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AFCRC-TN.57-376 IN&f ffUTE ©P N'-



oi J[ ^ JX ^ Institute of Mathematical Sciences
^^ C C C X X "^ Division of Electromagnetic Research


Acoustic Torques and Forces on Disks


CONTRACT No. AF 19(604)1717
APRIL, 1957


Institute of Mathematical Sciences

Division of Electromagnetic Research

Research Report No, EM-lOU

Joseoh B. Keller

(y^JosejA B, Keller

Morris Kline
Project Director

Apill, 1957

The research reported in this document has been sponsored by
the Air Force Cambridge Research Center, Air Research and
Development Command, under Contract No. AF 19(60U)1717.

New York, 19^7

- i -

The time-average forces and torques exerted by a plane sound wave
upon fixed rlf^id disks of various shapes are calculated. Results are given
for disks bounded by smooth closed convex curves. These results are then
specialined to ellipses and circles. Results are also given for infinitely


long thin strips. The results are all valid for ka large, where k » -^ ,
X being the wavelength and a being a typical dimension of the disk. The
oscillatory behavior of the torque as a function of ka and the occurrence
of numerous equilibrium positions are interesting consequences of these

Table of Contents


1, Introduction 1

2, The torque on a strip 3

3, The torque on a circular disk 6
Appendix 9

References 13

Figures 15

1 -

1. Introduction

'/Then a sound wave strikes an object it exerts a periodic force and
torque on the object. In this paper we calculate the mean values of these
quantities for certain objects wliich are thin, rigid and immobile. We call
such objects disks. We vd.ll consider disks whose rims are smooth closed convex
curves, and also disks which are infinitely long strips. The circular Rayleigh
disk"- -' is included. Our results are based upon the 'geometrical theory of
diffraction' 1- J, a new method for solving diffraction problems. This method
applies to situations in which the wavelength X is small compared to the obstacle
dimensions. However it also worics well for wavelengths as large as the obstacle

This defines the range in which our present results applyj nolo that
high-frequency sound is included in this range.

Most of the previous theoretical work on this subject has dealt with the
low-frequency case, in which the wavelength is large compared to the obstacle
dimensions. W. iTdnigi- -l calculated the torques on an ellipsoid and on a circular
disk as well as the attractive or repiilsive force between two spheres'- -^ in the
limit of infinite wavelength. M. Kotani'- -' extended the calculation of the
torque on a circular disk to include several powers of X" , L. V. KingL-"
calculated the force on a sphere to several terms in X" . Ke also calculated to
the same order the force on a circular disk d'"* to a normally incident '^lane wave,
taking account of the motion of the disk' - '. This same force wav; measured by R. '.
Boyle and J. F. Leimaiini-'^-' . King'-'^^also treated the torque on a circular disk, ps

did Kotani, but took account the disk's own motion, again to several terms in X


■food'- -' also indicated how the torque on a circular disk is moaified by xu

motion and checked his results by experimenting on disks in water, N, Kawai'
extended the calculation of the torque on a circular disk to somewhat smaller
values of X. He fo\md that the torque reversed in sense, i.e, changed sign, when

- 2 -


X became less than about .h of the disk circumference. H, Levine'- -^ obtained an
expression for the force on a circular disk due to a normally incident plane
wave. It is valid for long waves and is apparently also quite good for short ones.
He also determined the torque on an infinitely long strip for short wavelengths ^ ■'J.
This torque oscillates aroiind the value zero as X varies.

The present calculation begins with two formulas for the force and torque,
given without proof by H. Levine and J. Schwinger^ -' . The first of these is

(1) F « * cos a cr(a).

In (1) F denotes the normal force on a disk due to an incident plane wave of (real)
pressure amplitude P, a is the angle between the normal to the disk and the
direction of propagation of the incident wave, cr (a) is the scattering cross
section of the disk for this direction of incidence, p is the density of the
surrounding medium arid c is the sound speed in it. Equation (l) applies to an
infinitely long strip if F and a are defined to be the force and scattering
cross section per unit length.

The formula for the torque is, in modified notation ^ J,

(2) T - - p^Re[f^(ci,0,a)}.

In (2) T is the torque around an axis perpendicular to the plane containing
the direction of propagation of the incident plane wave and the normal to th«
disk. If this axis i« the z-axis then the propagation direction lies in the xy-plane
and the disk lies in the yz-plane. The angle between the propagation direction
and the positive x-axis is denoted by a. Then Pf(0,O,a) is the complex amplitude
of the wave scattered in a direction in the xy-plane making the angle with the
positive X-axis, The angular frequency of the incident wave is o> and the torqoe
T is positive if it tends to inereeae a.

- 3 -

To make the above definitions more precise we note that the incident
pressure p and the scattered pressure p are given by


p^ - Rejp e^^^ cos a + y sin a)-irot|

r . i(kr-cot) ]
p^^Re|Pf(0,X,a). 'i^^^ I

If the disk is an infinitely long strip parallel to the z-axis, fw(o,0,a) in
(2) must be replaced by f^(a,a) which is defined by

(5) Pg ~ Re|p f(j2f,a)}. -/^ e^ ^"^"^ .

Furthermore T then denotes torque per unit length.

Equations (1) and (2) are derived in the Appendix to this paper for
both finite disks and infinite strips. The derivation is patterned after that
of H. Levine, who derived (2) for the case of an infinite strip'- ■' and (l)
for finite disks with a « I- -I . Equation (l) for finite disks with a =
also follows from an identity derived by P.J, Westervelt'- -^ * L K This same
identity has also been derived by O.K. Mawardi'- -• v)M also gives additional
references to work on 'radiation pressure'.

2. The torque on a strip

The scattering amplitude f(0,;C,a) or f(0,a) for a rigid disk is the
negative of the corresponding amplitude for an aperture of the same size and shape
in a soft screen. This applies in the range < ^ < |- and follows from the
rigorous form of Babinet's principle (cf. [l?] ,p.39). From thla fact and the cross
section theorem we find that cr(a) - 2€r (a), where 0, so they are alternately
stable and unstable. To find these values we may rewrite (13) in the form

(Hi) T - ■■ 1^ ^^^ =75 sin(2ka - f) sin(2ka sin a)

Tt^/2p^c2(ka)3/2cos a L ^

+ sin a cos(2ka - ^)cos(2ka sin a) ,

Thus the equilibrium values of a, other than a = C, are the solutions of the

(... tan(2ka sin a) cot(2ka - ^)
^^^^ 2ka sin a " S^S

The stability of a - is deteimined by the sign of

(16) T^(0) - y^^ I 575 [2ka sin(2ka - J) + co3(2ka - f)j .

Although, as is well known, a = is stable for ka very small (when our formulas
do not apply), we see that it becomes alternately unstable and stable as ka
increases. When a = becomes unstable, there are two stable equilibria near
a = 0.

3. The torque on a circular disk

In order to calcvilate T for a disk bounded by a smooth ccnrex
curve we begin with (5?) of [2] » This equation gives u^, the fi«»ld idiich

- 7 -

results when an incident field of vuiit amplitude is doubly diffracted at the edges of
the complementary aperture in a soft screen. Therefore when the spheric&l wav©
factor is removed, this equation will yield -fj(0,X,a), the contribution of
doubly diffracted rays to the scattered amplitude. By using this result in (2)
we will obtain the contribution of such rays to the torque. It is to be expected
that, as in the case of the strip, these rays will yield the main contribution
to the torque. Therefore we will not bother to evaluate the contributions of
other rays.

Since the leading term in k of the derivative f , comes from the phase
factor, it suffices to evaluate the amplitude in (5?) in the forward direction


Let us apply (18) to a disk which is symmetric in the y- and z-axes.
Then only two doubly diffracted raj's go in the forward direction. They come
from rays incident at the two points on the edge in the plane z » 0. Furthermore
p - |, Pp - Pq - p, T^(P) - d^ = 0, s-r - i I sin 0, T(P) - 1 | sin o, and
B = (1± sin a)(2pp - d[l* sin aj). Here the positive signs apply to one ray and
the negative signs to the other. Inserting these values into (18) yields



3in[kd(l-»-3in a) •*• e+1

(1+sin a)^/^|2p-d(l+sin a)|^/^

3in[jcd(l-sin a) + e_l

(l-sin a)^/2|2p-d(l-sin a)!^/^

In (19) d is the width of the disk along the y-axis and p is the radius of
curvature of the edge at the y-axis, Ihe symbol c+ is equal to - ^ or accord-
ing as 2p - d(l * sin a) is negative or positive.

Suppose the disk is an ellipse with major axis of length 2a along the
y-axis and minor axis of length 2ea along the z-axis. Then d ■ 2a and p = e a.

Since e < 1 it follows that 2p - d(l+sin a) = 2a(c -l-sin a) < 0, so e_|_ » - | ,


while the value of e_ depends upon the sign of e + sin o - 1. Let us suppose


that this is positive, which requires sin a > 1 - e . Then e_ = 0, and (19) becomes

(20) T


co3[2ka(l-t-sin a)] ^ sinC2ka(l-sin a)]

(1+sin a)^/2^^g^ a-e^)^/^ (l-sin a)^/2(_3^^gjj^ ^\ ^2^1/2

In the special case of a circle, e = l,so sin a > 1 - e » and (20) yields

(21) T =


——5 5 —

p c (ka) (sin a)


cos C2ka(l+3in g )3 ^ sin]^a( l-sin a)]

(1 + sin a)


(1 - sin a)


This result applies for a / 0, ^ . The -torque tends to zero like (ka)" as ka
becomes infinite, but in an oscillatory manner. For fixed ka there are equilibrium
values of a which are alternately stable and unstable. They are obtained by
setting T = in (21). As in the case of the strip, the torque is larger than
the force on the circular disk for ka very small, but the reverse is true for
ka large.

- 9 -
APPENDIX! Derivation of the Basic Formulas (l) and (2)

Let us denote by T(x,t) the real time dependent velocity potential of

an acoustic motion at the point x » (x,y,z). The pressure p(x,t), to terra of

second order in T, is given by"- •'

Here p ani p denote respectively the constant pressure and density in the
undisturbed medium and c is the sound speed in it. If the motion is periodic
(hamonic) in time with angular frequency co, then Y can be expressed in terms
of the complex time independent velocity potential Y(x) by the relation

(>^2) Y(x,t) = Re e"^'^ T(x).

Upon inserting (A2) into(Al) and averaging over a period, we find that the
average pressure p(x) is given by

(A3) P(x) . p^ + ^ [kV- VT-Vf*].

In (A3) k - I .

The time average force F in the positive x-direction on a disk S
lying in the yz- plane is


F - j [p(0-,y,z) - p(0+,y,z)]dydz.
The time average torque T around the z-axis is

T - y[p(0-,y,z) - p(0+,y,z)]dydz,

(A5) ,

We will now express F and T in terms of Y by means of (A3), However, we first
note that Y ■ on the disk since it is rigid and immobile. We also make use


of the following well-known formula (cf, [I'l] , p, 39):

(A6) l(0-,y,z) + T(0+,y,z) = 2¥^(0,y,z).

Here I (x) is the velocity potential of the incident wave, which we assume to be

(A7) Y (x) o -Z_ e^^^ cos a + y sin a)^
o p^co

The real constant P denotes the pressure amplitude of the incident wave and a
is the angle between the direction of propagation of the incident wave and the
noiroal to the disk.

When (A3) is inserted into {t\h) and (a5) certain differences occxir
in the integrands. By means of the preceding observations these differences
can be simplified to

(A8) (Tf*)_ - (TT*)^ = BbUI 6T),

(A9) (VT-Vy*)_ - (VY.VT*)^ - ReCVgTo'Vg 5Y) - Re(T^ CT ).


In these formulas the subscript + or - indicates that a quantity is to be
evaluated at x = 0+ or x = 0-, The operator V^ - (g=, ■gr) is the gradient in
the yz-jxLane and 51 is the difference

(AlO) 6T(y,z) - T(0-,y,z) - l(0+,y,z).

When these formulas are used in (aU) and (a5) they become

(All) F - ^ Beije'^ =^ ""[k^ 5T+ik sin a BT^Jdydzj,

(A12) T = Is Rejr ye-^ ^^" '^[k^ 6T + ik sin a 5Ty]dydzl ,

11 -

These formulas simplify further tf the second tena in each integral
is integrated by parts with respect to y and the fact that 6T - at the edge
is used. They then reduce to

(A13) F = ^^^ Relf e-^ ^^" ^ 6T dydJ ^

(AU,) T = I3 Rej I e-^ ^^ ^ 6T[yk2cos2a - lie gin ajdydz

We now make use of the fact that 1 can be written in terms of 6T as

(Al$) I(x) " T^(x) - r6T(y',z') G^,(x,y,z;x',y' ,0)dy'dz' .

In (Al5) G denotes the Green's function

Let us denote the spherical coordinates of the point x by (r,{i^,X) with the
X-axis as the polar axis, as the co-latitude and 9c as the azimuth. Then
for r large compared to the dimensions of the disk (/.15) becomes

(AI7) Y(x) -^ fjx) * 2^ ik cos j2f [5T(x',y)e-^^y'^^'^ cos 7C * z'sin ain^)^^,^^,


The coefficient of -ke ^/2nT is defined to be the amplitude of the scattered
wave corresponding to the incident direction a. We write it as

(A18) -Lf(0,^,a) =-i^°2j |6T(x',y')e-^^y'^^"^°°^^*^'^^" are unstable. The curve
does not apply near a - ^.

Figure 3

The torq-je T on a
circular disk of radius
a as a f\inction cf the
angle_of incidence a for
- ? are

unstable. The curve
does not acply near
a - C and a - n/g.

10 20 30 40 50 60 70 80 90







12 18







22 26 30

5ie torcr-e per unit length
T on a strip of width 2«
as a function of ka for
tt = J(US°}. The vertical
scale is the value of

Tne curve is based upon
aq-iation (13). rhe values
of c at which T » are
eq-ailibriuiB angles. Tlnse
f-^r which T_ < are


and those for wr-ich
are unstable. The

s'xrre does not aprly for
ka sr^all.

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