Notes on Classical Sonic Boom Theory

Harvey S. H. Lam

Professor Emeritus

Department of Mechanical and Aerospace EngineeringPrinceton University, Princeton, NJ 08544

U. S. A.

email: lam@princeton.eduhttp://www.princeton.edu/∼lam

March 22, 2001

Abstract

A supersonic aircraft generates a wave system which propagatesaway from the aircraft. This wave disturbance in the far-field is calledthe sonic boom. The classical sonic boom theory is capable of pre-dicting the strength of the sonic boom in the far-field of a uniformatmosphere as a function of the weight and the length of the aircraft,

and the geometry of the wings and fuselage. Recently, the questionis asked: if one is allowed to inject disturbances ahead of the air-craft (by energy addition using lasers or electron beams), could onesubstantially reduce the strength of the sonic boom in the far-field?

These notes briefly summarizes the classical sonic boom theory in

a uniform atmosphere.1 It is hoped that it can be helpful to thosewho are assessing the question of current interest. The latest versioncan be found on the internet at

http://www.princeton.edu/∼mae/SHL/SonicBoom.pdf

1A version dated earlier than March 20, 2001 was distributed to a limited number ofpeople. It contains a minor error: the expression for κ(Mo) was incorrect. If you have acopy of the earlier version, please destroy it.

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1 Introduction

The continuity equation is:

1

ρ

Dρ

Dt+∇ · q = 0 (1)

where q is the fluid velocity vector. Assuming thermodynamic equi-librium, we have:

Dp

Dt= a2Dρ

Dt+

(∂p

∂s

)

ρ

Ds

Dt= a2 Dρ

Dt+ O(∆s) (2)

where a is the isentropic speed of sound:

a2 = (∂p/∂ρ)s = γRT = γp/ρ. (3)

The O(∆s) term represents the effects of local increase of fluid entropy(along a streamline). For example, localized heat injection ahead of

the aircraft, using electron beams or lasers, could be represented bydelta functions. Using (2) in (1), with the help of (3), we have:

1

ρa2

Dp

Dt+∇ · q = O(∆s). (4)

The momentum equation is:

Dq

Dt= −1

ρ∇p + ν∇2q (5)

where ν is the kinematic viscosity. We are temporarily keeping theviscous term (even though it is negligible outside of shock waves) inanticipation of later developments. We shall assume that the incomingfreestream has uniform entropy, and that the disturbance are “small.”Under conditions of practical interest, the flow is expected to be ho-

mentropic and irrotational.We now introduce the perturbation velocity potential φ:

q = Uo(ex +∇φ). (6)

Equation (4) becomes:

1

ρa2

Dp

Dt+∇2φ = O(∆s) (7)

2

For steady flow, the substantial derivative is simply:

D

Dt= q · ∇. (8)

We can compute the substantial derivative of p using the momentumequation as follows:

Dp

Dt=

ρq · ∇p′

ρ(9)

= ρq ·

−q·∇q︷ ︸︸ ︷q× (∇× q)−∇(

q · q2

)+ν∇2q

(10)

= ρq ·(−q · ∇q + ν∇2q

)(11)

We now use cylindrical coordinates (x, r, θ), and denote the compo-nents of the velocity vector by (u, v′,$′). We shall assume that in thefar-field the θ dependence and $ are both negligible. We have:

1

ρ

Dp

Dt≈ −u(u

∂u

∂x+ v′

∂v′

∂x)− v′u

∂u

∂r+ νu∇2u + O(...) (12)

where O(...) represents all “higher order” terms, including those causedby local entropy production ∆s.

The leading order approximation (keeping only linear terms) to

(12) is:1

ρ

Dp

Dt≈ −U 2

o∂2φ

∂x2+ O(...) (13)

Using this in (7), we have the linear governing equation for φ:

(1−M2o )

∂2φ

∂x2+

1

r

∂

∂r

(r∂φ

∂r

)+ λ∇2

(∂φ

∂x

)= O(...) (14)

where λ, which has the dimension of length (of the order of a few

mean free paths in units of L), is defined by

λ ≡ νUo

a2o

, (15)

and Mo is the flight Mach number:

Mo ≡Uo

ao. (16)

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The next approximation to (12) (neglecting third order and higherterms) is:

1

ρ

Dp

Dt≈ −u2 ∂2φ

∂x2− 2U 2

o

∂φ

∂r

∂2φ

∂x∂r+ νUo∇2

(∂φ

∂x

)+ O(...) (17)

Equation (14) can now be written as:

(M2 − 1)∂2φ

∂x2− 1

r

∂

∂r

(r∂φ

∂r

)+ 2M2

o∂φ

∂r

∂2φ

∂x∂r= λ∇2

(∂φ

∂x

)+ O(...).

(18)This is now a nonlinear PDE. In addition to the new last term onthe left hand side, the other nonlinear term comes from M (in the

paranthesis of the first term); it is the local flow Mach number, u/a,and not Mo. The small difference between these two Mach numbersis crucial to the far-field theory.

2 Characteristic Coordinates

We now introduce new independent variables (ξ, η):

ξ = x− r√

M2o − 1, (19)

η = x + r√

M2o − 1, (20)

and consider the flow field in the neighborhood of the Mach cone in

the far-field:ξ = O(1), η >> 1. (21)

We further assume that the dependence of φ on η for fixed ξ is weak,so that taking partial derivative with respect to η lowers the order ofmagnitude of the entity. We thus have:

∂

∂x=

∂

∂ξ+

︷︸︸︷∂

∂η(22)

∂2

∂x2=

∂2

∂ξ2+ 2

∂2

∂ξ∂η+

︷︸︸︷∂2

∂η2(23)

∂

∂r=

√M2

o − 1

− ∂

∂ξ+

︷︸︸︷∂

∂η

(24)

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∂2

∂r2= (M2

o − 1)

∂2

∂ξ2− 2

∂2

∂ξ∂η+

︷︸︸︷∂2

∂η2

(25)

∂2

∂x∂r=

√M2

o − 1

− ∂2

∂ξ2+

︷︸︸︷∂2

∂η2

(26)

Neglecting terms with overbrace in the far field, we can show that theleading order is:2

∂P

∂η+

P

2η− κP

∂P

∂ξ= ε

∂2P

∂ξ2+ O(...,

1

η2) (27)

where

P = −∂φ

∂x=

p′

ρoU2o, (28)

κ =M4

o (γ + 1)

4(M2o − 1)

= κ(Mo), ε =λ

4(M2o − 1)

. (29)

Now we further introduce new dependent and independent vari-ables:

X = ξ, τ =√

2η, P =Φ(X, τ ; θ)

κτ. (30)

The final leading order far-field equation for Φ is:

∂Φ

∂τ− Φ

∂Φ

∂X= ετ

∂2Φ

∂X2+ (...) (31)

which is the well known Burgers’ equation (which usually omits the τdependence on the right hand side).

To simplify the notations, we shall apply the initial condition atτ = 0 (instead of τ = O(1)):

Φ(X, 0; θ) = f(X). (32)

2Physically, the “new” approximation being made here is: only outgoing waves propa-gating along ξ=cosntat are being considered. All terms responsible for waves propagatingalong η=constant are neglected. The original derivation was given by G. B. Whithamin Proc. Roy. Soc. A201, 89, (1950). The derivation of κ(Mo) requires much algebra,particularly in getting the relation M2 = M2

o + M2o (2 + (γ − 1)M2

o )∂φ/∂x via the energyequation. When Sir Lighthill presented the derivation in Volume 6 of the Princeton HighSpeed Series, General Theory of High Speed Aerodynamics edited by W. R. Sears, he saidon page 436: “(Whitham) showed that the equation of continuity can be approximatedwith some appearance of reason ...”.

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where f(X) for (31) is to be derived from classical linear wing theoryvalid in the near field3, and it is expected to depend on Mo, thegeometry and the appropriately normalized weight of the aircraft and

the azimuthal angle θ. The needed “boundary conditions” are thevanishing of Φ for X < 0 upstream of the Mach cone from the forward“tip,” and X > 1, (sufficiently) far downstream of the Mach cone fromthe “tail” of the aircraft.

As it now stands, the far-field solution Φ(X, τ >> 1; θ) is governed

by (31) and the only input information needed is f(X). The questionis:

Can one introduce some disturbances in front of f(X) (causedby ∆s due to energy addition) so that the far-field solutionΦ(X, τ >> 1; θ) is substantially weaker than having theoriginal f(X) alone?

3 Properties of Burgers’ Equation

Burgers’ equation has some well-known properties.

3.1 Exact solutions

The following are exact solutions which can easily be verified:

Φ = C = constant (33)

Φ =X∗ −X

τ − τ∗, X∗, τ∗ = constants. (34)

3.2 Exact Inviscid Solutions

The coefficient ετ of the second order term in the Burgers’ equation is,

from the practical point of view, always a small number even when weconsider τ to be “large.” When the right hand side of (31) is neglected,we have the iniviscid Burgers’ equation which can be solved exactly.The result is difficult to express in equations, but it is simple to statein words:

A solution point (Φ,X) on the Φ, X plane moves with con-stant horizontal velocity −Φ.

3See G. N. Ward’s book on Linearized Wing Theory.

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Thus, positive solution points moves to the left, while negative solutionpoints move to the right. The exact solution as given by (34) shows thestraight solution line rotates (about X∗ with increasing time) counter-

clockwise when τ > τ∗, and clockwise when τ < τ∗. Note that whenτ∗ is positive (the slope of the solution line is positive), the solutionbecomes singular (i.e. a shock forms) at τ = τ∗.

3.3 Conservation Laws and Shocks

Integrating (31) with respect to X while holding τ fixed between X =A and X = B, we have:

d

dt

∫ B

AΦ(X, τ ; θ)dX =

[Φ2

2+ ετ

∂Φ

∂X

]X=B

X=A

. (35)

• Since the boundary condition for Φ requires that Φ vanishes atupstream and downstream infinity, we have:

∫ −∞

∞Φ(X, τ ; θ)dX = I∞ = constant (36)

• If the viscous term is neglected, i.e. in the ετ → 0 limit, thearea under the solution curve between two zero crossing points(A and B) is conserved:

∫ B

AΦ(X, τ ; θ)dX = I(A,B) = constant (37)

provided Φ(X = A) = Φ(X = B) = 0, even if one or moreshocks (see next bullet) are present between A and B.

• If we change to a moving frame (Z, τ) with constant velocty Us

in the −X direction,

Z = X + Usτ, τ = τ. (38)

(31) becomes:

∂Φ

∂τ− (Φ −Us)

∂Φ

∂Z= ετ

∂2Φ

∂Z2+ (...). (39)

Integrating this equation with respect to Z while holding τ con-

stant as before, we have:

d

dt

∫ B

AΦ(Z, τ ; θ)dZ =

[(Φ −Us)

2

2+ ετ

∂Φ

∂X

]Z=B

Z=A

. (40)

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Now, let A and B straddles a discontinuity (i.e. a shock). Theleft hand side vanishes in the limit as B → A; hence the righthand side must also vanish. Neglecting the viscous term, we

obtain the “jump condition” for a finite strength shock wave:

Us =ΦA + ΦB

2. (41)

In other words, a discontinuity moves with velocity equal to the

average values of Φ straddling the discontinuity. If Us is positive,it moves to the left, otherwise it moves to the right.

• Given a continuous initial condition f(X) which is positive (with-out loss of generality) between two zero crossing points (XL, XR).An incipient shock will emerge when τ reaches the reciprocal of

the maximum positive slope of f(X) inbetween. Marking thepoints which will become the front and the rear of an emergedshock at some later time τ = τ∗ by (XA,ΦA) and (XB, ΦB) ona f(X) vs. X plot, it can be shown that XA, XB and τ∗ arerelated by: ∫ XB

XA

(f(X)− X −X∗

τ∗

)dX = 0 (42)

where

τ∗ =ΦB −ΦA

XB −XA> 0 (43)

X∗ = XA −ΦA(Xb −XA)

ΦB −ΦA, (44)

and ΦA = f(XA), ΦB = f(XB). Geometrically, (42) says thatthe net area between f(X) and a straight line (with positiveslope) joining the front and rear points of a shock (when τ =τ∗ > 0) is zero.

3.4 The N-Wave

Consider the initial condition:

f(X ; θ) = σ(1

2−X), |X | ≤ 1, σ > 0, (45)

f(X ; θ) = 0, |X| > 1, (46)

where σ is some function of Mo, weight and geometry of the aircraftand the azimuthal angle θ. In other words, the initial condition looks

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like the capital letter N: there is a shock at X = −1, followed by anexpansion fan (represented by a straight line with negative slope), andthen followed by another (and relatively weaker) shock at X = +1.

Comparing this initial condition with the known exact solution (34),we have X∗ = 0.5 and τ∗ = −1/σ < 0.

Simply calculation yields the following conserved quantities:

I∞ = σ, (47)

I(−1,1/2) =9σ

8, (48)

I(1/2,1) = −σ

8(49)

The solution is given by a straight line which rotates counterclock-wise,and is straddled by two shock waves which moves away from thecenter. The position of the shock wave is easily determined by honor-ing the conservation laws:

Φ(X, τ ; θ) =12 −X

τ + 1/σ, XF (τ) ≤ X ≤ XR(τ), (50)

where XF (τ) and XR(τ) are the positions of the front and rear shocks(determined by honoring the conservation laws):

XF (τ) =1− 3

√1 + στ

2, (51)

XR(τ) =1 +

√1 + στ

2. (52)

The front and rear shock strengths are thus given by:

ΦF = Φ(XF , τ ; θ) =3σ

2√

1 + στ, (53)

ΦR = Φ(XR, τ ; θ) = − σ

2√

1 + στ. (54)

Note that both ΦF and ΦR decay with increasing τ . The total distancebetween the front and rear shock is:

XR −XF = 2√

1 + στ. (55)

The evolution of the N-wave can thus be described as follows:

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The null point separating the front compressive lobe fromthe rear expansive lobe, X = 0.5, remains fixed in time.The linear part of the solution rotates counterclockwise, its

slope decreasing with time. The shocks move away from thenull point parabolically. The strength of the front shock,as represented by PF , has the following τ dependence:

PF =Φ(XF (τ), τ ; θ)

κτ=

3σ

2κ

1

τ√

1 + στ(56)

Now, τ is related to r by:

τ =√

2η =

√2(x + r

√M2

o − 1) (57)

=

√2(ξ + 2r

√M2

o − 1) (58)

≈ 2(M2o − 1)1/4r1/2 (59)

So the r dependence of ΦF is given approximately by:4

PF =3σ

4κ

1

(M2o − 1)1/4r1/2

√1 + 2σ(M 2

o − 1)1/4r1/2(60)

Thus, the decay of ΦF with respect to r starts with r−1/2 inthe intermediate far-field as given by the linear theory, andgradually shifts over to the r−3/4 power for asymptoticallylarge r for the far far-field via nonlinear effects.

The quantitative behavior of the rear shock is entirely sim-ilar.

It is interesting to note that if some disturbance were intro-duced in front of the compressive lobe, the far-field solution

for the compressive lobe is not affected until its front shockcatches up with the injected disturbance, while the far fieldsolution for the expansive lobe is totally unaffected.

3.5 What Happens When ετ = O(1)?

We then have the full (viscous) Burgers Equation. The discontinuitiesare now smooth transitions (with thickness O(ετ)). While I∞ remains

conserved, the area under two zero-crossing points of f(X) will nolonger be conserved—it will decay.

4Note that both σ and κ depends on Mo. The dependence of the amplitude of ΦF onMo is not explicitly shown here.

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4 Wave Drag and f (X ; θ)

The wave drag Dw of a supersonic aircraft can be computed fromthe perturbation x-momentum flux out of a large cylindrical control

volume centered at the aircraft:

Dwave = − limr→∞

∫ +∞

−∞

∫ 2π

0ρou

′v′rdθdx (61)

where v′ here represents radial perturbation velocity. Using the linearintermediate field solution, we have:

Dwave ∝ ρU 2o A

∫ +∞

−∞

∫ 2π

0f2(X; θ)dθdX (62)

where A has the dimension of area (when f and X are dimensionless).

The important point is: Dw is always positive, and any increase of theoriginal f(X ; θ) will always increase the wave drag.

In additional to the wave drag, there is also a perturbation x-momentum flux out of the rear control surface of this big cylindricalcontrol volume. But this “wake” drag, which includes the vorticity

(induced) drag due to lift, contributes nothing to the sonic boom.

5 Discussion

In practical application of this, one must remove the uniform atmo-

sphere simplification. This generalization introduces much mathemat-ical difficulty, for the effects of defraction involves ray tracing whichis essentially numerical. For the issues being addressed here, it is rea-sonable to assume that any effort that wish to claim a potential ofsuccess for the non-uniform atmosphere case must pass the uniform

atmosphere test first.If one takes the position that the above (classical) sonic boom

theory is essentially correct, then it appears that the route to achievesonic boom reduction is through modification of f(X).

See http://www.princeton.edu/∼mae/SHL/Heat.pdf to see a

discussion on the wave drage generated by volume heat addition ina supersonic flow field and its cancellation by a “wake thrust” in thewake.

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6 Epiloque

Everything I know on sonic boom theory was taught to me by Profes-

sor Wallace D. Hayes in his lengendary course 523, 524 (Fundamentalsof Gas Dynamics and Advanced Gas Dynamics) in 1954-1955. I recallthe elegance and pride of authorship of Wally’s lectures on the sonicboom theory, including his famed “equivalent bodies of revolution,”the concept that for a whole aircraft, the radiation (and wave drag)

at each azimuthal angle θ can be represented by an equivalent bodyof revolution, its radius of revolution is computed by integrating thesources along the intersection of the forward Mach cone from the pointof observation with the actual aircraft itself. This concept was laterexperimentally verified by Whitcomb, and was the theoretical foun-

dation for the “coke bottle” shape of some supersonic fighters andbombers and the lumpy first class lounge of the Boeing 747.

J. M. Burgers came to Princeton in either 1955 or 1956 to give aBaetjer Seminar on his recent work on what is now called the Burger’sequation (the application in his mind was turbulence). I vividly recall

the vivid childish glee on his face when he presented the “equal area”algorithm for locating emerging discontinuities.

Wally passed away peacefully in March, 2001. I dedicate thesenotes to him.

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