent of fast signal processing

methods.

p t1 p t2

pt p t p t 12 1 2

u tp p

rdr

t

1 1 2

r

I ut p t t

tptptp 2212

1

t

dtr

tptptu

0

21

From Euler’s momentum

equation Spatial average

SPECTRAL FORMULATION OF INTENSITY FOR

USE WITH FFT ANALYSERS

The spectral (i.e., frequency) domain equivalent expression of

I ut p t t

r

G

12Im

I

G E p p12 1 2 * p p t e dti t1 2 1 2, ,

dtet ti II

is given by

where G12 is the pressure cross spectrum

and

A COMMERCIAL SOUND INTENSITY PROBE

EXAMPLE OF SOUND INTENSITY FIELDS

SIDE OF CAR AT 100Hz

EXAMPLE OF SOUND INTENSITY FIELDS

CELLO AT 160, 315 and 630Hz

EXAMPLE OF SOUND INTENSITY FIELDS

PISTON IN A BAFFLE AT ka = 2 and ka = 25.

ERRORS IN THE TWO-MICROPHONE SOUND INTENSITY

TECHNIQUE

Principal sources of error in the measurement of sound intensity using

the two-microphone technique in approximate order of importance are:

Bias (Systematic) Errors

a Finite difference and sum approximation error - increases with

increasing frequency and microphone separation distance r.

b Probe diffraction effects - Imposes an upper frequency limit on

their use.

c Phase mismatch error - Transducer and conditioning channel

mismatch. Must ensure phase matching as close as possible - reason why

intensity probe are disproportionately more expensive than single sensors

Random Errors

a Spectral estimation errors due to inadequate time average.

Normalized error , where B is the measurement bandwidth and T

is the effective analysis time window.

21

~

BT

Sound Power

SOURCE QUANTIFICATION The noise received at a location depends on the source strength and

also on the transmission of sound to that location.

Transmission path

LW Lp

ReceiverSource

Noise control strategies can be divided into:

• Reductions at source

• Reductions in the transmission path between source and receiver

In both cases it is useful to quantify the source, independent of its location.

For this we need a suitable measurement quantity representing the source

strength, not a receiver quantity. It should therefore be

• A property of the source alone, independent of its location

• Representative of the sound from the whole source

• Related to the receiver quantity

OBJECTIVES OF SOURCE QUANTIFICATION

Why do we want to quantify the source output?

1. to compare different machines or plant for user selection

2. for a manufacturer to check acceptability of components from sub-

suppliers

3. for source labelling

4. to check that the machine complies with regulatory or legal

requirements

5. for predicting the sound pressure at an operator position (for

assessing hearing hazard) or in the neighbourhood (environmental

impact)

6. to identify source mechanisms (diagnostics)

7. to understand the physics of the source in order to develop models

for the purpose of improving the design

as input to models of transmission paths for noise control

by reducing transmission

SOUND POWER MEASUREMENT

In view of the importance of sound power as a measure of source

‘strength’, its accurate measurement is extremely important. The

commonest techniques for measuring sound power may be organized as

follows: Sound Power Measurement

Techniques

Direct Indirect

Free Field

Technique

Diffuse FieldTechnique

Sound PressureMeasurements

Sound IntensityMeasurements

In-situ

Source

Substitution

Source Surface

Vibration MeasurementTechnique

DEFINITION OF SOUND POWER:

INTEGRAL FORMULATION

S

I

n

W dSS

I n. This integral expression follows from

Gauss’s theorem

The choice of control surface S is

arbitrary, as long as it completely

encloses the source

Time-stationary sources do not

contribute to the integral

Expression assumes that intensity can

be measured directly

SOURCE POWER IS WEAKLY AFFECTED BY ITS

ENVIRONMENT

It is important to be aware that strong reflections back on the source, for

example when the source is situated close to a reflecting surface, may alter

the source radiation resistance and hence increase (or decrease) its sound

power output. The acoustic behavior environment may therefore modify the

source power output although this is generally a weak effect at mid to high

frequencies.

Example

0 2 4 6 8 10 12 14

0.8

1

1.2

1.4

1.6

1.8

2

2kd

FREE-FIELD (or ANECHOIC CHAMBER) TECHNIQUE

(ISO 3745 (3744)) A measurement surface is constructed around the source and divided

into N segments. It is assumed that in the absence of reflections the

intensity may be deduced from the acoustic pressure

(i.e. LI = LP)

This assumes that the wave fronts are

planar, or spherical, and lie normal to

the measurement surface. The sound

power then follows as

W p c Si ii

N

2

1

/

Advantages: Very simple to implement

Disadvantages: Requires costly use of anechoic chamber. Measurements

cannot be made in-situ. Makes potentially very erroneous assumptions about

the radiated field.

DIFFUSE FIELD TECHNIQUE

(ISO 3741 (3742 - 1/2)) Here, it is assumed that under steady state conditions the rate of sound

power input by the source to the room equals the power dissipated by the

walls. From previous results

W I Ad I p cd 2 4/

where is the space-averaged mean square pressure in the room and

A is the random incidence (Sabine) absorption estimated from the rate of

decay of following transient excitation of the sound field via the

relationship, A=0.161T60/V, where T60 is the time taken for the sound field to

decay by 60dB and V is the room volume.

p2

Advantages: Simple to implement. Uses only measurements of acoustic

pressure.

Disadvantages: Assumes ‘large room’ acoustics, which implies high

frequencies or large rooms. Potentially costly.

p2

INTENSITY-BASED METHODS

(ISO 9614-1 and 9614-2)

Here the normal component of sound intensity normal to a hypothetical

surface enclosing the source is measured directly by the use of a sound

intensity probe.

W I Sni ii

N

1

The intensity estimate at each segment may be made by

either

(i). Point-sampling. A single intensity measurement

at the centre of each segment (ISO9614 – 1)

(ii). Scanning the intensity probe over the each

segment surface (ISO9614 – 2)

SOURCE SUBSTITUTION METHOD

(ISO 3747)

A special reference source of known radiated sound power spectrum

(determined by, for example, one of the methods above) is located in the

position (or as close as possible) to the source under test. Measurements

of the sound pressure level with the reference source are compared with

those due to the source under test. The ratio between sound power and

space average mean-squared pressure is assumed to be identical for

both sources.

L LW Wref

L Lp p ref

Advantages: Very simple to implement.

Disadvantages: Potentially large errors for sources of high directionality in

highly reverberant enclosures. Large errors may also arise

from interference in the pressure measurements from

coherent extraneous sources.

SOURCE SURFACE VIBRATION MEASUREMENT

Radiated sound power is inferred directly from the space-averaged mean

square surface vibration velocity. This can be measured using an

accelerometer, or by using a non-contact sensor such as a laser velocimeter

or a volume velocity transducer.

W v S crad 2

Radiation efficiency differs for different structures and is frequency

dependent. It must be obtained from predicted and measured values

published in the literature.

Advantages: Simple to implement. Offers a non-contact measurement

technique. Technique may be performed in-situ with

extraneous source operating simultaneously.

Disadvantages: Assumes radiation efficiency is known which is unlikely for

complex structures. Requires complicated structure to be

decomposed, arbitrarily, into simpler sub-structures.