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New York University

Institute of Mathematical Sciences
25 Waverly Place, Nemc York 3, N.Y.

Keller, Joseph B.

Geometrical acoustics I:
The theory of weak shock waves

ira^-^TYU 188
January 1953


Joseph B. Keller

I. Introduction

II. Discontinuity Conditions in Continuum Mechanics

A. Fundamental Equations

B. Shocks, Contact Discontinuities and Phase Change Fronts

C. Perfect Fluids

III. Geometrical Acoustics - Weak Shocks

A. The Variational or Acoustic Equations

B. Acoustic Discontinuities

C. Wavefronts and Rays

D. Variation of Shock Strength Along a Ray

E. Reflection and Transmission of an Acoustic Shock

at a Contact Discontinuity

F. Expansion Ratio for Straight Rays

G. Example: The Shock Tube

This report represents results obtained at the Institute
for Mathematics and Mechanics, Mew York University, under
the auspices of Contract Nonr-285( 02) .


INSTITUTE or ■ ' ■ ■'" • ' \T1CAL SCIENCES

25 Waverly t'lace, New York 3, N. Y.


(^-eometrical Acoustics I: The Theory of Weak Shock Waves''

Jose^oh B. Keller

I . Introduction

One of the main difficulties in the theoretical analysis
of shock wave problems is that the shock front and the flox-j
behind it must be determined simultaneously since they affect
each other. However, for very weak (acoustic) shocks the
interaction is so slight that the motion of the shock front and
the variation in its strength can be determined indeuendently
of the rest of the flow. This fact is deduced in the present
report, and the main conseauences of it are also determined.
The result is that weak shocks can be analyzed by the methods
of geometrical optics, and for this reason the theory of weak
shocks may be called geometrical acoustics. Another acoustical
phenomenon- -the propagation of periodic sound waves of high
frequency - can also be analyzed by the methods of geometrical
acoustics, and this analysis will be presented in another

Although it has often been noticed that certain shock
wave phenomena could be described in optical terms, there does
not seem to be any complete derivation of geometrical acoustics
in the literature. The present derivation begins with a
deduction of the discontinuity conditions for curved
discontinuities in any continuous medliom, and continues with
the analysis of the possible discontinuities in arbitrary
media. Then the acoustic equations are deduced as the
variational equations of the nonlinear fluid dynamic equations.
In the same way the acoustic discontinuity conditions are
obtained, and it is shox^rn how the geometrical optics,
considerations of wavef rents, rays, etc. apply to the
determination of shock waves in perfect fluids. Finally as one
of the main results, the variation of shock strength along a ray

A revision of notes on lectures on Mechanics of Continuous
Media delivered at New York University in the fall of 19^9 and
at the University of California (Berkeley) in June 1951.


is determined. This is extended to the determination
of the reflection and transmission coefficients for shocks
incident on contact discontinuities (boundaries).

For the most effective use of the present theory it is
important to take account of diffracted rays and wavefronts,
which can bo done by slightly refining the methods of ordinary
geometrical optics. An explanation of this refinement will be
presented in a future report, although use has alroadj'" been
made of it (see, for example, Reflection and Diffraction of
Pulses by Wedges and Corners, Albert Blank and Joseph 3. Keller,
Comm. Pure Appl. Math., June 19'^1, Vol. IV, a'o. 2).

The present analysis is patterned after that of
R. K. Luneberg's study of the propagation of discontinuities in
electromagnetic theory (see, for example, his Mathcaatical
Theory of Optics, Brown University or his Propagation of
Electromagnetic Waves, Now York University) .

1 1 . Discontinuity Condi t ion s In Cont inuuzn ¥c ch anics

A. Fundamental Equations

The fundamental equations of continuum mechanics, in
their integral form, are

(1) ^ P^V + I p(u^-v^)dS^ = Mass




'T^,.-pu^(u.-v. ; ]dS . Linear
'•' J J- J Komcntijm

(3) ^J p(e+^u^)dV=j pf^u^dV Energy


+ j [T.,u.-q.-p(e + |u2)(u.-v.)]dS.



^J o(^i+l"i+2-^l+2^H+l)^^^ Angular

y j'lomentum


- P-^iH-l^l+2-^i+2^i4-inu-v.)]dS.

(5) ;^ psdV - ! [q.9"^+ ps(u.-v. ) ]dS. > Second Law

V s

(6) e - e(o,s) First Law

(7) e = £3

In these equations S(t) is the surface of an arbitrary
volume V(t), x. are Cartesian coordinates, i takes the va.lues
].,2,3 (mod 3)> p is density, u. is Darticle velocity, s is
entropy per unit volurae, e is internal energy, f. is external
force (both per unit mass), T.. is the stress tensor, q. is
the heat flow vector and 9 is temperature. The su'-aiiiation
convention is understood and all quantities are functions of
X., t. If the equation of S(t) is

(8) (i)(x^,t) =
then the unit normal v. has components

(9) ^ = }^^_(^2,)-V2 .

The velocity v. of the surface is then defined by

(10) ^i = -Mt^'i'x.^'"'^ • .


It is assumed that the normal v. Doints out of V, and that

1 - '

d^. = v.dS where dS is an element of area of S.


The preceding equations can be transformed into other
forms by means of the two transformation theorems

(11) ^J AdV =/A^dV +/Av.dS,

(12) |b, dV^/B.dS- .

V "^i s

For the validity of the first theorem it is necessary that A
be continuous, and wo will have to make use of this fact.
Applying (11) to the left side of (1) yields

(13) J p^c'^V +1 pv^dS^ +/ p(u^-v^)d [q^©~^+ ps(u^-v^) jgV^ .


In deriving (2,9) it is assi-uiied that v. points toward side one;
if not the inequality raust be reversed. The equation obtained
in the above manner from (k) is an iiMincdiate consequence of
[21].). Thus the discontinuity conditions implied by (l)-(5)
are (22)-(25). These conditions together with the differential
relations (15)-(19) hIso ii'n/oly the integral equations (l)-(5).
(The derivation of (25) requires a proof that the left side of
(5) computed for V is greater than the sum of the same
expressions computed for V-, and Vp . This follows directly
from (19) . )

3. Shocks, Contact Discontinuities and rhasc Chanf/O Fronts
The jump conditions (22) -(25) may be rewritten in the

(26) p-j^TJ^ = P2IT2 or 92^^ A u^ = -U^^P

(27) Pi^iAu. = v^AT^^.

(23) p^IJ^A(e+| ut;) - v^Z:i(T^.u^- q.) =0

(29) p-^U-^As + v^ A^q^O"-"-) >

Here \-jo have introduced the normal velocities TJ-, , Up of the
surface relative to the medium on sides one and tx^ro
respectively, and the ''jump" ^, by

(30) ^g ■= g-^ - g2

(31) Ui = (v.-u,^)v.

(32^ U = (v.-u,,)v.

2 ^ 1 i2 ' 1

If U, =0 the surface moves i/ith the medium and the
discontinuity is called a conHta^t discontinuity. In this
case, since p.. and pp are not zero, (26) -(29) become

(33) ^2 ^ ^ °^ v^Au^ =

O^l) V . at. . =

; 7.

(35) ^j^''\1 " "j-ij^'^i

(36) V. A(q,G"^) > G .

Equations (33) > (3^) shov; that the normal velocity and the
stress on the surfa'ce are continuous, while (35) shovjs that the
jurap in normal heat flu:c equals the jump in pov/er of the normal
stress .

If U-, 7^ 0, (27) may be written

(37) Au^ = (p-lU^)"^v^.^T.

If in addition. A, p f^ '.: the discontinuity is called a shock.
From (37), (26) we find U -i-^g ^^■
relative to the medium on side one

From (37)5 (26) we find U -^^jq normal velocity of the shock

(38) U, - 1 1


Prom (38) > (26) we find that Up is given by (3S) with sixb-
scripts one and two interchanged.

If U, 7^ C and A p = the surface is called a phase
change front. Equation (37) applies, while (26), (28) yield

(39) ^i-^^-j = '" I

v^(u T. -q.)
(1;0) Tj^ = -^ -J^^ •

Equation (39) sho^^fs that the normal velocity is continuous,
while (I4.O) gives the norntial velocity TJ-, of the surface relative
to the fluid, which is also eoual to Up. Such fronts occur in
melting, freezing, evaporation and condensation, etc.

C. Perfect Fluids

A perfect (or non-viscous) fluid is defined by

ikl) T. . = -p5 .

Here p is the thormodynamic pressure, defined by


(^2) p = - -^V = P^^n •

T^or incompressiblG media ( ii.2 ) is meaningless, and then p is
introduced as a new dependent fvuiction.

For a perfect fluid (l3) is automaticallj'' satisfied
while (16), (17) J (19) can be reduced to

(li.3) u. +U.U, + p' p = f.

1 ■ :i 1^, ' ^ X. 1
ox. 1




(■'4-5) q. ^V <


In deducing (1-1-5) j the assumption that © > is made use of.
At a contact discontinuity (3l(-)-(36) become

(1-I-6) /\v> =

ikl) V Aq . =

(I4-8) q^v. A(©"^) >

Thus p and the normal heat flux are continuous, while heat
flov7s from higher to lower torapcrature at a constant

At a shock (3?)j (3^) become

('-1-9) Au^ = -Vi(p-L-i)'^Ap


(50) IT = i -£__ .

At a phase change front (k9) applies, and this with (39)
implies that u. and p are continu.ous. Then (ii-0) becomes

-V . Aq .

(51) u = ._J._.....I .

-■- p, Ae


If no heat conduction occnor's, then a phase change front cannot
occur in a perfect fluid, since (5l) would imply ^& - and
then all quantities '.^j-otTld bo continuous. (IJ-. cannot vanish by
definition of a phase change front.)

Ill, Geoinetrical __Acoust i c s - Weak Shocks

A. The Variational or Acoustic Equations

Let us consider a one para:Tieter set of solutions of the
equations of continuum mechanics dcpoiadinn; upon a pararaetcr h
Wo assunie the solutions are diff erentiablo x-zith respect to yi
at yi= and we differentiate (6), (7), ($^-(|?^>, (i|.2) with
respect to yi at y^^ = 0, obtaining

(1) p^ + (piu + pu^)y^ =


1 ^ .

12) u. +U.U, + u.u. - - T. . + - 4 T. . = f.

1.11 11 oil d 1', X

t ^ X . ^ X.. ' ''x . p '^X .

«J J J J

/ N • • • 1 r

(3) e . + u.e + \i, e + - q , - T. .u. - T- ,u,. ]

^ -^ t IX. 1 X. p ■ "1 111 11 1

1 1 ' X . ^ X . - X

1 J J


■ ^^i

X .
1 J

- "^ [q. - T. .u, ] =
2 '■^1 ij 1

X. ■^ X.

ik) T. . = T ..
^ ■ iJ Ji

(5) e = Spp + e^s

(d) p = p + p„s
P 2

(7) e = e^p + ej

P o

In these equations a dot denotes differentiation with

respect to >^ . Those equations are linear in the derivatives
• • •

p, s, u. , etc., which arc called the variations of the corres-
ponding undifferentiated quantities, or the acoustic density,
entropy, velocity, etc. The equations are called the
variational or acoustic equations, and the undifferentiated


quantities arc called the basic flow. It is to be noted that
a solution of the homogeneous variational equations (with
f. = 0) is orivcn by the x. derivative of a solution of the
oriF;inal equations (with f . = 0) .

In order to differentiate the discontinuity conditions,
vje must take account of the fact that the discontinuity surface
may depend upon r\ . Thus, from II-(8) we have

(S) ^ + f, X.

Since only normal displace nents of the surface need be
considered, vie may satisfj'- (v6) by setting


(9) x^ = -^i'l'^^x.^'

Making use of (9), we obtain from (2©)-(aq)

* _T /?

(11) [p^u.Au. - v.AT,.]-~[p,U,Z^u,- v^AT,,] vj(^2 ,-1/2 ^

(12) [o^TJ-LA(e + J u^) - A(T. .u^- q.)v ]

- fPlUi^(e-^ I u2) - A(T. u. - q )v ]^^^ vji^l )-V2^


K k

The inequalities j'-ield no definite results i\?hen differentiated
ujiless the equalities hold for Y?= 0. Then thoj?- also hold for
the derivatives, unless only positive yi •'^^'^ considered, in
which case the inequalities '.lOld for the derivatives. Equations
(10) -(12) arc linear in the acoustic quantities, and may be
called the acoustic discontinuity conditions.

B. Acoustic Discontinuities

'•men the basic flow is continuous (i.e. p, = p.^j '^^2_ ~ '^2'
etc.), (10)-(12) simplify to

(13) pv^Au^ - -UAp


Hk) pUAu. - V .AT. .

J ""J

(15) pFA(e + u^.u^) = V . A(Tj[ .u^+ T^ .u^ - q.) .

By using (111), wg may simplify (15) to

(16) pUAe = V .(T. . Au. - Aq . ) .

If Tj = WO have an acoustic contact discontinuity, and
(13) , [Ik) , (16) become

(17) v^Au^ =

(18) V .AT. . =

(19) V . Aq. = .

If U 7^ but ^q - we have an acoustic phase change
front and (13) becomes (1?) v.rhile (I4), (I6) become

(20) Zi>u^ = (d-t)~^v.AT. .

(21) TJ = (oAe)'^v^(T^^AA^ - AQj) .

If U j^ 0, Z^ p ^ wo have an acoustic shock and (Ik)
becomes ("lO) » which, with (13) yields

(22) TJ = + ._^J._....U .

'' A p

In a perfect fluid the following further simplifications

(23) Z!s.p = contact discontinuity
( 2'-!-) -^"i ~ -(p"^')' v.A.p shock or nhase front

25) LT = i ( ^ shock



(26) IT = -(pAe)~ v.Aq- phase front

(27) pU9As = -v.Aq.. any discontinuity

The last equation follows from II-(7), II- ( 1+2 ^f'S' i^ (l6).

If we nov; consider a perfect fluid without heat flo^^r
(q. = G) then phase fronts become impossible, and at a shock

(27) yields

(28) As = .

TJslnp- (28) in (5) yields Ap - P Ap, and thus frora (25) we
have for the velocity of an acoustic shock

(29) n = ±^-p- = i c .

The quantity c = Jp is called the sound speed, and it may vary
with position and time since it depends upon the basic flow.

If the definition of U, (31), and II-(9),, II-(IO) are
used in (29) we find

(30) (tt + ^^i'i^x )^ = ""^^^l ' •

This is a first order partial differential equation which must
be satisfied by an acoustic shock in a perfect ^ non-conducting
fluid. The coefficients depend upon the basic flow.

If (|), =^ 0, we may write the equation cj) = in the form

(31) t = W(x^) .
Then (30) becomes an equation for W

(32) (u W^ - 1)2 = c^l .

'^ J ' J

If u. = 0, (32) becomes the oiconal equation

(33) \ = ('v^ )^ = W^ +i/ +\'jI .

c J 12 3


If u . = constant, (32) can be reduced to (33) by making use of


the change of variable x. = x.-u.t. The equations (32), (33)
a-PPly only when c and u. are independent of t, which v;e will
henceforth assuiTie to be the case.

G. Wave fronts and Rays

The acoustic shock surfaces W = constant v;ill nox^j be
called wavefronts because the inte.cration of equations such as
(32) has been studied in georaetrical optics wnere that term is
used. There is, of course, a complete theory of such equations
and we will merely specialize it to this case.

First we define the wave normal p. by

{3h-) Pi = w^^ .

Prom (32), {3k-), P- satisfies the qiiadratic equation

(35) ^^l^i " -"-^ ^ °~Pi •

At each point in space (35) defines a surface containinfr. the
endpoints of the possible wave normal d. . This is c^lDed the
surface of wave normals. The reciprocal of the radius from
the oric-in to this surface in any direction is the velocity v
in that direction of a possible vxavefront, since by I-(IO),
I-(31), I-(32)

(36) V = (p^)-^/2 ^

By examininp; (35) one finds that the wave normal surface
is a siirface of revolution about the direction u. . Its section
by a plane containing this direction is a hyperbola, parabola

or ellipse according as the basic flow is supersonic, sonic or

2 1/''
subsonic (i.e. accordinp-; a? u = (u.) -^ ^ c). If u. - the

-"■ _1 -'-
wave normal surface is a sphere of radius c . If Q denotes

the anr-le between p. and u. then in the subsonic case there is

one wave normal for each value of 6, and the wavefront velocity

v = c+u cos 0. The same applies to the sonic case, except

that there is no wave normal for & = n.


I.i the supersonic case there are two solutions p. for

-1 C 4-

each direction if C < < cos ( — ) , and then v = u cos *-) 1 c,
the upper sign applying to that branch of the surface nearer
the origin. V/hen cos" (^} < © < cos" (- ^) there is one
solution p, for each direction and v = c +u cos 9. If
cos" (— ) < 9 there is no solution. These results are readily
understood in terms of vectorial addition of ix. and v..
The Hamiltonian function H{x^,Pj^) is defined by

(37) H(x.,p ) = c^'p. - (l-u..pj

In ter;ns of this, certain curves x.((r) , which we call rays,
are introduced by the equations

dx —^

(33) ^ = X^Ip, = ^\ (c^p. +u.cjp^) .

?Iere X is an arbitrary non-zero factor. From the last

expression in (33) we see that the rays are orthogonal

trajectories of the ;>ravc-c'ronts if and only if u. is zero or

dx. ^

■oarallel to p., since then -r-— ■ is uarallul to p. which is a
- 1' dcr - -^1

wave normal .

Prom the general theory of first order equations it

follows that the rays satisfy Hamilton's equations, Lagrange's

equations, Fermat's principle, etc. Furthermore, by means of

the rays vjc can construct the solution of (32) corresponding

to any given initial data. The v/avefronts can -also be found

by Huyghen' s construction. The vector -r;— also lies on

Fresnel' s ray surface, which is reciprocal to the wave normal

sLirface .

D. Variation of Shock Strength Along a Ray

To determine the variation of shock strength (i.e. Ap?
Ap, etc.) along a ray, we must consider the acoustic equations
for a perfect, non-conducting fluid. 3y specializing (l)-(3)j
(6) or by differentiating I-(I|-3) , I-(;4.i;) we obtain

, ^c ,; i) . v">^ 1 ;■

T' .-'

''■[•:- i ■ ■■;


(39) p^ + (pu^+ pu^)



(JiO) u. + u U. + U.U. + D~ D^ - po" p., = f.

• •

(h-l)s, +U.S+U.S =0

t J X. J X.

ik?-) P = PsS+ PpP .

We now consider the quc'.ntltios u. (x . ,V./(x . ) ) = u. (x . ) ,
p(x./-J(x .) ) = p(x.) , s(x ./-/(x ) ) ='g(x.) and p(x .,>/(:,:.))= p(x. )


• •

I'liese arc the values of u., p, s and p on a wavGx'ront, and
there are two such functions vjhich we define in this way, one
corresponding to each side of the wavefront. The derivatives
of cither of these functions are given by

u. = ti. + u. W
1 1 1 , X .
X. ^^j t J

^^3) ^. = P^. + Pt '^.


^_ • •


Combining (39)-(l-!-2) with ([|3) yields

(I1J4.) p,[l-u.N ] + u. [-pVJ,^ ] = -pil. - p u. - pu - p u
1 t 1 X. 1 X. 1

' 1 t -^ J X

" ^- ^>p"^px.- Psp"\.^ pp'^'^x.- X ""r'-A

^ 1 1 ix.^'-^x.


- p'^^Po P - p'"^Po "^
^x. Px.
1 1


One such set of equations Is obtained on each side of
the wavefront. We now subtract each equation on side tv;o from
the corresponding equation on side one. Then remembering that
the basic flov; was assioi'ded to be continuous, and that
As = As" = 0, we obtain

(1+7) [1-u.lC ] A(Pi.) + [-pV/„ ]A(u. )

= -pAu. - P., All. -u. ^p-u.A?^^^

X. 1 X. 1

^ i "^ j t i

_2 -1 — -1 — _ _

= [p Px " P ^o ^Ap-P P Apx - ^i ^u.-u.Au.
i Px , P i ^x . -J -^ X

()49) [1-u.W.^ ]A(s.) = -s„ /^u.


Equations (!4.7)-('!-9} are a set of five linear inhomogeneou£
equations for the five quantities -Ap , , Au. and Z^ s^, . Let us


compare these equations with the shock conditions which are a
set of five homogeneous equations for Ap, Au. and As.
These latter equations can be rewritten in the form

(50) UAp + ov^Au^ =

(51) v^p /^p + pUAu^ -I- v^p^Ziks =

(52) pUe„As =



If (1+7) is multiplied by -^ , (1+G) by -2- and (Il9) by

]\-'^ Jvt

pe,, ^"k V 'k

- — ~ then the coefficient matrix on the left side becomes

identical with that of (50)-(52). Therefore ( [j.7 )-( 1-1-9 ) can have
a solution if and only if the right sides of these equations
satisfy a certain orthogonality condition, since by assumption


the homogeneous equations have a non-zero solution. To obtain
this condition we eliminate As^ from (i|-8) by means of [k9) >
and similarly eliminate As from (51) t)y means of (52). The
resulting sets of four equations in four unknovms vfill both
have the same coefficient ruatrix, which becomes symmetric if
we multiply (i|7) and (50) by p and (I4.8), (51) hy p. Moiij the
solvability condition for (l-!-7)j (50) is that the rinht side
be orthogonal to the solution of the transposed homogeneous
equations, which are just (SO) , (51) because of the syiTuictrj.
Thus we obtain the condition

( 53 ; -P^ ( P Au, + P„ ^ u. + u . Ap + u . A p,^ ) A p

1 1

2, r -2 -1 -, ^- -1 ^ -

+ P (Ip Pv " P P^ jAp-p^p Ap,

^i Px_. P



- u. Au.- u.Au. )Au. + U po^s Au.v.Au. = G

^x. ^ ^ -X. ^ ' ^ ^i ^ ^ ^
3 J "^

Equation (53) can be considerably simp].ified a;:id solved
explicitly. First, v/e note that A u.= Au. and Ap = Ap.
Then we eliminate A u. in terms of -A p by {2l\.) , and we

eliminate A p by the relation Ap = c Ap, which follows from
(5), (23). In this way we obtain from (53) an equation in
vjhich the only acoustic quantity a"opearing is Ap. This
equation is,, after dividing by ^p,

. u.(Ap) 2u Ape,.

v^-t-/ X^ IT 'x. 1 J 1 2


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