Noise Equations

p’/rho0c0

v’ =

p’ = sqrt(2)prms

p’ from prms

drho/dt + grad dot pU = 0

conservation of mass equation

drho’/dt + rho0 grad dot U = 0

linearized mass 

rho0 dU’/dt + gradp’ = 0

linearized momentum

1/c0(d²p’/dt) - grad²p’ = 0

wave equation

p’ = c0² rho’

p’ rho’ relation

p’(t) = sum Ancos(wnt + phin)

p’(t) summation

F(w) = int f(t)e^-iwt dt

fourier transform

e^iw0t - e^-iw0t

sin(w0t)

int f(t) delta(t-t0) dt = f(t0)

shifting property of dirac delta

F(w)e^-iwt0

fourier transform f(t-t0)

F(w-w0)

fourier transform f(t)e^iw0t

1/abs(a) F(w/a)

fourier transform f(at)

(iw)^nF(w)

Fourier transform dnf(t)/dtn

dnF(w)/dwn

fourier transform (-it)^nf(t)

1

fourier transform dirac delta

fupp = 2^(1/N)flow

1/n octave band

k²P(x,w) +grad²P(x,w) = 0

helmholtz equation

k = w/c0 = 2pif/c0 = 2pi/lambda

k in the helmholtz equation aka wave number

I = p’V’ = p’n’nhat = (p’)²nhat/rho0c0

acoustic intensity

Pac = surface int IdS

acoustic power

1/r

1-d solution decays with

1/sqrt(r )

cylindrical waves decay with

V’ = grad phi

assumption we make to get wave equation in velocity potential, no vorticity

Q(x)H(t-t0)

right side of mass equation for continuous sources starting at t0

dirac delta

derivative of heaviside

mass = Q

mass source effects equation monopole

momentum = F

momentum source effects dipole

dQ/dt - gradF

right side of wave equation with monopole and dipole sources 

p’(x,t) = 1/4pic0² volume int s(y, t- abs(x-y)/c0) dv/abs(x-y)

using free space greens function to find p’

d2rho’/dt2 - c0² d2rho’/dxidxi = d2Tij/dxidxj

lighthills wave equation

Tij = rhouiuj + (p’-c0²rho’)deltaij - tauij

lighthill stress tensor, turbulence, entropy fluct, viscous stresses