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LD3907 13 -2 Z H 5 S
•07 1
1951 Knudsen, John R 1916-
•K65 A study of effects of vlsooslty end
heat conductivity on the transmission
of sound v/aveo in a compressible fluid*
57»12p« diagre*
Thesis (Ph«D«) - N*Y.U., Graduate
School, 19S1#
C8I863
Sholt List
Xerox University Microfilms, Ann Arbor, Michigan 48106
THIS DISSERTATION HAS BEEN MICROFILMED EXACTLY AS RECEIVED.
V PRATT .TP
Y. v
Hrw T. ' i'k'T ; ’T
iifc.1 tv : •
A Study of Sffeets of Viscosity and Bo«t
Conductivity on the Transmission of Sound
Waves In a Compressible Fluid.
John H. Xhudsen
April 1, 1951
A dissertation in the departacnt of
Mathematics submitted in partial fu lfill
ment of the requirements forthe degree of
Doctor of Philosophy at Nev York Uniyerslty.
1 am deeply Indebted to Prof* K. 0. Zriedriohs
for hie peraonal Interest, guidance and extreme
petienoe during the course of development of
this dissertation* Also, my thanks to
Arof. J. J. stoker for his Interest and timely
suggestions in the preparation of the final oopy*
J* R. K.
CCHTBTTS
fart I. General Dlaouaalan.
1* The Statement of the Problem Page l
2. General Results 4
fart IX. Mathematical Analysis.
The D ifferential fquatlana 11
4. In itia l and Boundary Conditions 12
5. Solution of the Initial-Boundary Value Problem 13
&• o c « l/f; U(t) ■ 1 for 0 <. t 17
6-1. fixed x and Large t 19
6-8. Obaerrer x - xQ e0t for Large t £0
6-3. Obaerrer x « ko0^i + 0ot for Large t 85
7. l/f» 0{t) ■ 1 for 0 -■ t « ti» U(t) * 0 for t^<- t 20
7-1. fixed x and Large t 29
7-2. Obaerrer x * ko0/t + e0t for large t 30
0. ot • 0; U(t) * 1 for 0 <• t 32
8-1. fixed x and Large t 36
8-2. Obaerrer x « x0 + o0t for Large t 37
8-3. Obaerrer x « ko0JT o0t for Large t 42
9. oc» 0; U(t) ■ 1 for 0 t •< t^, U(t) ~ 0 for t^ *< t 45
9-1. fixed x and large t 46
8-2. Obaerrer x - kn0\ft + eQt for large t 46
10. at ■ oo (X ■ 0) 48
10-1. Tlxed x and large t SO
10-g. Obaerrer x * x0 •»- e0t for large t 50
10-3. Obaerrer x - ko0Jt + 0ot for large t 51
11. etm oo; U(t) « 1 for 0 < t ■< t lt U(t) = 0 for t 1<. t 58
18. Penalty and Tang>erature 53
18-1. Praaaure 57
.Appendix
PART I, aatgULL DISCUSSION
1. Ihe 3tatenant of the Problem.
In connection with the problem of transmission of sound
waves in a compressible fluid it is of interest to study the effects
of viscosity and heat conductivity, since these effeots are always
present in a real situation. Our purpose here is to study these
effects in the case of an ideal polytropic gas in the sem i-infinite
channel of Fig. 1. We assume, for the purpose of simplifying the
b z r
XIIo rX
Fig. 1
mathematical problem, that the gas is in itially at rest in the
channel with constant pressure, density and temperature throughout.
Imagine that we have constructed at the left end of the
channel a sort of fan or blower B directed to the right into the
channel and producing the same kind of gas as is in the channel.
The blower should operate in sueh a way that it generates at any
time t a flow of gas with a prescribed veloeity at z ® 0. The
introduction of this prescribed flow speed at the entranoe to the
channel has the desired effeot of produoing a wave motion in the
channel.
2
Assume the blower action to oauae a ware to be propagated
to the right into the undisturbed gas. What, then, can be said about
the form and amplitude of this wave and about the general velocity,
pressure, density and temperature distribution along the channel for
various values of the time t? For example, suppose that at a certain
time, say t = 0, the flow speed at the mouth of the channel is suddenly
raised by aotion of the blower from the value zero to a value, say 1,
which is then maintained indefinitely. A wave front w ill then be
propagated to the right moving into the original undisturbed gas ahead
of it and followed by a region of disturbed gas behind it. In the
rather triv ia l first approximation in which the differential equations
of motion are linearized and in whieh viscosity and heat conductivity
effecta are neglected the wave assumes the form of a so-called
"square wave" moving with the apeed of sound into the gas at rest
and followed by a steady state region with constant flow speed 1. If
viscosity and heat conductivity effects are not neglected the form of
the wave is distorted and the region behind it ia not a steady state.
It w ill be our purpose here to Investigate in a linear or "acoustic”
approximation the effecta of visoosity and heat conductivity on suoh
a wave motion.
One other simplifying feature is introduced, i.e ., the
boundary layer effeots along the walls of the channel w ill be
neglected. Thus the problem is one-dimensional.
Of some interest is a description of the flow as seen by
an observer located at a fixed point x * xQ for large valuea of
3
time t. Ibis description w ill show the lim it flow approached in the
channel behind the wave front aa time increases indefinitely and the
manner in which this lim it ia approached.
Of greater interest is the description of the wave front as
aeen by an observer moving with it. Accordingly, we introduce an
observer moving along the channel at a speed equal to the sound speed
of the gas at reat, that la, an observer moving according to the law
x s x0 + c0t. from the point of view of auoh an observer the velooity
and pressure of the gaa in the wave will depend upon ^*and A, the
ooeffioienta of vlacoaity and heat conductivity, respectively, and
upon the time t. In particular, we shall be concerned with the
appearance of the velooity and pressure aa a function of x0 for large
values of t.
An even better description of the wave is obtained by an
observer moving according to the law x = ke0/T + c0t. Inasmuch as a
fixed k interval w ill cover an increasingly large portion of the
x-axis such an observer aoqulrss more information concerning the wave.
Ibis particular observer is suggested by the mathematical analysis of
the problaa which is relatively simple in this case.
As far as the nature of the prescribed velocity at the left
end of the channel ia concerned we shall first set up the problem in
a general way with an arbitrary prescribed boundary oonditian at
x & 0. Them we shall discuss the special case mentioned aoove in
whioh, with the gaa in itially at rest in the channel, the velocity
at x s 0 is suddenly increased at time t * 0 from zero to same value,
4
say 1, which remains constant for a ll time. Also, as a more Interesting
oase we take that in which the velocity at x = 0 is increased frcm zero
to 1 at time t = 0 and then suddenly decreased to zero again at some
time t = tj_. We are interested, in particular, in the decay of the
wave in the latter case as t increases indefinitely.
With respect to values of ju. and X three situations will oe
discussed. These are the two extreme situations in which viscosity and
heat conductivity in turn do not appear and an intermediate case in
which where i is the gas constant. (When necessary to give
A W
i a numerical value we assume the gas to be air and set V = 1.4). The
oases yu_- 0 and X » 0, although artificial from a physical standpoint
present Interesting mathematical problems whose conclusions at least
approximate physical situations. The particular value -^-for ~ as an
intermediary oase is chosen for two reasons. for one thing it serves
to greatly simplify the mathematical analysis of the problem. Also,
for 'i * 1.4 it is quite close to the ratio — s 1 which Becker
X z l - 1
has shown to be a good approximation in a real physical situation.
Z* General Results.
In fart II w ill be found the mathematical solution of the
problem for the oaaea at. ■ 0, and oo with the two types of
wares described above as viewed by the several observers for large
values of t. We state at this point seme of the more significant
results for the oase oc » .
O
Let u, p and^o, a ll funotiana of x and t, be the velocity,
