Sound Intensity

Contents

Sound Intensity#

The intensity of a sound wave is defined as

(33)#I=1T0Tp(t)u(t)dt

the relation between particle velocity u(t) and pressure gradient is given by (Euler’s equation of motion)

(34)#p(t)=ρou(t)t

which expresses the observation that particle have a mass that will resist a change in speed under the influence of an applied force (inertial reaction).

planewave#

Assume a plane wave that propagates in the x-direction

(35)#p^=p+expi(ωtkx)

Euler equation

(36)#p+ρ0ut=0

in x-direction

(37)#px+ρ0ut=0

results in particle velocity

(38)#u^x=1iωρp^x=kωρp+expi(ωtkx)=p+ρcexpi(ωtkx)=p^ρc

The paricle velocity is obtained by integration and using the Euler approximation to the pressure gradient one gets

(39)#u(t)=u(0)1ρ0tp(τ)dτ

Using u(0)=0 one ontains fot the sound intensity

(40)#I=1T1ρ0Tp(t)(0tp(τ)dτ)dt

Using next the Fourier synthesis

(41)#p(t)=12πππP(ω)exp(iωt)dω

and with

(42)#0texp(iωτ)dτ=iω(exp(iωt)1)

and therefore

(43)#0tp(τ)dτ=i2πππ(P(ω)ω)(exp(iωt)1)dω

we obtain

(44)#I=iρ12π12πππππP(ω)(P(ω)ω)(1T0Texp(iωt)(exp(iωt)1)dt)dωdω

Assume the temporal ingration

(45)#1T0Texp(iωt)(exp(iωt)1)dt=δ(ω+ω=0)

evaluates to one if and only if ω+ω=0 and to zero otherwise, so that

(46)#I=iρ12π12πππππP(ω)(P(ω)ω)dωdω

or

(47)#I=iρ12πππP(ω)(P(ω)ω)dω

Approximating the sound pressure and pressure gradient by pressure measurements of 2 sensors labeled (1,2)

(48)#P(ω)=P2(ω)+P1(ω)2
(49)#P(ω)=P2(ω)P1(ω)d2,1
(50)#P(ω)P(ω)=12d2,1{(P2(ω)+P1(ω))(P2(ω)P1(ω))}
(51)#P(ω)P(ω)=12d2,1{|P2(ω)|2|P1(ω)|2+P2(ω)P1(ω)P2(ω)P1(ω)}

With |P2(ω)|2=|P1(ω)|2

(52)#P(ω)P(ω)=12d2,1{P2(ω)P1(ω)P2(ω)P1(ω)}

or

(53)#P(ω)P(ω)=id2,1Im{P2(ω)P1(ω)}

and finally the spectral sound intensity along sensor pair (1,2) is approximated by

(54)#I2,1(ω)=iρd2,1P(ω)ΔP(ω)ω=1ρd2,1Im{P2(ω)P1(ω)}ω