- Email: [email protected]
Linear damping models and causality in vibrations
J. Sound Vib. (1970) 13(4), 499-509
LETTERS TO THE EDITOR LINEAR DAMPING MODELS AND CAUSALITY IN VIBRATIONS As is well known, linear viscous and linear hysteretic models for damping generally have their most correct application in small restricted frequency ranges. In these ranges the damping coefficient is usually to be taken at that constant value which yields an energy loss-per-cycle equivalent to that of the more complex physical system being modeled, be it linear or otherwise. In overall use, then, the damping coefficient must be considered only as an average "local constant" which is appropriately changed for different frequency ranges of the system under study. That many systems can be "satisfactorily" modeled in this crude fashion is testimony to the remarkable care, physical sense and vigilance of analysts on the one hand, and of the sustained, relatively good, physical appeal of the linear models on the other. Use of any model out of context is obviously to be avoided, for then its equation rapidly loses meaning simply through failure of the solution to match physical results. With modesty, we do not in general pretend that a given mathematical model for damping is more than a poor crutch, yielding perhaps acceptable results in limited ranges, but certainly not implying any detailed explanation of the underlying physics. This is especially true with the current state of the art in damping as applied to most engineering problems. We consider for discussion a system truly linear in mass and stiffness but non-linear in damping. We assume, however, that damping force is small compared to other forces, so that a linear damping model may still be attempted as a replacement for the actual damping. Let such a system be described by
m.'~+f~(x,~,t) + kx =f(t),
(l)
where the damping force isfa. Althoughfa is small, we shall nonetheless be concerned expressly with its characteristics. We shall seek a value of the damping coefficient c such that fd in (1) may be replacedby the model
(2)
f~_c.~
with c a constant, seeking those conditions for which it will be an adequate model for purposes of describing the overall motion. We shall find that equation (2) is a reasonable model only wl~en the motion x contains a single predominant frequency. 1"here is difficulty, as in all problems concerned with the assessment of damping, in assigning a coefficient c without some prior knowledge of both the nature of the physical "damper" which is in action and the motion to be expected from the system, sincefn depends upon both. If some knowledge of the motion to be expected is assumed, as for example response x at some frequency to, then one method to assign a value to c is to take it equal to c, as defined by the following energy balance per cycle:
f fnS~dt=cz f o
o
499
-?2dt.
(3)
500
LETTERS TO TIlE ED1TOR
Another closely related method may be introduced by first taking the Fourier transform &equation (1); i.e. if f xe-t'~
= X(co)),
--gO
eo
f fde-I~
= F~(co),
~0
f f ( t ) e -"~' dt = F(co), --o0
then equation (1) becomes - , o 2 rex(co) + V~(co) + kX(co) = F(co).
(4)
Since the transform of equation (2) is F~(co) = icoc x (co),
(5)
we may by analogy, define a frequency-dependent damping coefficient as
c(co)
F~(co)
/coX(co)'
(6)
which may be interpreted as the ratio of{he complex amplitudes of force and velocity relative to frequency co. For the type of damper postulated in equation (1) this definition will range over all frequencies contained inJ~ and x for the particular characteristic response elicited b y f ( t ) in equation (I). If the actual response x of equation (1) is not known or anticipated, the devices (3) or (6) may not be directly usable. One method of obtaining the necessary information, however, is to design a separate test of the same damper for which equation (1) is written. This test subjects the damper alone to an en forced sinusoidal input x at some frequency co and measures the resulting damper forcef~. Tests of this variety are classical in the study of the properties of materials. Again using the device (6) we then define a test damping coefficient as
V~T(co) cT(co) = ico xTcco) '
(7)
where the subscript T refers to the results of the sinusoidal test. Changing the test frequency co successively we may plot the function cr(co). However, we note that for many real materials this definition of cr will also be dependent upon the amplitude x of the test. We now may compare the coefficients c(co) from equation (6) and cr(co) from equation (7). We note that in the event that, for each frequency, the ratio Fa/X is the san~e as Fdr/Xr (the case where the amplitude off~ is lhzear in the amplitude of x), the definitions (6) and (7) are equivalent. Note that iffd is non-linear in x this equivalence fails. The definition (3) also gives a real form in the linear case, and when x-motion is sinusoidal, of exactly the same thing as equations (6) or (7), so that in this case all three are equivalent: el = c = Cr, for any specified frequency co. All of the above are variants on conceptions which have long been in common use, especially in those areas of vibrations associated with the measurement of the damping properties of elastomers and the like over wide frequency ranges. (See for example references [I] to [4].) It should be noted that in actual cases it is not usually possible to concentrate attention uniquely upon damping; variable stiffness (and even mass) with frequency and amplitude
LETI"ERS TO TIlE EDITOR
501
are moi-e the rule than the exception. The so-called "complex moduli" are typical results obtained in actual physical cases. These will be referred to again at a later point. Either through energy loss equivalence or force-velocity amplitude-ratio equivalence, the damping coefficients alluded to are defined inherently in the frequency domain; hence the only possible sense in which a form like m 2 + c(oJ)2 + k x = f ( t ) ,
(8)
written in the time domain, can be interpreted is that c(~o) is a constant appropriate to a given frequency ~o. This means then also that the solution of (8) is appropriate only if it centers around a response of this frequency o~. The only use of c(co) as a variable function can of course be in the frequency domain; this would lead to a search for the solution of (4) with equations (6) or (7) defining c(oJ). Even when possible, however, such a solution of necessity destroys the useful simplicity of equation (1) wherefd has been replaced by equation (2), since the new solution will in general be complicated, in fact representative of an equation of higher order than equation (1). The question arises then as to what cases may be treated successfully with c a constant. The general answer is that if c takes on a value appropriate to the undamped resonant frequency ~o= a / k - ~ of(1), the results in the neighborhood of that frequency will be acceptable; for response near another frequency oJt, c = c(o~0 must be used. No single constant is really appropriate for a response covering a large range of frequencies. The reasonableness of using c = c(V'k[m) near the resonance of equation (1) m~iy be appreciated by considering the solution of equation (4). Using equation (6) the transform of x is given by
F(~o) X ( w ) = _oj2 m + io~c(oJ) + k"
(9)
The modulus of this expression will have the denominator
{(k - mo~2)~ + [~oc(~,)]2}tn.
(10)
The minimum of this expression will define the peak amplitude of x response. When, for example, to = v / k / m , the denominator reduces to ~oc(oJ), which will be relatively small for small damping and will define the height of the response peak. The actual minimum occurs where c(w) [c(oJ) + w dc(w)/dm] - 2m(k - m o J 2) = 0. (11) Thus, for small c and slowly varying derivative dc/&o, this result occurs somewhere near ~o = V ' k / m . To some extent then, these simple considerations in the frequency domain justify what vibrations analysts have already been doing for years. C randall [5] has discussed some of the matters touched upon above, particularly the definition (6), which we have pointed out is equivalent to the well-known energy loss definition (3) ivhenfa is linear with amplitude x and motion is sinusoidal. Reference [5] does not point o u t t h e distinction which may occur between equations (6) and (7) if Fd is nomlinear with X, in which case an apriori test cannot adequately define c(co) for arbitrary motion x. Crandall [5] mentions that (8) would be improper; defining a damping loss factor r/by kv(~)
c(~,)--I~1"
02)
Crandall [5] converts (4) with (6) to {-moJ2 + k[l + ir/(oJ) sgn oJ]} X(~) = F(oJ),
(13)
502
T_~ IERS TO TIlE EDITOR
(where sgn ~o refers to the sign of co, normally positive unless doubly infinite frequency domains are employed). When 7/(co) is taken as a constant in (13), the expression may easily be formally inverted to the time domain. However, inversion implicity assumes validity of the result over all frequencies, implying ~ = constant for all frequencies, a condition one may not be willing to accept, if only on physical grounds. The solution cannot therefore be considered valid outside its application to response x at the single frequency co for which ~/has been selected. There is, then, for ",7a constant appropriate to a single frequency co, a special form in the time domain which is still acceptable, namely that in which one may t a k e f ( t ) = f o e ~t and x - - x 0 e "~ with Xo as well as m and k in general complex: mS~+ k(1 +
i~)x
= f o e t~'.
(14)
This form, used in the classical airfoil flutter problem, remains consistent and applicable; in fact it is identical for this case to equations (1) and (2) with c = c(w) defined by equation (12) for the frequency co in question. Reference [5] nonetheless warns against this model (rather than against its use out of context). Using it with x r e a l , f = 8(t), and assuming ~ constant for all frequencies, a "causal" anomaly is there demonstrated, namely a response preceding the input. A much simpler demonstration of the anomalous nature of equation (I 4) out of context is afforded by reference [6], which points out that, for f = 0 and simple initial conditions, equation (14) cannot properly describe the physical case of decaying oscillations. The device k(l + it/) in equation (14) did not at all originate in the context of equation (13) but is of course directly related to the well-known complex modulus tests for materials. It was originally introduced into the flutter problem on physical grounds supported by such tests. Reference [5] therefore commits a historical distortion to regard the various treatments of equation (14) as "'ad hoc rationalizations" of an improperly inverted Fourier transform. Theodorsen and Garrick [7], following an uncited reference of Becket and Foeppl, introduced it as a device providing a damping force "proportional to displacement and in phase with velocity" in the flutter context. Therein its proper use is strictly delimited in reference [8]. In this role, in which the phase relationship between displacement and velocity is exactly preserved throughout the motion considered, the device has correctly served for over two decades of flutter analysis. As mentioned above, no realistic model of damping is as yet possible in the absence of experiment. This is due to the fact that the underlying physical effects in damping are only qualitatively or superficially understood. In general, if c(co) is defined by an experiment on a real material, as suggested earlier, the question of causality need not arise, since the basic causal relationships of the physics of the material are already satisfied. It is only when the effects of a hypothetical model are examined that causal anomalies may arise. Inherent in an experiment is not only the fact that an effect must follow a cause, but also the important quantitative relationships of the effect to the cause. What aretlikely to be noncausal in a hypothetical damping model are arbitrary (non-physical) definitions of its amplitude and phase variations with frequency. What is essentially wrong then with the damping model i~kx used out ofcontext is in part the arbitrary fixing oft/for all frequencies, but equally importantly the locking of its phase to that of io~xoe"~ in cases where this no longer represents the correct damping term. Thus we see that use of the classically acceptable model of the complex material modulus must also be confined to the context of motion representable by the form e ~'. Historically the data from tests have been obtained in this context. Either they are usedin the same context or a reinterpretation must be made to the context ofequations (3) or (7).
LETTER TO TIlE EDITOR
503
The foregoing remarks are offered in commentary upon some confusing points left open by reference [5].
Department of Civil and Geological Enghwering, Princeton University, Princeton, New Jersey 08540, U.S.A.
R . H . SCANLAN
Received 3 August 1970 REFERENCES
1. G. W. PAIN'~R 1951 Bull. Ant. Soc. Test. AIater. 177. The measurement of the dynamic modulus of elastomers by a vector subtraction method. 2. G. W. P~NTER 1958 S.A.E. natn. AIag., Los Angeles Paper 83B. Dynamic properties of BTR elastomer. 3. D. G. FLOMand A. M. BtmCnE 1959 J. appL Phys. 30, 11, 1725. Theory of rolling friction for spheres. 4. D. G. FLOM 19603". appL Phys. 31, 306. Rolling friction of polymeric materials. 5. S. H. ~ a g A L L 1970 J. Sound Vib. 11, 3. The role of damping in vibration theory. 6. R. H. SCANLANand A. MENDELSON1963 AIAA J. 1, 1938. Note on structural damping. 7. T. TItEODORSENand I. E. GARRICK1940 NACA tech. Rep. 685. Mechanism offlutter--a theoretical and experimental investigation of the flutter problem. 8. R. H. SCANLANand R. ROSENBA~t 1968 Aircraft Vibration and Flutter. New York: Dover Publications.
THE SOUND PRESSURE OF A UNIFORM, FINITE, PLANE SOURCE In his paper " N o t e on two common problems of sound propagation" E. J. Rath6 [1] solves the equation for the total sound pressure at a distance a from the centre of a uniform finite plane source x=e/2
( r.~.,.)tot~t --- 2
Y-cl2
-~edxdy~-~--~
2 x-O
y=O
by introducing x = a t a n ~, y = a tan/3, r = a/cos cq R = r/cos/3 without defining c~ or/3. (Note that in the original version Z0 had been replaced by I, presumably a printer's error.) Inspection of Figure 6 shows that c~ must be the angle between a and r, but iffl is the angle between r and R, then y does not equal a tanfl, butis in fact asee-tan]3. The integral then becomes insoluble algebraically, and has to be solved numerically for specific values of a, b and c. Where a is greater than b and c, the error in using Rath6's approximate solution p
z
( r.m.s.)total
=
IVZ0 tan_l(c/2a ) tan_1(b/2a ) 77DC
is negligible, but closer to the source it becomes significant. During the course o f an investigation into the noise radiation from large plane surfaces, six distinct cases have been considered in detail, and the sound pressure level obtained from the approximate algebraic solution to the integral has been compared with the level given by numerical integration. The results are presented as a collection of "error curves" which can be used to correct the sound pressure levels obtained from the approximate algebraic integration. 34
LETTERS TO THE EDITOR LINEAR DAMPING MODELS AND CAUSALITY IN VIBRATIONS As is well known, linear viscous and linear hysteretic models for damping generally have their most correct application in small restricted frequency ranges. In these ranges the damping coefficient is usually to be taken at that constant value which yields an energy loss-per-cycle equivalent to that of the more complex physical system being modeled, be it linear or otherwise. In overall use, then, the damping coefficient must be considered only as an average "local constant" which is appropriately changed for different frequency ranges of the system under study. That many systems can be "satisfactorily" modeled in this crude fashion is testimony to the remarkable care, physical sense and vigilance of analysts on the one hand, and of the sustained, relatively good, physical appeal of the linear models on the other. Use of any model out of context is obviously to be avoided, for then its equation rapidly loses meaning simply through failure of the solution to match physical results. With modesty, we do not in general pretend that a given mathematical model for damping is more than a poor crutch, yielding perhaps acceptable results in limited ranges, but certainly not implying any detailed explanation of the underlying physics. This is especially true with the current state of the art in damping as applied to most engineering problems. We consider for discussion a system truly linear in mass and stiffness but non-linear in damping. We assume, however, that damping force is small compared to other forces, so that a linear damping model may still be attempted as a replacement for the actual damping. Let such a system be described by
m.'~+f~(x,~,t) + kx =f(t),
(l)
where the damping force isfa. Althoughfa is small, we shall nonetheless be concerned expressly with its characteristics. We shall seek a value of the damping coefficient c such that fd in (1) may be replacedby the model
(2)
f~_c.~
with c a constant, seeking those conditions for which it will be an adequate model for purposes of describing the overall motion. We shall find that equation (2) is a reasonable model only wl~en the motion x contains a single predominant frequency. 1"here is difficulty, as in all problems concerned with the assessment of damping, in assigning a coefficient c without some prior knowledge of both the nature of the physical "damper" which is in action and the motion to be expected from the system, sincefn depends upon both. If some knowledge of the motion to be expected is assumed, as for example response x at some frequency to, then one method to assign a value to c is to take it equal to c, as defined by the following energy balance per cycle:
f fnS~dt=cz f o
o
499
-?2dt.
(3)
500
LETTERS TO TIlE ED1TOR
Another closely related method may be introduced by first taking the Fourier transform &equation (1); i.e. if f xe-t'~
= X(co)),
--gO
eo
f fde-I~
= F~(co),
~0
f f ( t ) e -"~' dt = F(co), --o0
then equation (1) becomes - , o 2 rex(co) + V~(co) + kX(co) = F(co).
(4)
Since the transform of equation (2) is F~(co) = icoc x (co),
(5)
we may by analogy, define a frequency-dependent damping coefficient as
c(co)
F~(co)
/coX(co)'
(6)
which may be interpreted as the ratio of{he complex amplitudes of force and velocity relative to frequency co. For the type of damper postulated in equation (1) this definition will range over all frequencies contained inJ~ and x for the particular characteristic response elicited b y f ( t ) in equation (I). If the actual response x of equation (1) is not known or anticipated, the devices (3) or (6) may not be directly usable. One method of obtaining the necessary information, however, is to design a separate test of the same damper for which equation (1) is written. This test subjects the damper alone to an en forced sinusoidal input x at some frequency co and measures the resulting damper forcef~. Tests of this variety are classical in the study of the properties of materials. Again using the device (6) we then define a test damping coefficient as
V~T(co) cT(co) = ico xTcco) '
(7)
where the subscript T refers to the results of the sinusoidal test. Changing the test frequency co successively we may plot the function cr(co). However, we note that for many real materials this definition of cr will also be dependent upon the amplitude x of the test. We now may compare the coefficients c(co) from equation (6) and cr(co) from equation (7). We note that in the event that, for each frequency, the ratio Fa/X is the san~e as Fdr/Xr (the case where the amplitude off~ is lhzear in the amplitude of x), the definitions (6) and (7) are equivalent. Note that iffd is non-linear in x this equivalence fails. The definition (3) also gives a real form in the linear case, and when x-motion is sinusoidal, of exactly the same thing as equations (6) or (7), so that in this case all three are equivalent: el = c = Cr, for any specified frequency co. All of the above are variants on conceptions which have long been in common use, especially in those areas of vibrations associated with the measurement of the damping properties of elastomers and the like over wide frequency ranges. (See for example references [I] to [4].) It should be noted that in actual cases it is not usually possible to concentrate attention uniquely upon damping; variable stiffness (and even mass) with frequency and amplitude
LETI"ERS TO TIlE EDITOR
501
are moi-e the rule than the exception. The so-called "complex moduli" are typical results obtained in actual physical cases. These will be referred to again at a later point. Either through energy loss equivalence or force-velocity amplitude-ratio equivalence, the damping coefficients alluded to are defined inherently in the frequency domain; hence the only possible sense in which a form like m 2 + c(oJ)2 + k x = f ( t ) ,
(8)
written in the time domain, can be interpreted is that c(~o) is a constant appropriate to a given frequency ~o. This means then also that the solution of (8) is appropriate only if it centers around a response of this frequency o~. The only use of c(co) as a variable function can of course be in the frequency domain; this would lead to a search for the solution of (4) with equations (6) or (7) defining c(oJ). Even when possible, however, such a solution of necessity destroys the useful simplicity of equation (1) wherefd has been replaced by equation (2), since the new solution will in general be complicated, in fact representative of an equation of higher order than equation (1). The question arises then as to what cases may be treated successfully with c a constant. The general answer is that if c takes on a value appropriate to the undamped resonant frequency ~o= a / k - ~ of(1), the results in the neighborhood of that frequency will be acceptable; for response near another frequency oJt, c = c(o~0 must be used. No single constant is really appropriate for a response covering a large range of frequencies. The reasonableness of using c = c(V'k[m) near the resonance of equation (1) m~iy be appreciated by considering the solution of equation (4). Using equation (6) the transform of x is given by
F(~o) X ( w ) = _oj2 m + io~c(oJ) + k"
(9)
The modulus of this expression will have the denominator
{(k - mo~2)~ + [~oc(~,)]2}tn.
(10)
The minimum of this expression will define the peak amplitude of x response. When, for example, to = v / k / m , the denominator reduces to ~oc(oJ), which will be relatively small for small damping and will define the height of the response peak. The actual minimum occurs where c(w) [c(oJ) + w dc(w)/dm] - 2m(k - m o J 2) = 0. (11) Thus, for small c and slowly varying derivative dc/&o, this result occurs somewhere near ~o = V ' k / m . To some extent then, these simple considerations in the frequency domain justify what vibrations analysts have already been doing for years. C randall [5] has discussed some of the matters touched upon above, particularly the definition (6), which we have pointed out is equivalent to the well-known energy loss definition (3) ivhenfa is linear with amplitude x and motion is sinusoidal. Reference [5] does not point o u t t h e distinction which may occur between equations (6) and (7) if Fd is nomlinear with X, in which case an apriori test cannot adequately define c(co) for arbitrary motion x. Crandall [5] mentions that (8) would be improper; defining a damping loss factor r/by kv(~)
c(~,)--I~1"
02)
Crandall [5] converts (4) with (6) to {-moJ2 + k[l + ir/(oJ) sgn oJ]} X(~) = F(oJ),
(13)
502
T_~ IERS TO TIlE EDITOR
(where sgn ~o refers to the sign of co, normally positive unless doubly infinite frequency domains are employed). When 7/(co) is taken as a constant in (13), the expression may easily be formally inverted to the time domain. However, inversion implicity assumes validity of the result over all frequencies, implying ~ = constant for all frequencies, a condition one may not be willing to accept, if only on physical grounds. The solution cannot therefore be considered valid outside its application to response x at the single frequency co for which ~/has been selected. There is, then, for ",7a constant appropriate to a single frequency co, a special form in the time domain which is still acceptable, namely that in which one may t a k e f ( t ) = f o e ~t and x - - x 0 e "~ with Xo as well as m and k in general complex: mS~+ k(1 +
i~)x
= f o e t~'.
(14)
This form, used in the classical airfoil flutter problem, remains consistent and applicable; in fact it is identical for this case to equations (1) and (2) with c = c(w) defined by equation (12) for the frequency co in question. Reference [5] nonetheless warns against this model (rather than against its use out of context). Using it with x r e a l , f = 8(t), and assuming ~ constant for all frequencies, a "causal" anomaly is there demonstrated, namely a response preceding the input. A much simpler demonstration of the anomalous nature of equation (I 4) out of context is afforded by reference [6], which points out that, for f = 0 and simple initial conditions, equation (14) cannot properly describe the physical case of decaying oscillations. The device k(l + it/) in equation (14) did not at all originate in the context of equation (13) but is of course directly related to the well-known complex modulus tests for materials. It was originally introduced into the flutter problem on physical grounds supported by such tests. Reference [5] therefore commits a historical distortion to regard the various treatments of equation (14) as "'ad hoc rationalizations" of an improperly inverted Fourier transform. Theodorsen and Garrick [7], following an uncited reference of Becket and Foeppl, introduced it as a device providing a damping force "proportional to displacement and in phase with velocity" in the flutter context. Therein its proper use is strictly delimited in reference [8]. In this role, in which the phase relationship between displacement and velocity is exactly preserved throughout the motion considered, the device has correctly served for over two decades of flutter analysis. As mentioned above, no realistic model of damping is as yet possible in the absence of experiment. This is due to the fact that the underlying physical effects in damping are only qualitatively or superficially understood. In general, if c(co) is defined by an experiment on a real material, as suggested earlier, the question of causality need not arise, since the basic causal relationships of the physics of the material are already satisfied. It is only when the effects of a hypothetical model are examined that causal anomalies may arise. Inherent in an experiment is not only the fact that an effect must follow a cause, but also the important quantitative relationships of the effect to the cause. What aretlikely to be noncausal in a hypothetical damping model are arbitrary (non-physical) definitions of its amplitude and phase variations with frequency. What is essentially wrong then with the damping model i~kx used out ofcontext is in part the arbitrary fixing oft/for all frequencies, but equally importantly the locking of its phase to that of io~xoe"~ in cases where this no longer represents the correct damping term. Thus we see that use of the classically acceptable model of the complex material modulus must also be confined to the context of motion representable by the form e ~'. Historically the data from tests have been obtained in this context. Either they are usedin the same context or a reinterpretation must be made to the context ofequations (3) or (7).
LETTER TO TIlE EDITOR
503
The foregoing remarks are offered in commentary upon some confusing points left open by reference [5].
Department of Civil and Geological Enghwering, Princeton University, Princeton, New Jersey 08540, U.S.A.
R . H . SCANLAN
Received 3 August 1970 REFERENCES
1. G. W. PAIN'~R 1951 Bull. Ant. Soc. Test. AIater. 177. The measurement of the dynamic modulus of elastomers by a vector subtraction method. 2. G. W. P~NTER 1958 S.A.E. natn. AIag., Los Angeles Paper 83B. Dynamic properties of BTR elastomer. 3. D. G. FLOMand A. M. BtmCnE 1959 J. appL Phys. 30, 11, 1725. Theory of rolling friction for spheres. 4. D. G. FLOM 19603". appL Phys. 31, 306. Rolling friction of polymeric materials. 5. S. H. ~ a g A L L 1970 J. Sound Vib. 11, 3. The role of damping in vibration theory. 6. R. H. SCANLANand A. MENDELSON1963 AIAA J. 1, 1938. Note on structural damping. 7. T. TItEODORSENand I. E. GARRICK1940 NACA tech. Rep. 685. Mechanism offlutter--a theoretical and experimental investigation of the flutter problem. 8. R. H. SCANLANand R. ROSENBA~t 1968 Aircraft Vibration and Flutter. New York: Dover Publications.
THE SOUND PRESSURE OF A UNIFORM, FINITE, PLANE SOURCE In his paper " N o t e on two common problems of sound propagation" E. J. Rath6 [1] solves the equation for the total sound pressure at a distance a from the centre of a uniform finite plane source x=e/2
( r.~.,.)tot~t --- 2
Y-cl2
-~edxdy~-~--~
2 x-O
y=O
by introducing x = a t a n ~, y = a tan/3, r = a/cos cq R = r/cos/3 without defining c~ or/3. (Note that in the original version Z0 had been replaced by I, presumably a printer's error.) Inspection of Figure 6 shows that c~ must be the angle between a and r, but iffl is the angle between r and R, then y does not equal a tanfl, butis in fact asee-tan]3. The integral then becomes insoluble algebraically, and has to be solved numerically for specific values of a, b and c. Where a is greater than b and c, the error in using Rath6's approximate solution p
z
( r.m.s.)total
=
IVZ0 tan_l(c/2a ) tan_1(b/2a ) 77DC
is negligible, but closer to the source it becomes significant. During the course o f an investigation into the noise radiation from large plane surfaces, six distinct cases have been considered in detail, and the sound pressure level obtained from the approximate algebraic solution to the integral has been compared with the level given by numerical integration. The results are presented as a collection of "error curves" which can be used to correct the sound pressure levels obtained from the approximate algebraic integration. 34