Sound intensity and power

Professor Phil Joseph


. 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 , are not conserved.

. Performance of noise control systems often specified in terms of energy, e.g., , absorption coefficient. INSTANTANEOUS INTENSITY

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

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

t is given by It  ptut

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


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

Net rate of change of energy = Rate of energy in – Rate of energy out E I I  xy  x xy  y xy t x t In 3 - dimensions E I I I  .It  0 .I  x  y  z t x x x RELATIONSHI[P BETWEEN SOUND IINTENSITY AND SOURCE SOUND POWER

Applying Gauss's divergence theorem

.A dV  A.nˆ dS V S to E  .I  0 t



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 I represents a net overall flow of energy and is called active intensity. A value   I 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.




U O S Reactive intensity


Interfering monopoles Intensity at the microphones is

I  p1t p2tu1t u2t

 p1tu1t p2 tu2 t p1tu2 t p2 tu1t

Sum of intensities This term represents generated by each correlations. It is zero for source individually. statistically unrelated Intensities are therefore source strengths, i.e., if the not generally additive. 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. It  ptut p1t p2t

1 pt2p1tp2t t p t  p t p t  1 p t  p t  1  2     2  21   2   ut   dt  0r t Spatial average 1 Fromp1 Euler’sp2 momentum urt  equation d  r r SPECTRAL FORMULATION OF INTENSITY FOR USE WITH FFT ANALYSERS

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

 I   Iteit dt It  ptut  is given by ImG  I   12 r

where G12 is the pressure cross spectrum

 * p   p t eit dt G12   Ep1p2 and 1,2   1,2   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.  1 Normalized error~  BT  , 2 where B is the measurement bandwidth and T is the effective analysis time window. 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.

Source Transmission path Receiver

LW Lp 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

Sound Pressure Sound Intensity Source Source Surface Measurements Measurements Substitution Vibration Measurement Technique

In-situ Free Field Diffuse Field Technique Technique DEFINITION OF SOUND POWER: INTEGRAL FORMULATION

. This integral expression follows from W   I.n dS Gauss’s theorem S . The choice of control surface S is S 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 n I 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 2 1.8






0 2 4 6 8 10 12 14 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

N 2 W   pi / c Si (i.e. LI = LP) i1  This assumes that the wave fronts are planar, or spherical, and lie normal to the measurement surface. The sound power then follows as 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 2 W  Id A Id  p / 4c

where p 2 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. Advantages: Simple to implement. Uses only measurements of acoustic pressure.

Disadvantages: Assumes ‘large room’ acoustics, which implies high frequencies or large rooms. Potentially costly. 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. N W  Ini Si i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.

LW  LWref  

  Lp  Lp 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. 2 W  rad v Sc

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.