Introduction
Basal melt water is widely considered to have a major influence upon the sliding behavior of temperate glaciers. Numerous field studies have demonstrated links between variations in the subglacial drainage network and variations in sliding speed, with enhanced sliding during the melt season a nearly ubiquitous observation. Nonetheless, there has been a lack of consensus as to whether sliding-speed variations are due primarily to the amount of melt water or to the water pressure. Much of this controversy centers on issues of geometry: just how is melt water at the glacier bed distributed? Is melt water transported primarily through conduits, through a widespread, thin sheet, or through permeable till? How might the flow configuration be related to the glacier’s dynamic behavior?
Analyses have been developed to describe various aspects of subglacial water flow, both for sheet flow (Reference WeertmanWeertman, 1972; Reference WalderWalder, 1982; Reference Weertman and BirchfieldWeertman and Birchfield, 1983[b]) and for flow in channels incised into the basal ice (Reference RöthlisbergerRöthlisberger, 1972; Reference ShreveShreve, 1972; Reference WeertmanWeertman, 1972; Reference NyeNye, 1976; Reference Spring and HutterSpring and Hutter, 1981). The subglacial drainage system is likely also to include cavities formed by “separation” of sliding ice from the glacier bed. Reference LliboutryLliboutry (1976, 1978, 1979) has suggested that cavities at temperate-glacier beds may be characterized as either “autonomous” storage sites or as forming part of an “interconnected hydraulic regime” draining melt water. Detailed geomorphic mapping of recently deglaciated bedrock surfaces (Reference WalderWalder and Hallet, 1979; Reference Hallet and AndersonHallet and Anderson, 1980) suggests that “interconnected” cavities form part of a network of anastomosing conduits, the other “segments” of the conduit network being channels incised into the basal ice (R channels) or into the bed (Nye channels). Recent field studies of glacier dynamics and hydraulics (e.g. Reference IkenIken and others, 1983; Reference KambKamb and others, 1985; Reference IkenIken and Bindschadler, 1986) also strongly suggest an important role for subglacial cavities in melt-water drainage. This paper describes a theoretical model developed to explain some aspects of water flow through subglacial cavities.
Analysis of Steady Flow in Cavities
I will focus attention on flow in cavities filled with water such that the water flows “under pressure”. Viscous dissipation in the flowing water produces frictional heat that tends to melt the cavity walls; at the same time, the cavity will tend to collapse as long as the water pressure is less than the ice-overburden pressure. This is quite analogous to the situation in an R channel (e.g. Reference RöthlisbergerRöthlisberger, 1972, p. 178). The cavity will in addition tend to be “opened” by continual sliding of the glacier over its bed. In the steady state, the opening and closing processes must balance. It will turn out that melting is often of secondary importance, compared to sliding, as a cavity-enlarging process. This leads to very significant differences between the hydraulic behavior of R channels and that of cavities. In particular, the strong tendency for “stream capture” and flow localization in a few large cavities, so characteristic of R channels (Reference RöthlisbergerRöthlisberger, 1972; Reference ShreveShreve, 1972), should not exist in a cavity-dominated drainage system.
A major consideration in the present analysis is an idealization of cavity geometry motivated by observations of exhumed melt-water drainage systems on recently deglaciated bedrock (Reference WalderWalder and Hallet, 1979; Reference Hallet and AndersonHallet and Anderson, 1980; and my observations at Findelengletscher and elsewhere). I assume that cavities are typically elongated along the lee sides of bedrock protuberances, with water typically flowing through cavities in a direction oblique to the local ice-flow direction. The elongated, cavity-like conduits may be linked to each other by short R channels and Nye channels; furthermore, cavity cross-sections may change along the water-flow direction due to variations in bed topography (cf. Reference KambKamb, in press). The result is a complex, tortuous drainage system that largely “avoids” regions of the bed where the normal stress exerted by the ice is large (Fig. la). This model differs significantly from the hypothetical situation of cavities joined along the ice-flow direction by R channels and Nye channels, much like beads on a string (Fig. lb). The latter scenario requires that channels traverse high-pressure regions on the stoss sides of bed protuberances.
a. Schematic plan view of basal drainage network. Conduits largely “avoid” high-pressure regions of the bed (hatched ). b. Hypothetical basal drainage system in which channels and cavities connect only along the ice-flow direction, with channels crossing high-pressure regions (hatched). This is probably not a physically plausible configuration.
The model of the subglacial conduit system in Figure la is clearly a simplification that leaves out important features of the “real” system. Perhaps the most significant departure from reality is the implicit assumption that the water flow is essentially rectilinear. Real cavities may have more than one “entrance” and “exit”, as well as “dead-end” regions that function essentially as storage sites (cf. Reference WalderWalder and Hallet, 1979, fig. 7). Modeling the flow in such a drainage system might be possible by using numerical techniques analogous to those developed for simulating flow in porous media (e.g. Reference DullienDullien, [c1979]; Reference Seeburger and NurSeeburger and Nur, 1984; Reference YaleYale, unpublished). In the present study, however, 1 am seeking only to elucidate basic features of cavity hydraulics; these should not be grossly affected by the geometrical simplifications introduced into the analysis.
Cavity size
I now derive an approximate expression for steady-state cavity size in the interconnected regime. (The development here parallels that of Reference WeertmanWeertman and Birchfield (1983[a], p. 24–25), who considered the case of autonomous cavities.) For clarity, cavities will be assumed to form on the lee sides of “steps” of height R on the bed and to be much longer along the water-flow direction than in any transverse direction (Fig. 2). I will define an “effective cavity-closure rate” ue as the difference between the rate of closure u c by creep and the rate of melting um of the walls. This effective closure rate will be balanced with the rate of cavity opening by sliding to find the cavity “length” L, defined in Figure 2.
Idealized cavity geometry.
The creep-closure rate will be approximately (cf. Reference NyeNye, 1953)
where pe = Pi – pw is the effective pressure, with being approximately the ice-overburden pressure and pw the water pressure in the cavity; n and A are flow-law parameters for ice (Reference NyeNye, 1953). The melting rate um is given by
where Q is the volumetric flux of water through the cavity; G is the hydraulic gradient (units of Pa/m), assumed constant along the water-flow path; vi is the specific volume of ice; P is the perimeter of the cavity roof for a section transverse to the water-flow direction; and H is the latent heat of fusion per unit mass. The factor (1-γ) ≈ 0.68 corrects for the energy required to maintain the water at the pressure melting-point (Reference RöthlisbergerRöthlisberger, 1972, p. 179). Implicit in Equation (2) are two assumptions; first, the rate of melting of the cavity roof is spatially uniform; secondly, any frictional heat that flows into the glacier bed eventually causes melting of basal ice in the vicinity of the cavity.
Combining Equations (1) and (2) gives
For simplicity, I will follow Reference LliboutryLliboutry (1978) and assume the cavity roof is elliptical in cross-section. P will then generally be given by an elliptic integral, but assuming “well-developed” cavities such that L < c. 3R, one may approximate (cf. Reference SelbySelby, [C1972], p. 18)
where k1 = π/23/2 ≈ 1.1. Note from Equation (3) that, for sufficiently low effective pressure pe , the effective closure rate would vanish; hence, no stable, finite-sized cavity could exist. This point will be elaborated later.
Consider now the role of sliding in keeping a cavity open. The time T c that any “particle” of ice in the cavity roof is separated from the bed is given approximately by
In this same period of time, the “particle” will have moved (by definition) a distance L along the ice-flow direction. For sliding speed us, therefore,
so combining Equations (3) and (6),
For the approximate case of “long” cavities, substitution of Equation (4) into Equation (7) and a bit of re-arrangement leads to the result:
where 
An interesting feature of this result is that, in comparison to sliding, melting will often have a rather minor effect on cavity length. This is indicated in Table I, in which values of 
are given for several plausible values of flux Q, hydraulic gradient G (cf. Appendix), and cavity height R. Unless the discharge through a single cavity is a significant fraction of the total melt-water discharge from the glacier, 
will usually be small in comparison with us, the sliding speed. Cavity length will then be given approximately by
Effective Melting Rate 
m
, IN m/a, for Flow in Cavities of Height R
= 0.1 m. for 
<< us, The Sliding Speed, Melting is of Negligible Importance in Determining Cavity Size
Reference KambKamb and others (1985) have also argued that the geometry of interconnected cavities will be primarily determined by water-pressure effects rather than by melting.
Stability characteristics of a cavity-dominated drainage system
An interesting result bearing on the question of stability of interconnected cavities also follows from Equation (8). Noting first that the cavity’s cross-sectional area is approximately equal to π RL/4, one may write
Assuming turbulent flow, s may also be related to discharge Q by the Manning formula (cf. Reference NyeNye, 1976, p. 189), which for the cavity may be shown to take the form
where 
with m the Manning roughness coefficient, vw the specific volume of water, and g the acceleration of gravity. It should be stressed that Equation (10) assumes G to be constant, i.e. there are no constrictions in the cavity. Combining Equations (9) and (10) results in
For comparison, the corresponding expression for a steady-state R channel may be written as (cf. Röthlisbereer 1972 p. 180)
Equation (11) has two important consequences. First, it shows that in the steady state, the greater the discharge, the greater must be the pressure gradient, for fixed pe This is precisely opposite to the steady-state behavior of R channels (Reference RöthlisbergerRöthlisberger, 1972, p. 180) and has important consequences for evolution of the drainage network. Should two cavities empty into a common “sink”, then up-stream of that point, water pressure in the cavity carrying the greater discharge will be more than that in the cavity carrying the lesser discharge. Hence, “capture” of melt water by a few main conduits is strongly disfavored for the sort of interconnected cavities envisaged here. A system of relatively small conduits distributed over much of the glacier bed would be stable, in accordance with conclusions reached by Reference KambKamb and others (1985). This theoretical prediction is in good accord with field studies of former subglacial drainage conduits exposed on recently deglaciated bedrock (Reference WalderWalder and Hallet, 1979; Reference Hallet and AndersonHallet and Anderson, 1980). These studies clearly indicated that there had been no long-term tendency for development of arborescence in drainage systems dominated by cavities and Nye channels.
I have previously alluded to the second major consequence of Equation (11), namely, that for pe below a certain value, there will be no stable, finite-sized interconnected cavities, i.e. the effective cavity closure rate ue will vanish. It may be shown that this condition is equivalent to setting the denominator of Equation (11) equal to zero, “Permissible” steady-state values of p e are therefore given by the expression
where
The lower limit on allowable values of p
e
, which I will denote by pe
min
, should be in the range of physically reasonable values. To calculate Pe
min
, I take the flow-law parameters n and A to be 3 and 5.14 – 10–7Pa s1/3
, respectively (Reference LliboutryLliboutry, 1983, p. 220). R is probably on the order of 0.1 m, a value suggested by measurements of uplift of glacier surfaces (Reference IkenIken and others, 1983; Reference KambKamb and others, 1985; Reference IkenIken and Bindschadler, 1986) and by observations of deglaciated bedrock (Reference WalderWalder and Reference Hallet and AndersonHallet, 1979; Reference Hallet and AndersonHallet and Anderson, 1980); the weak dependence of pe
min on R (pe
min α R
2/9
for n
= 3) means that a ten-fold change in R will change p
e
min
by a factor of only about 1.7. The Manning roughness is probably on the order of 0.1m–1/3 s (Reference RöthlisbergerRöthlisberger, 1972, p. 181, table I). Estimation of the hydraulic potential gradient G is somewhat more problematical. The tortuous flow paths in an interconnected cavity system will reduce G below the values of c. 102–103Pa/m that would likely obtain in R channels along the ice-flow direction (e.g. Reference WeertmanWeertman, 1972, p. 293). Moreover, the likely existence of constrictions in the cavity network may have a large effect on G in the “wide” cavities on which we are focusing; as shown in the Appendix, much of the hydraulic potential drop may occur in fairly short constrictions (as also stressed by Reference KambKamb (in press)). I therefore estimate G to be c. 10—102 Pa/m in the “wide” cavities. Using the above estimates and taking 
one finds
The upper bound on this estimate seems somewhat large -certainly, smaller values of pe have been measured at glaciers that are apparently not behaving unstably (e.g. Engelhardt and others, 1978) - but the estimated values of pe min are certainly in the range of physically plausible values.
The notion of a water-pressure threshold at which interconnected cavities would become unstable was apparently first advanced by Reference IkenIken (1981, p. 34). On the basis of simple force-balance arguments rather than any explicit model of cavity formation, Iken gave the minimum effective pressure for cavity stability on a “wavy” bed as
where τ is the basal shear stress and δmax is “the angle between the steepest tangent on the stoss faces of bed obstacles and the mean bed slope”. For τ on the order of 0.1 MPa (1 bar) and δmax no more than a few tens of degrees, pe mm from this criterion would probably be less than c. 1 MPa (10 bar).
Reference KambKamb (in press) has proposed that an interconnected cavity system will be unstable to pressure perturbations at sufficiently high water pressure. Based on a formal analysis of cavity formation rather more complex than presented above, Reference KambKamb predicted that (in our notation)
where ηi is the effective linear viscosity of the ice. Equation (13 and Kamb’s expression (Equation (15)) are similar, though not identical, for n = I (in which case A is equal to ηi). One obvious difference between the two expressions is that sliding speed u s does not enter explicitly into Equation (13). This might be related to the fact that, unlike Reference KambKamb, I have not explicitly examined the hydraulics of the constrictions between cavities. Reference KambKamb has argued that such constrictions (which he calls “wave cavities”) differ significantly in their hydraulic characteristics from the larger “step cavities” (cavities typically formed at the lee of bed-slope discontinuities) that I have analyzed here.
Despite uncertainty over the detailed nature of the parameters controlling the instability, it does seem clear that a system of interconnected cavities will be unstable at sufficiently high water pressure. It seems probable that, if the water pressure in an interconnected cavity system rose to values high enough to stop cavity closure, the amount of ice-bedrock separation would increase rapidly, lowering the effective bed roughness and causing accelerated sliding (Reference IkenIken, 1981; Reference IkenIken and others, 1983; Reference IkenIken and Bindschadler, 1986). It is likely that flow through the interconnected cavities could be “short-circuited” by rapidly growing R channels oriented along the ice-flow direction. “Leakage” of water from cavities by way of a thick water sheet is less likely because of adverse pressure gradients near cavity margins (cf. Reference RöthlisbergerRöthlisberger and Reference IkenIken, 1982); moreover, any sheet-like leakage would be unstable relative to channelized leakage (Reference WalderWalder, 1982).
I suggest that, once the “short-circuits” develop, the drainage network will evolve along one of two paths. In the first, which is related to the scenario proposed by Reference KambKamb (in press) in relation to glacier-surge termination, the newly created R channels rapidly grow and capture the melt water previously carried by the interconnected cavities. An arborescent network of R channels develops. This is probable only if the total melt-water discharge through these unfavorably situated R channels (which must cross high-pressure regions of the bed) remains very large, leading to the high melting rates necessary to overcome the relatively high rates of creep closure. If the melt-water discharge is not sufficiently high, the drainage system will evolve differently; either newly formed R channels will be squeezed shut before an arborescent R-channel network has time to develop, or such a network may form, but only temporarily. Collapse of these R channels will essentially restore the former interconnected cavity system. This latter evolutionary path resembles that suggested by Reference IkenIken and others (1983, p. 36). It is also noteworthy that Reference IkenIken and Bindschadler (1986) found the us versus pw relationship at Findelengletscher is essentially time-invariant, despite the occurrence of several large perturbations caused by down-glacier propagating disturbances in the subglacial drainage network. Apparently, this network, which Reference IkenIken and Bindschadler concluded is cavity-dominated, was able to recover from the effects of the high-pressure disturbances.
Flow rates in the subglacial drainage system
The rate of movement of melt water through the subglacial drainage system, as determined from dye-tracer studies, has been used to infer characteristics of that system (e.g. Reference StenborgStenborg, 1969; Reference KrimmelKrimmel and others, 1973; Reference BurkimsherBurkimsher, 1983). It is important to recognize that flow rates may be quite different for various drainage configurations.
The mean flow rate Ū for a conduit may generally be defined as
where Q is the volumetric flux and s is the cross-sectional area. For a cavity, Equation (10) implies that
Before estimating numerical values for Ūcavity , it is critical to recognize that any physically plausible cavity system will have constrictions in which the flow cross-section is greatly reduced (cf. Reference KambKamb, in press; see also field studies by Reference WalderWalder and Reference Hallet and AndersonHallet (1979) and by Reference Hallet and AndersonHallet and Anderson (1980)). As shown in the Appendix, the occurrence of such constrictions nearly always leads to the condition
where the subscript “wide” refers to the relatively wide sections of the cavity. I therefore restrict the discussion to those sections. It is also shown in the Appendix that the hydraulic gradient in the wide sections satisfies the condition
for an R channel oriented along the ice-flow direction. It is plausible that Gwide is one or two orders of magnitude smaller than Gchannel .
Table II gives values of Ūcavity for plausible values of Gcavity and R with the other parameter values as before. It is clear that the mean flow rate in a cavity system may be
Mean Flow Rate in Subglacial Cavities as a Function of Hydraulic Gradient G and Cavity Height R. Except for Very Large Cavities at Implausibly Large Hydraulic Gradients, Mean Flow Rates are Much Less than for R Channels at the Same C and with the same Conduit Cross-Sectional Area
on the order of 0.1 m/s or less for many plausible situations. Such values are definitely on the lower end of the range of Ū values inferred from dye-tracer studies (Reference BurkimsherBurkimsher, 1983, p. 404). Substantially higher Ū values are likely in an R-channel system, as I show next
For a semi-circular R channel, one may use the Manning equation as expressed by Reference NyeNye (1976, p. 189) to write
Consider now the relative values of Ū for a cavity and R channel of the same cross-sectional area S. For fluxes low enough that melting plays a negligible role in determining cavity size. Equation (9) indicates that
Substituting Equation (21) into Equation (20) and comparing to Equation (17), one finds
Suppose that Gcavity/Gchannel = 0.05, R = 0.1 m, u s = 10 m/a, and that k1, n, and A have values as given above. One then calculates
where p e is expressed in MPa. For plausible effective pressures in the range 0.1-2 MPa (1-20 bar).
The relative fluxes Qcavity / Qchannel will be equal to Ūcavity channel’ because equal cross-sections have been assumed for the two types of conduit.
One may conclude from this development that the mean flow rate in a cavity system may be much less than in an R-channel system and that, for equal melt-water discharges, a cavity-dominated system must contain many more conduits than a system dominated by R channels. These conclusions accord with those reached by Reference KambKamb and others (1985, p. 476-78) in their study of the surge of Variegated Glacier. Their interpretation of the measured change in Ū (from surging to non-surging states) as reflecting a change in the subglacial hydraulic system from cavity-dominated to channel-dominated is consistent with the analysis presented above.
Discussion: cavity-channel interaction
It is interesting to consider how a channel-dominated basal drainage system might change spatially to a cavity-dominated system. Suppose that an R channel “entering” a bed region in which cavities exist is successively “tapped” by a number N of such cavities. I have demonstrated above that a cavity of cross-sectional area comparable to that of a channel will typically carry a much smaller discharge (by one or two orders of magnitude) than the channel. Hence, unless the cavities are quite large compared to the channels or a very large number of cavities exist and tap the channels, some channels should “survive” in cavity-dominated regions. Surviving channels might shrink in cross-section in response to decreased discharge, or might pass into open-channel flow sufficiently near the glacier margins (e.g. Reference HookeHooke, 1984).
The notion of coexisting basal cavities and R channels is supported by some field observations. Mapping of de-glaciated bedrock by Reference WalderWalder and Reference Hallet and AndersonHallet (1979) indicated that the basal drainage system beneath the cirque glacier studied contained both cavities and Nye channels. The latter would almost certainly have been accompanied by R channels if oriented along the ice-flow direction (the most common orientation); some of those trending oblique to the ice-flow direction might also have been accompanied by R channels (cf. Reference HookeHooke, 1984).
Basal zones containing both R channels and interconnected cavities might exhibit somewhat unstable behavior, due to the contrasting Q versus pe relationships of the two types of conduit (cf. Equations (11) and (12)). Consider, for example, the response to an increase in melt-water discharge. In order for cavities to handle the additional melt water, they would have to enlarge under enhanced water pressure. Channels would also have to enlarge, initially at enhanced p w (Reference RöthlisbergerRöthlisberger, 1972, p. 198) but would stabilize at a lower p w than initially. The channels might then tend to “recapture” melt-water discharge from the cavities. In the extreme case, R channels would grow and recapture all discharge from the cavities. Behavior along these lines might be involved in the termination of glacier surges (Reference KambKamb and others, 1985). In any case, it seems probable that glacier-bed regions containing both cavities and R channels will show fluctuations in water pressure and sliding speed as cavities and channels expand and contract in response to changing melt-water discharge. It is tempting to speculate that such phenomena are related to water-pressure and sliding-speed fluctuations observed at various glaciers (e.g. Engelhardt and others, 1978; Reference IkenIken and others, 1983; Reference IkenIken and Bindschadler, 1986)
Conclusions
The increasing recognition of the important role of subglacial cavities in glacier hydraulics and dynamics has motivated the analysis presented here. This simple analysis demonstrates that the hydraulics of subglacial cavities should be quite distinct from that of conduits incised into basal ice; specifically, the flux versus pressure relationships and the stability characteristics of the two drainage systems should be quite different. This work constitutes only an initial step in understanding the hydraulics of subglacial cavities. Additional developments must include considering more reasonable geometries for the cavity network as a whole; analysis of the behavior of the constrictions between cavities (Reference KambKamb, in press) must be coupled with realistic estimates of the distribution of such constrictions on real glacier beds. Reference IkenIken and Bindschadler (1986) and Reference KambKamb (in press) have suggested that the overall “connectivity” or the cavity system will vary with water pressure, with possibly important implications for the way in which dye tracers will move through the system.
The transient hydraulic response of subglacial cavities must also be analyzed. This analysis will necessarily be complicated by the need to treat the connectivity of the cavity system as time-dependent. Some beginnings at analyses of this type, with the aim of explaining the propagation of wave-like transients in the subglacial drainage system, have been recently reported by M. Fahnestock and B. Reference KambKamb (personal communication). Coupled with this work should be analyses of the transient glacier-sliding behavior induced by hydraulic transients. Such work should lead to a better understanding of the observed time differences between, say, peaks in water pressure and in sliding speed.
Acknowledgements
This work was begun in 1984 while I was with the Project Office-Glaciology of the U.S. Geological Survey, Tacoma, Washington. I am grateful to Dr M. Meier for providing me with the opportunity to spend several months with the Project Office, as well as for his comments on this work during early stages in its development. I am also pleased to thank C. Raymond, N. Humphrey, B. Reference Hallet and AndersonHallet, H. Röthlisberger, and an anonymous referee for comments on earlier versions of this manuscript.
Appendices
Appendix Hydraulic Gradient in Cavity System with Constrictions
In the main text, I derived an expression (Equation (11)) relating the flux Q to hydraulic gradient G, sliding speed us, cavity height R, effective pressure p e , and roughness m. If the water pressure is not so high that the cavities are nearly unstable, then a good approximation to Equation (11) is
where n and A are the flow-law parameters (Nye, 1953).
Consider now a cavity containing constrictions as shown in Figure 3. In the steady state, the flux must be the same in both the “wide” and “narrow” sections of the cavity, hence from Equation (A-l)
Schematic view of cavity network with constrictions.
where I assume m to be uniform along the cavity network. The relative values of the hydraulic gradients are therefore
The difference in p & between adjacent narrow and wide sections will not be large in comparison to pe, so to a good approximation
It is clear from this expression that the hydraulic gradient in narrow constrictions may easily be one or two orders of magnitude greater than in wide parts of a cavity.
To estimate the actual value of Gwide, consider again Figure 3. The total drop in hydraulic gradient ΔΦ through the combined narrow and wide parts of the cavity is
where lnarrow and l wide are the lengths of the respective cavity segments. The average hydraulic gradient <G> along the cavity network is therefore
or using Equation (A-5)
Now let Gi be the gradient that would apply for a conduit along the ice-flow direction, and let 
be the tortuosity of the actual cavity network, defined as
where li is defined in Figure 3. The average hydraulic gradient <G> along the cavity network may therefore also be expressed as
Equating the two expressions for <G> and using Equation (A-4), one finds after a bit of algebra:
Supposing, for example, that 
=2 (perhaps an underestimate), lnarrow/lwide =0.1, and Rwide/Rnarrow =4, finds
It seems clear that the hydraulic gradient in the wide sections of the cavity may be only a very small fraction of the hydraulic gradient that would obtain in conduits along the ice-flow direction.
The mean flow rate Ū through the cavity network may also be readily related to the flow rates in the different sections. The time required for a “parcel” of water to traverse the narrow section is
and similarly for the wide sections,
whrer U narrow and Uwide are the flow rates for the respective sections. The mean flow rate Ū is defined as
Straightforward algebra leads to the result
Uwide muStn be less than Unarrow , SO if 1 narrow /1 wide << 1 one finds that
