Sound Intensity

Sound Intensity#

The intensity of a sound wave is defined as

(33)#

I = \frac{1}{T} \int_{0}^{T} p (t) u (t) d t

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

(34)#

\nabla p (t) = - ρ_{o} \frac{\partial u (t)}{\partial 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)#

\hat{p} = p_{+} \exp i (ω t - k x)

Euler equation

(36)#

\nabla p + ρ_{0} \frac{\partial u}{\partial t} = 0

in x-direction

(37)#

\frac{\partial p}{\partial x} + ρ_{0} \frac{\partial u}{\partial t} = 0

results in particle velocity

(38)#

{\hat{u}}_{x} = - \frac{1}{i ω ρ} \frac{\partial \hat{p}}{\partial x} = \frac{k}{ω ρ} p_{+} \exp i (ω t - k x) = \frac{p_{+}}{ρ c} \exp i (ω t - k x) = \frac{\hat{p}}{ρ c}

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

(39)#

u (t) = u (0) - \frac{1}{ρ} \int_{0}^{t} \nabla p (τ) d τ

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

(40)#

I = - \frac{1}{T} \frac{1}{ρ} \int_{0}^{T} p (t) (\int_{0}^{t} \nabla p (τ) d τ) d t

Using next the Fourier synthesis

(41)#

p (t) = \frac{1}{2 π} \int_{- π}^{π} P (ω) \exp (- i ω t) d ω

and with

(42)#

\int_{0}^{t} \exp (- i ω τ) d τ = \frac{i}{ω} (\exp (- i ω t) - 1)

and therefore

(43)#

\int_{0}^{t} \nabla p (τ) d τ = \frac{i}{2 π} \int_{- π}^{π} (\frac{\nabla P (ω)}{ω}) (\exp (- i ω t) - 1) d ω

we obtain

(44)#

I = - \frac{i}{ρ} \frac{1}{2 π} \frac{1}{2 π} \int_{- π}^{π} \int_{- π}^{π} P (ω^{'}) (\frac{\nabla P (ω)}{ω}) (\frac{1}{T} \int_{0}^{T} \exp (- i ω^{'} t) (\exp (- i ω t) - 1) d t) d ω d ω^{'}

Assume the temporal ingration

(45)#

\frac{1}{T} \int_{0}^{T} \exp (- i ω^{'} t) (\exp (- i ω t) - 1) d t = δ (ω^{'} + ω = 0)

evaluates to one if and only if $ω^{'} + ω = 0$ and to zero otherwise, so that

(46)#

I = - \frac{i}{ρ} \frac{1}{2 π} \frac{1}{2 π} \int_{- π}^{π} \int_{- π}^{π} P (- ω) (\frac{\nabla P (ω)}{ω}) d ω d ω^{'}

or

(47)#

I = - \frac{i}{ρ} \frac{1}{2 π} \int_{- π}^{π} P^{*} (ω) (\frac{\nabla P (ω)}{ω}) d ω

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

(48)#

P (ω) = \frac{P_{2} (ω) + P_{1} (ω)}{2}

(49)#

\nabla P (ω) = \frac{P_{2} (ω) - P_{1} (ω)}{d_{2, 1}}

(50)#

P^{*} (ω) \nabla P (ω) = \frac{1}{2 d_{2, 1}} {(P_{2}^{*} (ω) + P_{1}^{*} (ω)) (P_{2} (ω) - P_{1} (ω))}

(51)#

P^{*} (ω) \nabla P (ω) = \frac{1}{2 d_{2, 1}} {| P_{2} (ω) |^{2} - | P_{1} (ω) |^{2} + P_{2} (ω) P_{1}^{*} (ω) - P_{2}^{*} (ω) P_{1} (ω)}

With $| P_{2} (ω) |^{2} = | P_{1} (ω) |^{2}$

(52)#

P^{*} (ω) \nabla P (ω) = \frac{1}{2 d_{2, 1}} {P_{2} (ω) P_{1}^{*} (ω) - P_{2}^{*} (ω) P_{1} (ω)}

or

(53)#

P^{*} (ω) \nabla P (ω) = \frac{i}{d_{2, 1}} Im {P_{2} (ω) P_{1}^{*} (ω)}

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

(54)#

I_{2, 1} (ω) = - \frac{i}{ρ d_{2, 1}} \frac{P^{*} (ω) Δ P (ω)}{ω} = - \frac{1}{ρ d_{2, 1}} \frac{Im {P_{2} (ω) P_{1}^{*} (ω)}}{ω}

Sound Intensity

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Sound Intensity#

planewave#