Fundamentals of Aeroacoustics A (very) basic lecture on sound and aerodynamics
Umberto Iemma
Department of Engineering, Roma Tre University via Vito Volterra 72, 00146, Roma, Italy
Peking University College of Engineering January 8, 2020 2 Sound in quiescent media 3 Sound in presence of flow 4 Lighthill’s Acoustic Analogy 5 Generalised functions and Ffowcs–Williams and Hawkings Equation 6 Maybe ... Introduction to the numerical challenge
Summary
1 Introduction
U. Iemma Fundamentals of Aeroacoustics 3 Sound in presence of flow 4 Lighthill’s Acoustic Analogy 5 Generalised functions and Ffowcs–Williams and Hawkings Equation 6 Maybe ... Introduction to the numerical challenge
Summary
1 Introduction 2 Sound in quiescent media
U. Iemma Fundamentals of Aeroacoustics 4 Lighthill’s Acoustic Analogy 5 Generalised functions and Ffowcs–Williams and Hawkings Equation 6 Maybe ... Introduction to the numerical challenge
Summary
1 Introduction 2 Sound in quiescent media 3 Sound in presence of flow
U. Iemma Fundamentals of Aeroacoustics 5 Generalised functions and Ffowcs–Williams and Hawkings Equation 6 Maybe ... Introduction to the numerical challenge
Summary
1 Introduction 2 Sound in quiescent media 3 Sound in presence of flow 4 Lighthill’s Acoustic Analogy
U. Iemma Fundamentals of Aeroacoustics 6 Maybe ... Introduction to the numerical challenge
Summary
1 Introduction 2 Sound in quiescent media 3 Sound in presence of flow 4 Lighthill’s Acoustic Analogy 5 Generalised functions and Ffowcs–Williams and Hawkings Equation
U. Iemma Fundamentals of Aeroacoustics Summary
1 Introduction 2 Sound in quiescent media 3 Sound in presence of flow 4 Lighthill’s Acoustic Analogy 5 Generalised functions and Ffowcs–Williams and Hawkings Equation 6 Maybe ... Introduction to the numerical challenge
U. Iemma Fundamentals of Aeroacoustics Introductory notes
In the following we will assume Familiarity with the differential calculus and tensor algebra Basic knowledge of continuum mechanics and fluiddynamics Familiarity with vector and tensor notations Fourier transform and frequency domain representation
U. Iemma Fundamentals of Aeroacoustics What is Acoustics ?
According to American National Standards Institute (ANSI) ... “Acoustics is the science of sound, including its production, transmission, and effects”
Lindsay’s acoustic diagram
U. Iemma Fundamentals of Aeroacoustics What is Acoustics ?
From the physical viewpoint, sounds is associated to the transmission of mechanical energy without a mean displacement of matter . . .
Acoustic waves propagate by means of the transfer of momentum through a zero–mean harmonic motion of the fluid particles. The amplitude of this motion is typically small ⇒ Linear governing equations
U. Iemma Fundamentals of Aeroacoustics What is Acoustics ?
From the perceptive viewpoint, human response to an acoustic perturbation is an extremely complex phenomenon . . .
U. Iemma Fundamentals of Aeroacoustics ... and Aeroacoustics ?
The definition of aeroacoustics is not so well established (no ANSI definition). A general statement widely accepted is: “Aerocoustics is the branch of acoustics that studies the sound generated by and propagating onto a flowing fluid.”
Generation can be induced by turbulence and/or interaction with moving or stationary solid boundaries.
It’s a complex physical phenomenon, extremely challenging from the theoretical and numerical point of view.
U. Iemma Fundamentals of Aeroacoustics ... and Aeroacoustics ?
Complex physics ⇒ Rich spectral content
Broadband noise from turbulence Tonal noise (harmonic sequance) from rotating machineries Tonal noise (inharmonic sequence) from shocks
U. Iemma Fundamentals of Aeroacoustics ACOUSTICS IN QUIESCENT MEDIA
U. Iemma Fundamentals of Aeroacoustics Conservation equations
1 Consider a small control volume 2 Evaluate time variation of mass in integral form n d ZZZ S ρ dV dt V
3 Evaluate time variation of momentum in integral form V d ZZZ ρ vdV dt V
4 indefinitely shrink V to obtain the local form of conservation laws.
U. Iemma Fundamentals of Aeroacoustics Dilatation rate
Conservation equations
Continuity equation The density rate of change of a fluid particle moving at speed v equals the opposite of the divergence of v.
1 Dρ = − ∇ · v ρ Dt
Density rate of change
U. Iemma Fundamentals of Aeroacoustics Conservation equations
Continuity equation The density rate of change of a fluid particle moving at speed v equals the opposite of the divergence of v.
1 Dρ = − ∇ · v ρ Dt
Density rate of change Dilatation rate
U. Iemma Fundamentals of Aeroacoustics Material time derivative
The material (or total) derivative of the scalar quantity f is defined as
Df ∂f = + v · ∇f Dt ∂t It represents the rate of change of f measured by an observer moving at speed v.
U. Iemma Fundamentals of Aeroacoustics Eulerian vs Lagrangian form
Lagrangian form 1 Dρ = −∇ · v ρ Dt Better insight of the physics. Incompressibility is associated to a divergence–free velocity field. Eulerian form ∂ρ + ∇ · (ρv) = 0 ∂t Fully conservative. Advantageous in many numerical applications and analytical manipulations.
U. Iemma Fundamentals of Aeroacoustics External stresses Πij = p δij − σij Body forces
Conservation equations
Conservation of momentum The acceleration of a fluid particle equals the forces acting on it divided by the fluid density.
Dv ρ = − ∇ · Π + f Dt
Acceleration
U. Iemma Fundamentals of Aeroacoustics Body forces
Conservation equations
Conservation of momentum The acceleration of a fluid particle equals the forces acting on it divided by the fluid density.
Dv ρ = − ∇ · Π + f Dt
Acceleration
External stresses Πij = p δij − σij
U. Iemma Fundamentals of Aeroacoustics Conservation equations
Conservation of momentum The acceleration of a fluid particle equals the forces acting on it divided by the fluid density.
Dv ρ = − ∇ · Π + f Dt
Acceleration
External stresses Πij = p δij − σij Body forces
U. Iemma Fundamentals of Aeroacoustics Conservation equations
Conservation of momentum Note that
Πij = p δij − σij (1)
p is the hydrostatic pressure σ is the viscous stress tensor The definition of Π into Cauchy law yields
Dv ρ = −∇p + ∇ · σ + f Dt
U. Iemma Fundamentals of Aeroacoustics Linearization
Assume small perturbations of a reference state p0, ρ0, v0
0 ρ = ρ0 + ρ 0 p = p0 + p 0 v = v0 + v
0 For each quantity • /•0 << 1. Substitute in 1 Dρ = −∇ · v + Q Continuity equation (with a mass source) ρ Dt Dv ρ = −∇p + ∇ · σ + f Momentum equation Dt
U. Iemma Fundamentals of Aeroacoustics Linearization - Medium at rest
Let’s start considering a stagnant, uniform fluid with ρ0 and p0 uniform and v0 = 0. Substitute into conservation equations to the first order in perturbation
∂ρ0 = −ρ ∇ · v0 ∂t 0 ∂v0 ρ = −∇p0 + ∇ · σ0 + f 0 ∂t Note that 0 1 No material derivatives appear since v0 = 0 and convective terms in v are higher order;
2 p0 and ρ0 could be not uniform also for a stagnant fluid (e.g., standard atmosphere)
U. Iemma Fundamentals of Aeroacoustics Wave equation - 1
∂ ∂ρ0 Time derivative of first equation + ρ ∇ · v0 = Q ∂t ∂t 0 Subtract − ∂v0 Divergence of second equation ∇ · ρ = −∇p0 + ∇ · σ0 + f 0 ∂t
∂2ρ0 ∂Q − ∇2p0 = −∇ · f − ∇ · ∇ · σ0 + ∂t2 ∂t
U. Iemma Fundamentals of Aeroacoustics Wave equation - 2
Assume an equation of state in the form
p = p(ρ, S)
The perturbation can be calculated as
0 ∂p 0 ∂p 0 p = ρ + S ∂ρ S ∂S ρ The speed of sound is defined as 2 ∂p c0 := ∂ρ S to yield
0 2 0 ∂p 0 p = c0 ρ + S ∂S ρ
U. Iemma Fundamentals of Aeroacoustics Wave equation - 3
The wave equation for the pressure can be finally obtained
1 ∂2p0 1 ∂p ∂2S0 ∂Q − ∇2p0 = − ∇ · f − ∇ · ∇ · σ0 + + 2 2 2 2 c0 ∂t c0 ∂S ρ ∂t ∂t
U. Iemma Fundamentals of Aeroacoustics Inhomogeneous body forces Non uniform stresses Entropy fluctuations Unsteady mass injection
Wave equation - 4
The wave equation for the pressure can be finally obtained
1 ∂2p0 1 ∂p ∂2S0 ∂Q − ∇2p0 = ∇ · f − ∇ · ∇ · σ0 + + 2 2 2 2 c0 ∂t c0 ∂S ρ ∂t ∂t
D’Alembert operator
U. Iemma Fundamentals of Aeroacoustics Non uniform stresses Entropy fluctuations Unsteady mass injection
Wave equation - 4
The wave equation for the pressure can be finally obtained
1 ∂2p0 1 ∂p ∂2S0 ∂Q − ∇2p0 = ∇ · f − ∇ · ∇ · σ0 + + 2 2 2 2 c0 ∂t c0 ∂S ρ ∂t ∂t
D’Alembert operator Inhomogeneous body forces
U. Iemma Fundamentals of Aeroacoustics Entropy fluctuations Unsteady mass injection
Wave equation - 4
The wave equation for the pressure can be finally obtained
1 ∂2p0 1 ∂p ∂2S0 ∂Q − ∇2p0 = ∇ · f − ∇ · ∇ · σ0 + + 2 2 2 2 c0 ∂t c0 ∂S ρ ∂t ∂t
D’Alembert operator Inhomogeneous body forces Non uniform stresses
U. Iemma Fundamentals of Aeroacoustics Unsteady mass injection
Wave equation - 4
The wave equation for the pressure can be finally obtained
1 ∂2p0 1 ∂p ∂2S0 ∂Q − ∇2p0 = ∇ · f − ∇ · ∇ · σ0 + + 2 2 2 2 c0 ∂t c0 ∂S ρ ∂t ∂t
D’Alembert operator Inhomogeneous body forces Non uniform stresses Entropy fluctuations
U. Iemma Fundamentals of Aeroacoustics Wave equation - 4
The wave equation for the pressure can be finally obtained
1 ∂2p0 1 ∂p ∂2S0 ∂Q − ∇2p0 = ∇ · f − ∇ · ∇ · σ0 + + 2 2 2 2 c0 ∂t c0 ∂S ρ ∂t ∂t
D’Alembert operator Inhomogeneous body forces Non uniform stresses Entropy fluctuations Unsteady mass injection
U. Iemma Fundamentals of Aeroacoustics Dipole Quadrupole
Wave equation - 5
Source terms can be classified on the basis of their radiation propeties
1 ∂2p0 1 ∂p ∂2S0 ∂Q − ∇2p0 = − + − ∇ · f − ∇ · ∇ · σ0 2 2 2 2 c0 ∂t c0 ∂S ρ ∂t ∂t
Monopole
U. Iemma Fundamentals of Aeroacoustics Quadrupole
Wave equation - 5
Source terms can be classified on the basis of their radiation propeties
1 ∂2p0 1 ∂p ∂2S0 ∂Q − ∇2p0 = − + − ∇ · f − ∇ · ∇ · σ0 2 2 2 2 c0 ∂t c0 ∂S ρ ∂t ∂t
Monopole Dipole
U. Iemma Fundamentals of Aeroacoustics Wave equation - 5
Source terms can be classified on the basis of their radiation propeties
1 ∂2p0 1 ∂p ∂2S0 ∂Q − ∇2p0 = − + − ∇ · f − ∇ · ∇ · σ0 2 2 2 2 c0 ∂t c0 ∂S ρ ∂t ∂t
Monopole Dipole Quadrupole
U. Iemma Fundamentals of Aeroacoustics Monopoles
Isotropic radiation, decay with 1/r
U. Iemma Fundamentals of Aeroacoustics Dipoles
Dual–lobes radiation, decay with 1/r 2
U. Iemma Fundamentals of Aeroacoustics Quadrupoles
Four–lobes radiation, decay with 1/r 3
U. Iemma Fundamentals of Aeroacoustics Perfect fluid
Now introduce the assumptions 1 No body forces 2 Inviscid fluid 3 Isentropic flow Wave Equation
1 ∂2p0 − ∇2p0 = Q˙ 2 2 c0 ∂t Fourier transform ⇒ Helmholtz equation
k2p˜0 + ∇2p˜0 = j ω Q˜
U. Iemma Fundamentals of Aeroacoustics Semi–infinite duct. Add linearised momentum
0 0 0 1 x ρ0v˙x + px = 0 ⇒ vx (x, t) = pf t − ρ0c0 c0
Specific impedance
Simple closed–form solutions
1D problem
0 2 0 0 x x p¨ − c0 pxx = 0 ⇒ p (x, t) = pf t − + pb t + c0 c0
U. Iemma Fundamentals of Aeroacoustics Simple closed–form solutions
1D problem
0 2 0 0 x x p¨ − c0 pxx = 0 ⇒ p (x, t) = pf t − + pb t + c0 c0
Semi–infinite duct. Add linearised momentum
0 0 0 1 x ρ0v˙x + px = 0 ⇒ vx (x, t) = pf t − ρ0c0 c0
Specific impedance
U. Iemma Fundamentals of Aeroacoustics Free space Green’s function - TD
Solution of the non–homogeneous WE 1 ∂2G − ∇2G = δ(x − y)δ(t − τ) with p = O(r −1) 2 2 c0 ∂t In 3D δ(t − τ − r/c) G(x, t, y, τ) = = G δ(t − τ − r/c) 4πr 0
Acoustic Delay !
U. Iemma Fundamentals of Aeroacoustics Free space Green’s function - FD
In frequency domain
k2G˜ + ∇2G˜ = δ(x − y)
In 3D −e−i k r G˜(x, y, k) = = G e−i k r 4πr 0 The Green’s function, i.e., the fundamental solution of the problem, is used to represent the solution in integro–differential form. EXTREMELY POWERFUL TOOL !
U. Iemma Fundamentals of Aeroacoustics Boundary Integral Equation formulation
Boundaries at finite distance Imagine to have a domain V bounded by boundaries at finite distance (Obstacles, boundaries of closed cavities ...) ⇒ ∂V = SB ∪ S∞
U. Iemma Fundamentals of Aeroacoustics Boundary Integral Equation formulation
Time Domain
1 ∂2p0 p’ eq. multiplied by G G − ∇2p0 = Q˙ 2 2 c0 ∂t Subtract − 1 ∂2G G eq. multiplied by p’ p0 − ∇2G = δ(x − y)δ(t − τ) 2 2 c0 ∂t Z t Z Integrate over time and space t0 V
Z t Z Z t Z 1 Z t Z 0 0 ¨ ¨0 2 0 0 2 p (y, t) = QGdV(x)dτ + 2 p G − G p dV(x)dτ + G∇ p − p ∇ G dV(x)dτ t0 V t0 V c0 t0 V
Integrate by parts, apply Gauss theorem and homogeneous initial conditions...
Z Z 0 ! 0 ∂p 0 ∂G0 p (y, t) = [Q]r/c G0dV(x) + G0 − p r/c dV(x) V ∂V ∂n r/c ∂n
U. Iemma Fundamentals of Aeroacoustics Boundary Integral Equation formulation
Frequency Domain h i HE multiplied byG˜ G˜ k2p˜0 + ∇2p˜0 = j ω Q˜ Subtract − h i G eq. multiplied by˜p0 p˜0 k2G˜ + ∇2G˜ = δ(x − y) Z Integrate over space V
Z Z p˜0(y, t) = Q˜ G˜dV(x) + G˜∇2p˜0 − p˜0∇2G˜ dV(x) V V
Integrate by parts, apply Gauss theorem ...
! Z Z ∂p˜0 ∂G˜ p˜0(y, k) = Q˜ G˜dV(x) + G˜ − p˜0 dV(x) V ∂V ∂n ∂n
U. Iemma Fundamentals of Aeroacoustics SOUND IN FLOWS
U. Iemma Fundamentals of Aeroacoustics 2 Exchange of loads with steady and moving boundaries 3 Unsteady thermodynamics (not covered today) 4 Unsteady non–linear effects (not covered today) 5 Convection
1 Structured and unstructured vorticity
What’s the physics behind ?
U. Iemma Fundamentals of Aeroacoustics 3 Unsteady thermodynamics (not covered today) 4 Unsteady non–linear effects (not covered today) 5 Convection
What’s the physics behind ?
1 Structured and unstructured vorticity 2 Exchange of loads with steady and moving boundaries
U. Iemma Fundamentals of Aeroacoustics 4 Unsteady non–linear effects (not covered today) 5 Convection
What’s the physics behind ?
1 Structured and unstructured vorticity 2 Exchange of loads with steady and moving boundaries 3 Unsteady thermodynamics (not covered today)
U. Iemma Fundamentals of Aeroacoustics 5 Convection
What’s the physics behind ?
1 Structured and unstructured vorticity 2 Exchange of loads with steady and moving boundaries 3 Unsteady thermodynamics (not covered today) 4 Unsteady non–linear effects (not covered today)
U. Iemma Fundamentals of Aeroacoustics What’s the physics behind ?
1 Structured and unstructured vorticity 2 Exchange of loads with steady and moving boundaries 3 Unsteady thermodynamics (not covered today) 4 Unsteady non–linear effects (not covered today) 5 Convection
U. Iemma Fundamentals of Aeroacoustics Effect of flow
So, let’s remove the assumption of quiescent medium and repeat the procedure already followed. For an inviscid flow we obtain the set of Linearised Euler Equations (LEE) Lagrangian form
Dρ0 + ρ ∇ · v0 + v0 · ∇ρ + ρ0∇ · v = 0 Dt 0 0 0 Dv0 ρ + ∇p0 + ρ v0 · ∇v + ρ0v · ∇v = 0 0 Dt 0 0 0 0
Eulerian form
∂ρ0 + ∇ · ρ0v + v0ρ = 0 ∂t 0 0 ∂v0 ρ + v · ∇v0 + v0 · ∇v + ρ0v · ∇v + ∇p0 = 0 0 ∂t 0 0 0 0
U. Iemma Fundamentals of Aeroacoustics The convective wave equation
LEE and scalar WE LEE are currently widely used for numerical simulation (typically, FEM or FDM) for they capability to capture the effect of non–uniform, compressible aerodynamics.
Nonetheless, it is possible to derive scalar wave equations capable to take into account aerodynamic convection under specific additional conditions.
Without entering into analytical details, we will give some example.
U. Iemma Fundamentals of Aeroacoustics The convective wave equation
Asymptotic stream + aerodynamic perturbation
A first reformulation can be done by setting v0 = U∞ + u. After some manipulation we can write
∂ 2 + U · ∇ p0 − c 2 ∇2p0 = q(x) ∂t ∞ 0
where all the effects of flow non–uniformity are collected in q(x) and moved to the RHS.
U. Iemma Fundamentals of Aeroacoustics The convective wave equation
u = 0 If the aerodynamics reduces to (or can be approximated by) a uniform stream, we obtain
∂ 2 + U · ∇ p0 − c 2 ∇2p0 = 0 ∂t ∞ 0
This formulation is widely used in many engineering application.
U. Iemma Fundamentals of Aeroacoustics The convective wave equation
Convective Green’s function (frequency domain) The Green’s function of this equation has the form
−1 j k (ˆr+M∞·r) q Gˆ(x, y, k) = e β2 ,r ˆ = β2r 2 + (M · r)2, β2 = 1 − M2 4πrˆ ∞ ∞ The corresponding Boundary Integral Formulation in frequency domain is
I 0 0 ˆ ∂p n 0 E(y)p (y) = G − M∞M∞ · ∇p dS S ∂n I " ˆ !# 0 n ˆ n ˆ ∂G + p 2jkM∞G + M∞M∞ · ∇G − dS S ∂n
U. Iemma Fundamentals of Aeroacoustics The convective wave equation - Potential Aerodynamics
Let’s assume that ∇ × v0 = 0 ⇒ v0(x) = ∇Φ(x) In addition, let’s consider the acoustic potential v0 = ∇ϕ as unknown. It can be easily shown that LEE reduce to ∂ ρ0 ∂ϕ ρ0 ∂ϕ − 2 + v0 · ∇ϕ + ∇ · ρ0 ∇ϕ − 2 v0 + v0 · ∇ϕ = S ∂t c0 ∂t c0 ∂t with
1 2 γ−1 2 2 %0 γ − 1 v0 c0 γ − 1 v0 = 1 − 2 v0 · ∇ϕ + , 2 = 1 − 2 v0 · ∇ϕ + ρ∞ c∞ 2 c∞ c∞ 2
Widely used in FEM implementation of the CWE.
U. Iemma Fundamentals of Aeroacoustics Doppler effect
Let’s consider a plane wave propagating in a uniform stream with speed v and two observers, one at rest and the other one comoving with the stream.
0 0 0 In the moving system p(x0, t0) ≡ ei(k·x −ω t ), with x0 = x − vt, t0 = t. It follows
0 0 0 p(x, t) = ei(k·x−ω t −k·vt) = ei(k·x−ω t) with ω = ω0 (1 − k · v) = ω0 (1 + Mcosθ)
U. Iemma Fundamentals of Aeroacoustics Doppler effect
For an isotropic point source ...
U. Iemma Fundamentals of Aeroacoustics The Lighthill’s analogy
Recast the equation describing the generation and propagation of noise by unsteady flows in the form of non–homogeneous wave equation 1 Take time derivative of continuity 2 Subtract divergence of momentum equation 2 2 3 Subtract c0 ∇ ρ from both sides
∂2ρ ∂2T − c2∇2ρ = ij where T = ρv v + (p − c2ρ)δ 2 0 ij i j 0 ij ∂t ∂xi ∂xj
SOUND AND PSEUDO–SOUND !!
U. Iemma Fundamentals of Aeroacoustics Sound and pseudo–sound !!
Convected vs propagating pressure fluctuations
Separation of the two effects from measures and simulations in the near field is one of the hot topics !!
U. Iemma Fundamentals of Aeroacoustics Moving Boundaries: Ffowcs–Williams ad Hawkings Eqaution
The problem is reformulated making use of generalised functions.
Generalised functions (or distributions) extend the concept of differentiable function also in those cases where classic derivarives do not exist. USEFUL TOOL TO TREAT DISCONTINUITIES.
2 Dirac delta function δ(x) = lim e−nx pn/π n→∞
Z ∞ δ(x − a)g(x)dx = g(a) −∞ Z ∞ X g(xk ) δ(h(x))g(x)dx = 0 xk = roots ofh |h (xk |) −∞ k
U. Iemma Fundamentals of Aeroacoustics Moving Boundaries: Ffowcs–Williams ad Hawkings Eqaution
The problem is reformulated making use of generalised functions.
Generalised functions (or distributions) extend the concept of differentiable function also in those cases where classic derivarives do not exist. USEFUL TOOL TO TREAT DISCONTINUITIES.
2 2 Heaviside function H(x) = lim 1/2(tanh(nx) + 1)e−x /n n→∞
0 if x < 0 H(x) = 1/2 if x = 0 1 if x > 0 d d dH df df H(x) = δ(x), H(f (x)) = = δ(f ) dx dx df dx dx
U. Iemma Fundamentals of Aeroacoustics Moving Boundaries: Ffowcs–Williams ad Hawkings Eqaution
The geometry of a moving, solid boundary is described by the function F (x, t) = 0 with F < 0 inside it
Define the Heaviside function H(F (x, t)). It follows that
∇H = δ(F ) ∇F H˙ = δ(F ) F˙
U. Iemma Fundamentals of Aeroacoustics Moving Boundaries: Ffowcs–Williams ad Hawkings Eqaution
1 Mass and momentum conservation laws are multiplied by H(F ) 2 The differentiation properties of H(F) are applied
∂2ρ0H ∂ − c 2∇2(ρ0H) = ∇ · [∇ · (T H)] − ∇ [(p I + σ) ∇F δ(F )] + ρ F˙ δ(F ) ∂t2 0 ∂t 0
The motion of the boundary appears explicitly (and elegantly) as F˙ = vB · n
Typically solved through an integral formulation...
U. Iemma Fundamentals of Aeroacoustics Moving Boundaries: Ffowcs–Williams ad Hawkings Eqaution
Typically used in integral form...
with Vn = F˙ = vB · n
Using the properties of generalised functions yields a retarded integro–differential equation
U. Iemma Fundamentals of Aeroacoustics Essential Bibliography
1 Allan D. Pierce, Acoustics: An Introduction to Its Physical Principles and Applications, 3rd edition, Springer, 2019, ISBN 3030112144. 2 Philip M. Morse and K. Uno Ingard, Theoretical Acoustics, Princeton University Press, 1986, ISBN 0691024014 3 S. W. Rienstra and A. Hirschberg, An Introduction to Acoustics, Eindhoven University of Technology, Eindhoven, 2009. Available online. 4 S. W. Rienstra and A. Hirschberg, An Introduction to Aeroacoustics, Eindhoven University of Technology, Eindhoven, 2004. Available online. 5 D.G. Crighton, A.P. Dowling, J.E. Ffowcs Williams, M. Heckl and F.G. Leppington, Modern Methods in Analytical Acoustics, Springer-Verlag London, 1992. 6 U. Iemma and V. Marchese, AcouSTO: User Manual and Book Of Tutorials, http://acousto.sourceforge.net
U. Iemma Fundamentals of Aeroacoustics Acknowledgments
Thank you for your attention!
U. Iemma Fundamentals of Aeroacoustics