Deriving the Acoustic Wave Equation from General Relativity

M. J. Gagen

Physics Department, The University of Queensland, QLD 4072 Australia Email: [email protected] URL: http://www.physics.uq.edu.au/people/gagen/ (Dated: July 27, 2004) Physics is often taught in such a way as to minimize the overall unity and coherency of the field. For instance, the fields of general relativity and special relativity seem to apply to astrophysics and cosmology without making much connection with fluid mechanics and acoustic theory for instance. In reality, all of these apparently distinct fields are one and the same. We demonstrate this by deriving the acoustic wave equation from general relativistic principles. In essence, even the simplest every-day acoustic theory is an approximation to the full general relativistic equations governing the motion of the universe at large.

I. EINSTEIN’S FIELD EQUATIONS

Einstein’s field equations lie at the core of general relativity and specify the relationship between the curvature of spacetime and the density of energy-momentum. In essence,curved spacetime tells matter and energy how to move, while the matter and energy tell spacetime how to curve [1]. The equations themselves are:

Gµν = −8πGTµν . (1) Here, T αβ is the energy-momentum tensor, G is the Gravitational constant while Einstein’s constant [2],the speed of light c is set to unity. In addition,

Gµν = Rµν − 1/2gµν R (2) written in terms of the curvature scalar µκ R = g Rµκ. (3) and the Ricci tensor λ Rµκ = Rµλκ (4) dependent on the Riemann-Christoffel curvature tensor which specifies spacetime curvature via λ λ ∂Γµν ∂Γµκ Rλ = − +Γη Γλ − Γη Γλ , (5) µνκ ∂xκ ∂xν µν κη µκ µη in terms of affine connections

1 ∂gκν ∂gκµ ∂gµν Γλ = gλκ[ + − ] (6) µν 2 ∂xµ ∂xν ∂xκ which are themselves specified in terms of the metric gµν detailing the local characteristics of spacetime at each spacetime point. As mentioned,the Einstein equations specify forces acting on the energy-momentum distribution,and changes in this distribution must satisfy various conservation laws. These include the non-destruction of energy-momentum at any point αβ T ;β =0. (7) and the conservation of mass-energy α J ;α = 0 (8) where J α is the four-vector particle flux.

II. RELATIVISTIC HYDRODYNAMICS

Acoustic waves are normally modelled using perfect fluid (or gas). Further,gravitational fields play little part in the description of sound waves and it suffices to ignore gravitational effects. This allows setting αβ gαβ = ηαβ = η = diag (−1, 1, 1, 1) (9) 2 so the motion of the fluid is described by the conservation equations. The energy-momentum tensor for a perfect relativistic fluid is

T µν = ηµν p +(p + ρ) U µU ν (10) where the velocity four-vector is

U α =(γ,γv) (11) √ 2 with v =(vx,vy,vz)andγ =1/ 1 − v in geometrized units setting the speed of light to unity. Here, p and ρ are the momentarily comoving frame fluid pressure and energy density respectively [3]. A perfect gas has particle flux

J α = N α = nU α (12) where n is the momentarily comoving frame particle number [3]. The conservation equations for energy-momentum and particle flux,after some algebra,are then

D (γn)=0 2 D γ (p + ρ) − ∂tp =0
 D γ2(p + ρ)v + ∇p =0, (13) with D = ∂t + ∇.v and ∇ =(∂x,∂y,∂z). These equations give respectively,particle number conservation,mass continuity and momentum conservation. These equations specify the speed of light as the absolute cosmic speed limit via terms like

a γ ∂tvx = 0 (14) for some index a. As velocity in the x direction increases towards the speed of light vx → c = 1 in the units chosen here,then γ →∞. Yet the equality to zero is strict here despite the growing value of γ which necessarily requires that the rate of change of velocity with time goes to zero ∂tvx → 0asvx → c. Because of this,the velocity can never actually get to the speed of light.

III. NON-RELATIVISTIC FLUID MECHANICS

The above equations can be simplified in the non-relativistic or low velocity limit. In this regime we set v  c =1 the speed of light, γ → 1, p ≈ ρv2,and p + ρ → p. These assumptions give the velocity four vector as

U α =(1, v) (15) and the energy-momentum tensor as

T 00 = ρ T 0j = ρvj T ij = ρvivj + diag (p, p, p) (16) where ρ and p are now the usual fluid density and pressure. Assuming an adiabatic fluid with zero conductivity [4,5] in two dimensions for simplicity,the conservation equations then become the inviscid and dimensionless Euler equations

∂tU + ∂xF + ∂χG = 0 (17)

ρ ρvx 2 γ U =(ρvx ) F =(ρvx + ρ /γ ) ρvy ρvxvy . ρvy G =( ρvxvy ) 2 γ ρvy + ρ /γ

Here and we show mass continuity (top line) and momentum conservation in the x and y directions. Adiabatic perfect gases with p = ργ (γ =1.4) are considered and all variables are dimensionless with dimensioned (primed) variables being given by x = xL, v = va0, p = pp0, ρ = ρρ0 and t = tL/a0 with L being some convenient length parameter 2 and a0 = γp0/ρ0 giving the local speed of sound. 3

IV. ACOUSTIC WAVE EQUATION

The acoustic wave equation is derived from the Euler equations by simply differentiating with respect to time giving γ γ     ∂xxρ ∂yyρ 2 2 0=∂ttρ − − − ∂xx ρv − ∂xy (ρvxvy) − ∂yx (ρvxvy) − ∂yy ρv . (18) γ γ x y

In non-dimensional units,and for an adiabatic fluid, p = ργ (γ =1.4),a Taylor series expansion in pressure gives

p =1+p1 (19) and ∂ρ ∂2ρ (p − 1)2 ρ =1+ |0(p − 1) + |0 ∂p ∂p2 2 1 1 2 =1+p1/γ + ( − 1)p . (20) 2γ γ 1

This Taylor expansion is valid only when p1  1 and we can truncate the expansion at the first term to give small amplitude acoustic theory. The error involved in this truncation equates to the second term and is non-negligible −2 when pressure fluctuations are greater than about p1 > 10 . If we further assume small amplitude pressure disturbance,and small fluid velocities vx,vy  1 the speed of sound, and substitute these results in the differentiated Euler equations,then we derive the acoustic wave equation

∂ttp1 − ∂xxp1 − ∂yyp1. (21)

V. ACOUSTIC MONOPOLE THEORY

For many situations,the acoustic wave equation is to complicated to solve,and further approximations are intro- duced. Acoustic monopole theory considers a vibrating point source modeled as a small spherical emitter of radius R undergoing small oscillations much less than the radius

Rδt˙  R. (22)

The resulting volume changes are then

V˙ =4πR2R˙ (23) which cause spherically symmetric pressure waves to propagate into the surrounding medium. These pressure waves decay as 1/r and have the usual functional dependence on (r − a0t) required to satisfy the wave equation and to describe delayed waves travelling outwards from the sphere at speed a0. Thus,at the point ( r, t),the pressure has functional dependence

p¯(r − a0t) p1(r, t) ∝ . (24) r In spherical coordinates with approximately constant fluid density the conservation of momentum equation is

ρ∂tvr = −∂rp1/γ (25)

2 where we set vr = 0 for low fluid velocities. Substituting Eq. (24) here gives

p¯(r − a0t) ρ∂tvr ∝−∂r γr p¯ ∂rp¯ ∝ − . (26) γr2 γr This solution applies at the surface of the sphere where r = R is small and assumed to be much smaller than the wavelength of sound emitted,allowing us to ignore the second term on the RHS. Noting that

vr = R˙ (27) and

R¨ ≈ V/¨ (4πR2) (28) 4 we have

ρV¨ p¯ ≈ . (29) 4πR2 γR2

In turn,this allows us to solve for ¯p and to substitute this back into Eq. (24) to obtain

γρV¨ p1(r, t) ≈ (30) 4πr relating the pressure wave emitted by a monopole source to the second derivative of the volume change of that source. Dipole and quadropole sources are then constructed from appropriate arrays of monopoles [6]. This equation has been widely applied to,for instance,modelling care tire noise. In this field,it is also commonly assumed that volume changes are simple harmonic,allowing the further substitution

V¨ ∝ V (t). (31)

[1] S. Weinberg. Gravitation and Cosmology: Principles and applications of the general theory of relativity. John Wiley and Sons, New York, 1972. [2] K. Brecher of Boston Univeristy suggested in April 2000 that Einstein be recognized by naming the speed of light “Einstein’s constant”. See News in Brief, Scientific American, July (2000). [3] C. W. Misner, K. S. Thorne, and J. A. Wheeler. Gravitation. W. H. Freeman and Company, San Fransisco, 1973. [4] M. J. Zucrow and J. D. Hoffman. Gas Dynamics. John Wiley and Sons, New York, 1976. [5]Z.U.A.Warsi.Fluid Dynamics: Theoretical and computational approaches. CRC Press, Boca Raton, 1993. [6] P. M. Morse and K. Uno Ingard. Theoretical Acoustics. McGraw-Hill, New York, 1968.