Transmission of sonic boom pressure through glass window pane

Benveniste, J. E.

Cheng, D. H.

English

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Transmission of Sonic Boom Pressure
through a Window Vane
Report. No, e
Grant 1\1 r, - 33 - 013 - 039
r
tional Aeronautics and Space Administration
t
Novembe r 1970
i
9
.	 j
 
44	 CY>
jot
SCHOOL OF ENGINEERING
DEPARTMENT OF CIVIL, ENGINEERING
CO Sk1lf^.'7
10	 (ACCESSION NUMBER) 	 (THRU)
O (A. )	 (CODE)J
Q (NASA CR OR TMX OR A NUMBER)
	
(CATEGORY)	
A
4
Ci
a1
1. Introduction
in previous studies C of the effects of sonic booms on
structures, the structural elements were assumed to be subjected to
the outside overpressure. Actually, a certain amount of energy is
transmitted. to the interior of the structure and exterior structural
elements are subjected to a net pressure equal to the difference between
exterior and interior overpressure.
The computation of the actual net pressure would be a hopeless
task since the interior overpressures depend on the multiple reflections
and the absorption of sound waves bpi", walls, floo.^.s aad ceilings. As
an approximation of the situation near a window, the case of an elastic
plate set in an rigid, infinite baffle and subjected to an N-shaped
sonic boom normal to its plane was studied. The plate was assumed
square and simply supported along its edges
Related studies in the literature are by Morse (2) in the
study of the transmission of monochromatic vibrations f rom one acous-
tic medium to another through a circular membrane set in an infinite
baffle and by Q'Callahan and Madden (3)
 in the study of the interaction
of a free stream of an ideal gas with a circular membrane set in an
infinite rigid wall. The former study was by a variational method
while the latter used finite difference equations.
C
22. Nomenclature
- area of square plate
I
a	 .-
Y
radius of circular plate
c	 = speed of sound in air
speed of sound, in pla'.e material 	 Ycg
D	 = f lexura l rigidity of plate
E	 = modulus of elasticity of plate
9 n (t) =time factor in normal series expansion
H(t)= unit step function	 t
h	 - plate thickness	 }'
k	 - TTwave number in air
L side of square plate
k
p(t)= free- f ielO pressure
P
	
=0 maximum overpressure in p(t)
Pint= interior pressure a
pext= exterior pressure
9	 _ pext -' pint
q	 =
mn
coeff icient of
	 q	 in normal series expansion
R
R	 = 'G/T1 f
r	 _ distance between 2 points in plate
T period of plate
	 E
f	 t	 - time
w	
= plate deflection
Wmn ^x^ ^') _ normal function
.
X, y coordinates
6t = time increment
wave length
f3
yr = Poisson's ratio
TT;, = power transmitted to the interior
TO = power impinging on plate
^a
density of air
9
density of plate material
-^ = sonic boom duration
W circular frequency
Ulm
 
n^
circular frequency in m, nth mcde
r4
3, Formulation of the Problem
The equation of motion of the plate is taken according to
the elementary theory of plates. (4)
.r
where w is the plate deflection, D is the stiffness of the plate,
h its thickness and Sg the density of its material and q = pext
- pint the net pressure equal to outside pressure minus inside
pressure.
The net pressure is given in terms of the pressure rulse
p (t) and the deflection w (x, y, t) through the equation lil
(x	 2 (t^	 ppW (X)y, - r-c d.x 'd.u3'	 (2)
where A
is the density of air, c is the speed of sound,, and where the
a
integration is over the area A of the plate. The pressure
pulse will be taken as the familiar N wave ( Fig. 1)
	
Pi t )	 Pn	 H Cz t	 (4)
where H (t) is the unit step function
The initial conditions on the plate will be taken as
w (x Y, 0 ) = w (X, y, O) 	 n	 (5)
while the boundary conditions are (see fig. 2)
	
w (o, o,t )	 w(o, L, t )	 w(L,o,t)	 w(L,L,t)	 o	 (^)
4. Solution
A numerica l
 
solution of the problem is obtained by using
finite differences on the time variable and the normal mode
expansion for the space variable. The solution is taken in
the form	 (7)WN) t)` z WI'V)Of ,y) 9 Ct)
C11 See Appendix for derivation
er
5
where Winn (x,Y) represent the normal mode
soh vh^,X sin n
L
with m, n integers
Using the mod.al representation (7) and. the orthoganality
of the normal modes, the equation of motion of the plate re-
duces to the O.D. E.
m r► 	 S
(X, Y)where Wmn is the circular frequency associated with wmn ,.I
and
4 ^ 
^t
which, by the use of (2) can be written ass
9MV = T^"*) —
6	 / ..
44	 S^V) VM MX soh	 s^^ut-► k's^v,s^ 9us ' r^c dxd akdZ^	 (9)
^, s	 ^	 L	 ^-	 ^	 r	 y y
where
p if both m, n oddmn	 n hnh	 (10)
0	 otherwise
and where r is givF.1rx by (3)
In obtainiing a numerical solution equation (7)
is _solved by finite diZferences ands in using (9) to compute
gmn)the summation is truk,cated while the integration is replaced
by a summation,
a
1	 .
5. Numerical Results
The case considered was that of a glass pane with
u	
Lh - 250. The relative densities of glass and air: were taken
as 2.5 and 0.00129 respectively. Numerical results were
obtained, for two particular values of R T	 viz R 11i
R a. In both cases the series was truncated: to its first
term. In solving the ordinary differential equation (7) by
finite differences the time step was taken as t _ 1 	 L10 c:
i.e. one tenth of the time required for sound to sweep the
plate. The following recurrence formulae were used. 5)  	 z
rr	
I^
+g(t-24b)
	
w	 (ll)	 ^
combined with simple two-term formula , for the first two steps.
The results are presented, in the graphs of Figs. 3
and 4 in which the values of 	 q%A)	 and	 (t) P11? (o
L 111(o )	 9, ►are plotted as functions of
	
Results were obtained forT si	 ,
values of t up to -2.0 since little departure from periodicity
li
is expected beyond this value.
The computations were carried. out at the City College
Computation Center on a IBM 360-50. The most time-consuming
part of the computation was the !evaluation of the quadruple
integral occurring in equation (9). The total computation time
for each case was ten minutes,
OF
6
j"
^	 .
6, Discu^ssran
It is clear from the graphs of Fig, 3 that the
mGi ,jor ec^ntribution to transmtted pressure occurs immediately
following the d.iecontinuties in the pressure :pulse. A
comparison. of transmitted. pressure. due to an N^wave and tY^iat
due to a step pulse with the same initial pressure rise 3,s
shown in Fig, 3; the two cures ire hardly ds^:ingushable.
The transmitted pressure seems to build; up almost
linearly to its maximum at time 0,7 ^ which is approximately
the. time required fora sound: wave to travel from the corner
to the center of the square plate. The magnitude of the
^naximu^n transmitted ressu a can b -eu hl checked bp	 r	 ^^^' g y	 Y
assuming that the acceleration r^maix^s constant and equal to
q^o^	 and that the plate moves as a rigid: . circular disc
4h
of radius L	 Application of tYie integral in (2) .yields trie2
expression (12)
? P4w^.	 ^ p ^ 9 ^^ L q^^>	 quo) ^ 2 pa
^^ h;,
.:
:For the numera,cal values used,. this is equal. to O,,1^9(0) which
is of the same order of magnitude ae the value of 0.07(^(0^ ob-
taned.from the graph,	 If instead of assuming that..the
plate moves. as a rigid disc, a cosine law of acceleration vari-^
ation is assumed with maximum. at the:-center and varnishing at
,^	 ,^
. ?	 ,,
	
. .;	 the edges, a value closer to the computed value is obtained viz
	
s	 ^
2 P^^ w _ ? 
^^. ^ q ^o^	 q ^o^ ^ 2p^	 (13 ^
^r ^^ h
yie ding a value of 0^085(^(0)
a
k'
" i,
u
^''
^^
t
Y
w
The magnitude of the: computed. maximum value of the
transmitted pressure is quite s^^all (0 07 9 (p^-= 0. t3'^ppo ) eorres-•
2
pondi.ng to a power transmssio,^^ ^,oss of about ^$ d8. Yet actua3^
measureznenta^^ ^ of pressures behind.. plate windows subjected to .^
sonic booms seem to i^z^d.:tcate transmitted .pressures of the order caf
t:
^.0, 5po ox transmission losses of about 6 d:3,
Tn investigating thy: reason for the da.screpancy, the
transmissioc^- lase for monochromat^.c pulses in two models will be com-
pared with. the txansmission loss of the f^.exural panel here considered,
The first. model. shown. in F^.g. 5 is the one dimensional
	
r.	 slab of thickness h with the pre sure pulse transrnitt^d from
one .side. to the other. The power ratio • n^^rr^	 is gven^^ ^
by the followa.ng approximate formula, valid if, as we assume that..
the wavelengEh in the slab is .large compared. to h.
	
a	
4
^'	
1.	
4 SaC 	 ^ 14 )
	
z	
sa
	
'^	 Say.
	
_	 .
^^	 For frequencies betwc^tn 20 and a00 H2 and a glass thickness of 1^4
	
p ^	 in. , the transmission loss according to thi. formula wca^uld vary
^^
between 7 . F d.B and 27 , 6 dB,
t
	
^'	 Thee second model shown in Figs. ^ is a rigid. circular
t
-piston of radius a set in plane baffle. Usr:g the asymptotic
'	 , " ^	 expressions of the impedance for high. and:: low frequencies ^^ ^` the
:,<
,^'
•,	 following expre ions are found for the power. ratio
,,
	
-;	 For ka = Ga >j ^
	
-^
TS' _	 ^ ^	
J _ _ 4 S^^ 2	 ^ 15 a ,
...
^	 h
Sa	 n ka
_. __
u^.,^.	 __^ ._
rFor ka = ^a ^^, l	 (sad ^a
z.
4	 _	 g
^	 (15b^
^..	 2 sp a. 3rr
Tt is int^sr+^stihc^ to note that the. power ratio f ear the rigid piston
tends to the exprresson t4r the one--dimensional slab ^o,r high fre-
r^uencies and to an expression s^.m3,lar to that of. Eqs. (12) & (13 )
for low frequencies. Zt thus appears that for low frequencies,
t'he approxi^tate formula (13) and therefore the calcu^.Lated trane-
miffed pressures are verified, Fc^r higher frequencies the tran®-
mission loss. seems not to depend on the
^/h
Thus the lower transmis ion. l^
Edwards Pair Farce Base tests (6) mush bc^
hood that in the test ^xame houses much
through the roof and wall panels rather
ratio ^/h brut rather an
saes mt^asured at tfis
attributed to tk^e ll^el^
of the energy is transmitted
than through the windows..
	.^ z	 An indirect proof of the above statement lies in the almost non-
	
r	 existent correlation-between the p y"late gYass window displacement
4{
	,^	 (which. were measured- by strain gages) and the internal pressure
	
^^	 (which was monitored, by a microphor ►^) in its immedia tc^ ,vi°cirxty C 6
	
i 	 u
^	 +	 s.
(Fag,.. G23; G24; G25') subsequent to thc^ passing of :press°are w-ave
,
One last remark may be m^^^de concernng_^q. 12: zt
	
,`	 might at_ ^^.rst seem. strange 'that tY^^e st ffness of the plate .does
f
	°'^;	 not. enter this formula, If, howevt,^r, it is remembered that the
	
^^	
^
	
^^	 derivation of the formula assumed .ta constant acceleration. po
^^ h
during the: .
 time it takes the sound to travel -the span L, then it
is seen that the stiffness will ncrt affect'. the result provided
	
i	 _
that the fundamental period Uf ,tha^ pa rye T^^ i s large comps r^Qd to
c	 This seems to he the:.. case foz^ ordinary` window panes
_.
	
_g	
_^ _	 _.
s
t^. U
price ^.^ rec^^^ceo, after same z^anipxx^.ation, ^o
...
	
-- -^--^	 ,^ ^	 (16
Cg
where	 ^,^
^^ 1_
{
or, substituting numerical values for e glass;
h ^^ 1 4 , '3	 ^ 1?
.,
Ir
^	 ! 	
♦ 	
^.,
i,'
r::	
._.
+n^i. ..
+ti
z	 °
^ t
^', eQnclu S ons
The pressure transm^.tted through a square window for a
normal- NWwaue has been. computed. under the assumption that the win^-
v	
dow is ^e^^ in a xig,i^ b^^f1e, Foxe usual. d^.mer^sor^s file pressura
tra nsrni^t.:ed had small .mac^xaitude (about ^,0;35t^(Q)^.
A simple ap^aroxmate formula, was de^relope^^ (Eq. l3) which
can b^ used in most cases commonly ^ncountered^ in practice.
The computed transm3.^:ted pressure are much less than:.
tY,e internal prc^ss^ure measured in Frame houses during sonic boom
experiments. Tt is believed that the discrepancy is due to trans-
m^,ssion of pressure through the areas o^ wall and r^ao as well as
w ^. ndow s
r;
;-
R
,.
_^
I^
	 ,,
^
,.
..
.. ^^.
^^
_.
^^
n
4	 *	 e
J
^^
8. References
^. ,	 Cheng, D, H. end Benveniste, ^'. E,
Dy^lamic Response o^ Structural, Elements Exposed to Sonic Booms
NASA CR-1281, Marc'^n 1969,.
2. Morse, P,M, and Inc^ard, K.U, 	 °
Theoretical Acoustics
McGraw Hi 11, 1968.
3. O^ Callahan, J.C, and Madden R.
The ^:nteraction of a compressible fluid and an ela$tically
deforming boundary,
P^^per No, 70-75 AIAA 8th Aerospace Sciences Meeting,
January 1970.
4,
	
	 Timoshenko S, and Woinowsky-Krieger, S,
Theory of Plates anc^ Shells
y
	 2nd Bdition, Mc-Grave .Hill.; ].959,
!^
5.	 Craric^,^11 S.
Engineering Analyrsis
McGraw Hill, 1955
6^,
	
	 Sonic Boom Tests a Edwards Air Force Base
Interim Report 1967.
7.
	
	 Kinder L.E. and Frey A,R.
Fundamental of Acoustics.
x13
Append ix
Derivat on of Equation (2)
Let tine Fourier transform; of a time-dependent quantitx
;^(^) be denoted by
00
	
,^	
_ ^
	 - ^w
x (w^	 x ^^) e, d.t
_^
Then, in the frequency domain ^J , the transmission problem
shown in Fig.. 2 reduces to finding 
^^Aext (x, , f^ z, W) def fined fc^r z < 0,
pint. (x, y, ^, ^) defined for z > 0, which satisfy
A -The. reduced wave equat,ionz
:^ -^ The following boundary conditiv ►n .for z = 0
.^.. _ .^.
	 ^. 3
^a ^..
	 ^a ^.	 o ^ ^ ^ ^.	 c 19
^(^
Z	 ,^Z	 otherwise
-.
-• xhe following rad^.ation conditions: The solution
^.	
npint corresponds to an outgoing wave while the solution p ert con-
tams the wave. p(G,^)e^i^'z travelling along the positive z direction,
a . wave travelling along the negative z direction, and an outdoing wave;..
Using -the. properties of t1^e Green ftxn^ction for. a planes ^ 2
`;	 it can be seen. that all conditions A,B,C.are satisfied by 	 `
,x	 ,^	 ^	 - Lk Z ^
	 l ^t s	 2	 ^	 °^^	 ^	 ^
^^.x^ = ^^ W ) P^  ^ p ^W^ ^ ^- 4w	 ^cK^^^ ^^ e dx^,^^ coo)
	
zn p	 ,^
Lw Q
2^r ^	 y
a ^
^: ^3.
-a	 ...,^:	 . ^ .. . ^.	 _
^,1
5S
y
14
where R = (x ^ k ^ 12' ^ ^ ^ ^^ ^^`^+ "^^"	 and A is the area
of the sg,uare plat
Since the conditior,.s A, B, C define a prob^.em having a
unique solution ., the above formulas so^.ve the transmission problem
n3.n the frequency domain in terms of W'.
Py the definit^.on of q one finds from (^0
^	 ^^	 W k^ ^
	 ^`^ r I ^	 C21)
n	 ,^,	 ^
^^ x
` ^	 i^ rnm the well--known diFterentation and shifting proper-
ties raf t^^e Fourier transforms, eq. (2) . follows fx'c^m eq, . (21) ,
P,
t
r
D
F^.g. 1
	 N ^ wave
G	
.....^..Wov e. ^ row ^"
1•
i
_.,.. {
^^
_^	 ^
	
.^.	 c-^
f I
	 1	 ^^	 .-i
1	 a
^ ^	 ^	 -J	 !1•M
	
1	 r^	 _^`..._..__...^__
	 ....__	 ___	
.^	 Q
•	 ^^
tr,
	
1:,	 'L^
_ ,	 to
	
^	 i1-	
^
-j	 ^	 ^	 m
	
~f ♦ 	 11
	
'"'^	
.0	 Gr
I	 ^	 \ ^/	 ^-i^._.____...^^.__._.	 __......	 .._.	
_., t
	
_ -
	
__...^	
..__	 1.1
7	 \	 ^\	 /	 (J
--,	 Q'	 wJ
	
^	 r^
	
^	 ^	 ^
	
{	 N
'^`	 a
/^
	
^ r/	 ^	 ^
	
i I .=r
	
^ s	 Q)
.._^__	 ..	 ._..^^.._._,, ^Y___.^_,. ---
	 -^	 ---
	
J.^
t	 ^^ i I	 i	 Q	 ^{-^
^ 1	 ^ / ,' ^,	 ^	 Sri	 ^	 ^^
^^	
C
	IS ^ 	 i	 Sri
/ ^	 ^'_^I	 i^ -, ^	 C
	
^	 ^	 ^
^	 ^ ^^ I	 ^,	 i
^	 : ^,	 .N
^;	 ::.
^	 ^	 ^^	 ^,
	
.r	 J
Y^
	
L '	 L^	 +--{
ti
-^i
_..^ - — —^
^^
	
^	 ^	 ^^
f j^
	
U	 ^
:^
---_t^--^
	 - -	 -Z --
	
^	 .-I
	
a'	
^	 `°
	
i	 _
_ ^ _^ L
__ -*- - --
	 ^.
	
^,	 /	 i ^	 ,.,
	
^ ^	 \ ^-
	
1	 u.
i	 T` ^^
^:-^^^
..t.._......__1_.__ _ ,.^
	 ^
o	 c.^ ^
	 Lo	 •^_	 N
.-^	 o	 0	 o	 O
.-
^~
^^
-^-y
!^
I
.' ^^
w o
> . ^ ni
^ ;^
	_^  ^ 	 ^ O
I
O
► 	 I
—?—yG^,,,^' -- -^
^ ^
^	 ^ {I
^ ^^
s
	
- -	
,.^
	
^S	 oo	 G
	
O ^	 ^ ..-a
	
^	 e	 1
O	 c^J	 •^-
G O
^	 ^
t
i i	 ^^
v
m
s
ti0
u^
	
^^
7
u;
u^
v
a
^o
.,^
n
E-^
w
ri
^.
WavQ. ^roh a
^^
.,^:^...	 ^^.,.^,,. ,,...F^..^,...,....d.^............. 	 ,
h	 ^^.
p^
Fig, 5 pne^d^,^rtension^^. S^,a}^ a.n Acous^ie Medium:
^_.._^...,.^....
	 r..,...^ .,,^,.^, ., ^,..^.....^_
.,..r,.,	 ..	 ._.^ ^..,.....,^
P` s!- c^h
Fig. 6 Rlgc^ Circular Pietc^n in Ftig^.d .Baffle


Transmission of sonic boom pressure through a window pane

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