Theory of the Propagation of UHF Radio Waves 

in Coal Mine Tunnels 

ALFRED G. EXSLIE, ROBERT L. LAGACE, MEMBER, IEEE, AND PETER F. STRONG 

Abstract-The theoretical study of WF radio communication in 

coal mines, with particular reference to the rate of loss of signal 

strength along a tunnel, and from one tunnel to another around a 

comer is the concern of this paper. - Of prime interest are the nature - .... ... ... .-. .... ... ... 

... of the propagation mechanism and the prediction of the radio .... ... ... ... frequency that propagates with the smallest loss. The theoretical 

results are compared with published measurements. This work was 

part of an investigation of new ways to reach and extend two-way 

communications to the key individuals who are highly mobile 

within the sections and haulageways of coal mines. 

INTRODUCTION 

T FREQUENCIES in the range of 200-4000 MHz A the rock and coal bounding a coal mine tunnel act as 

relatively low-loss dielectrics with dielectric constants in 

the range 5-10. Under these conditions a reasonable 

hypothesis is that transmission takes the form of waveguide propagation in a tunnel, since the wavelengths of 

the UHF waves are smaller than the tunnel dimensions. 

An electromagnetic wave traveling along a rectangular 

tunnel in a dielectric medium can propagate in any one 

of a number of allowed waveguide modes. All of these 

modes are "lossy modes" owing to the fact that any part 

of the wave that impinges on a. wall of the tunnel is partially refracted into the surrounding dielectric and partially reflected back into the waveguide. The refracted 

part propagates away from the waveguide and represents 

a power loss. It is to be noted that the attenuation rates of 

the waveguide modes studied in this paper depend almost 

entirely on refraction loss, both for the dominant mode 

and higher modes excited by scattering, rather than on 

ohmic loss. The effect of ohmic loss due to the small conductivity of the surrounding material is found to be 

negligible at the frequencies of interest here (see Appendix D\ 

DJ. 

The study reported here is concerned with tunnels of 

rectangular cross section and the theory includes the 

case where the dielectric constant of the material on the 

side walls of the tunnel is different from that on top and 

bottom walls. The work, which has previously been reported [I] in summary form, extends the earlier theoretical 

Fig. 1. Wave guide geometry. 

~ork byAIarcatili and Schmeltzer r2] and by Glaser [3], 

which applies to waveguides of circular and parallel-plate 

geometry in a medium of uniform dielectric constant. We 

present in the body of the paper the main features of the 

propagation of UHF waves in tunnels. Details of the 

derivations are contained in the accompanying appendices. 

THE FUXDARIEXTAL (1,l) WAVEGUIDE MODES 

The propagation modes with the lowest attenuation 

rates in a rectangular tunnel in a dielectric medium are 

the two (1,l) modes, which have the electric field E 

polarized predominantly in the horizontal and vertical 

directions, respectively. We mill refer to these two modes 

as the EI, and E, modes. 

The main field components of the Eh mode in the tunnel 

axe 

E, = EO cos klx cos kpy exp ( - ikg) (1) 

123 H, = - EO cos klx cos kzy exp ( - iksz) 

w!4 (2) 

where the symbols have their customary meaning. The 

coordinate system is centered in the tunnel with x horizontal, y vertical, and x along the tunnel asshown in Fig. 1. 

In addition to these transverse field components there a,re 

small longitudina,l components of E, and Hz a.nd a small 

transverse component Hz. For the frequencies of interest 

here kl and Icz are small compared with 123, which means that 

the wave propagation is mostly in the z direction. From a 

geometrical optics point of view, the ray makes small 

gra,zing angles ~5th the tunnel m-alls. 

In the dielectric surrounding t,he tunnel the wave solu- - 

tion has the form of progressive waves in the transverse 

Manuscript received May 9, 1974; revised Kovember 14. 1974. nrell as the longitudind directions. ~h~ propagation This work was supported bj- the U. S. Bnreau of Mines, Pittsburgh Minine and Safetv Research Center. USBhI Contraqt 110122026. constant ks for the (1,l) mode is an eiaenvalue - determined 

The sews and co~clusions contained in this documer~t are those of by the boundary conditions of continuity ofthe tangential the authors and should not be interpreted as necessarily representing 

the official policies of the Interior Department's Bureau of Mines or components of E and H at the walls of the tunnel. Owing 

the C. S. G'overnment. 

The authors are with Arthur I). Little, Inc., Cambridge, 11%~. to the simple form of t,he wave given by (1) and (2) these 

02140. conditions can be satisfied only approximately. However, 
a good approximation to k3 is obtained. The imaginary 

part of kS, which arises owing to the leaky nature of the 

mode, gives the attenuation rate of the wave. The loss 

LEh in dB for the (1,l) Eh mode is given by 

where K1 is the dielectric const,ant of the side walls a,nd 

K2 of t,he roof and floor of the tunnel. The corresponding 

result for the (1,l) E, mode is 

The derivation of (3) and (4), along with the generalized expressions for higher modes, is given in Appendix 

A, which also gives the range of validity of the formulas. 

The losses calculated by (3) and (4) agree closely with 

those calculated by a ray approach (see Appendix C). 

Fig. 2 shows loss rates in dB/100 ft as functions of frequency calculated by (3) and (4) for the (1,l) En and E, 

modes in a tunnel of width 14 ft and height 7 ft, representative of a haulageway in a seam of high coal, and for 

K1 = Kz = 10, corresponding to coal on all the walls of 

the tunnel. It is seen that the loss rate is much greater for 

the E, mode. Fig. 3 shows the calculated Eh loss rate for a 

tunnel of half the height. The higher loss rate in the low 

coal tunnel is due to the effect of the dz3 term in (3). 

hleasurements of attenuation in straight coal mine 

tunnels, and also around corners into cross tunnels, have 

been made by Goddard [411 at frequencies of 200,415, and 

1000 MHz for various orientations of the transmitting 

and receiving dipole antennas. A striking feature of the 

results is that for 415 and 1000 MHz the rate of decay of 

signal strength in a straight tunnel (in dB/100 ft) is the 

same, mithin experimental error, for all orientations of the 

two antennas, i.e., horizontal-horizontal, vertical-horizontal, and vertical-vertical. At 200 hlHz, only VH and 

TTIT measurements for a few antenna separations are 

reported and the decay rates are somewhat different. For 

comparison nith the theoretical decay rates we shonr the 

experimental points on Fig. 2. The 415- and 1000-MHz 

points are for the HH orientation, which is almost the 

same as for the other orientations, while the 200-MHz 

point is the average of the VH and W decay rates. 

The conlparison shows clearly that the propagation in 

the tunnel is by the (1,l) Eh mode. However, the somewhat higher values of the experimental attenuation at 

41.5 and 1000 MHz suggest that some additional loss 

mechanism sets in at higher frequencies. This mechanism 

must also be able to account for the independence of loss 

rate on antenna orientation, since the theory up to this 

point predicts no transmission for the VH antenna 

arrangement. 

rill expe~imental reslllts and Fig. 4 in the present paper are taken from t.he work of Goddard, Collins Radio Co., Cedar Rapids, 

Iowa. 

15 

Lon in dB 

per 100 Feet 

8 Exp. Data 10oddard)@1 

Fig. 2. Refraction loss for Eh and E, modes in high-coal. 

Lou in dB 

per 100 Feet 

Mode 

K=10 

3.5ft. EII K=10 

14 fc. 

- Theory 

Frequency(MHz) 

Fig. 3. Refraction loss for 4 mode in low-coal. 

PROPAGATION MODEL 

The higher observed loss rate at the higher frequencies 

relative to the calculated Eh mode values, and the independence of the loss rate on antenna orientation can both 

be accounted for if one allows for scattering of the dominant (1,l) EJ, mode power by roughness and tilt of the 

tunnel walls. The scattered power goes into many higher 

modes and can be regarded as a "diffuseJJ radiation component that accompanies the Eh mode. This diffuse radiation component is in dynamical equilibrium with the En 

mode in t.he sense that its rate of generation by scattering 

of the Eh mode is balanced by its rate of loss by refraction 

int,o the surrounding dielectric. Since the diffuse compo- 
Fig. 4. Example of coal mine tunnels and cross cuts. 

nent consists of contributions from the (Ill) E, mode and 

many higher order waveguide modes, all of which have 

much higher refractive loss rates than the fundamental 

Eh mode, the dpamical balance point is such that the 

level of the diffuse component is many dB below that of 

the Eh mode at any point in the tunnel. 

Our propagation model, comprising the (1,l) Eh mode 

plus an equilibrium diffuse component, explains the discrepancy between theory and experiment in Fig. 2, since 

the loss due to scattering of the Eh mode is greater at 

1000 MHz than at 415 MHz owing to the larger effect of 

wall tilt at the higher frequency. The model accounts for 

the independence of loss rate on antenna orientation, since 

the loss rate is always that of the Eh mode, except for 

initial and final transition regions, no matter what the 

orientations of the two antennas may be. The transition 

regions, however, cause different insertion losses for the 

different antenna orientations. 

Further strong support for the theoretical model is provided by the discovery by Goddard [4] that a large loss 

in signal strength occurs when the receiving antenna is 

moved around a corner into a cross tunnel; and that the 

signal strength around the corner is independent of receiving antenna orientation. We interpret these findings as 

follows. The (1,l) Eh mode in the main tunnel consists of 

a highly collimated beam of radiation owing to the small 

ratio of wavelength to tunnel dimensions. Such a beam 

acts in essentially a geometrical optics way and is almost 

unaffected by the openings into the cross tunnel. The 

(1,l) mode therefore couples very weakly into the cross 

tunnel modes. In this respect the situation is very different from the case of crossed mono-mode metal waveguides. 

Furthermore if the Eh mode did couple signi6cantlg into 

the cross tunnel, by symmetry the radiation in the cross 

tunnel would also be horizontally polarized, contrary to 

the experimental results. 

The diffuse component in the main tunnel, on the other 

hand, covers a wide angular range and therefore couples 

strongly into the cross tunnel. i\/Ioreover, since it consists 

of both Eh and E, higher modes it is largely unpolarized. 

Therefore the fraction of the diffuse component that enters 

the cross tunnel is also largely unpolarized, in accord with 

the experimental results. 

Another experimental result is that the initial attenuation rate in the cross tunnel is much higher than the rate in 

the main tunnel. This is also in accord with the model 

since the diffuse radiation component has a much larger 

loss rate than the Eh mode owing to its steeper angles of 

incidence on the tunnel walls. 

THE DIFFUSE RADIATION COMPONENT 

Scattering of the (1,l) Eh mode into other modes to 

generate the "diffuse" cymponent occurs by two mechanisms: wall roughness and wall tilt, which coal mine tunnels 

exhibit to a marked degree. Fig. 4 is a plan view that 

depicts typical sidewall geometry of tunnels in room and 

pillar coal mines. The roof and floor of these tunnels also 

have long range tilt variations, since the tunnels are made 

to follow the undulations of the coal seam. 

Roughness is here regarded as local variations in the 

level of the &face relative to the mean level of the surface 

of a wall. For the case of a Gaussian distribution of the 

surface level, defined by a root-mean-square roughness h, 

the loss in dB by the Eh mode is given by the formula 

(see Appendix D) 

This is also the gain by the diffuse component due to 

roughness. 

It may seem surpAg that the loss due to roughness 

increases with wavelength. The reason is that, in a ra.y 

picture of a given mode, the grazing angle of the ray on the 

walls of the waveguide increases with wavelength. The 

loss varies directly with grazing angle and number of 

bounces per unit length, and inversely with wavelength. 

Increase in wavelength increases the ht two factors and 

decreases the third. The net effect is an increase of loss 

with wavelength. 

Long range tilt of the tunnel walls relative to the mean 

planes, which define the dimensions dl and d2 of the tunnel, 

causes power in the Eh mode to be deflect,ed away from 

the directions defined by the phase condition for the mode. 

One can calculate the average coupling factor of such deflected radiation back into the Eh mode and thereby hd the 

loss rate due to tilt. The result in dB is (see Appendix E) 

where 8 is the root mean-square tilt. Equation (6) also 

gives the rate at which the diffuse component gains power 

from the Eh mode as a result of the tilt. 

It is noted from (5) and (6) that roughness is most important at low frequencies while tilt is most important at 

high frequencies. 

Figure 5 shows the effect on the (1,l) Eh mode propagation of adding the loss rates due to roughness and tilt to 

the direct refraction loss given in Fig. 2. The curves are 

calculated for a root-mean-square roughness of 4 in and 

for various assumed values of 8. It is seen that a value 8 = 

lo gives good agreement with the experimental values. 

The effect of tilt is much greater than that of roughness 

in the frequency range of interest. 

Having determined the value of 8, for the assumed value 

of h, we can now find the intensity ratio of the diffuse com- 
15 

Loss in dB 

per 100 Feet 

RMS Roughness h=4 inches 

0 1 I I I I I I I I 

0 500 1000 1500 2000 2500 3000 3500 4000 

FrequencdMHz) 

Fig. 5. Resultant. propagation loss for Eh mode in high-coal (refraction, wall roughness, and t.ilt). 

ponent to the Eh mode from the equilibrium ba,lance 

equat,ion 

where Lhd is the loss rate from the Eh mode &to the diffuse 

component,, and Ld is t,he loss rate of the diffuse component, by refraction. To estimate Ld approximately, we take 

the loss ra,te to be that of an "average ray" of the diffuse 

component having direction cosines (l/d-3,1/a,l/d). 

Then 

where R, the Fresnel reflectance of the average ray for 

K1 = KZ = 10, has the value 0.28. Then for dl = 14 ft, 

dz = 7 ft, z = 100 ft, we find that Ld = 119 dB/100 ft. 

This value has to be corrected for the loss of diffuse radiat.ion into cross tunnels, which nre assume have t,he same 

dimensions as the main tunnel and occur every 75 ft. 

From relative area considerations me find that this loss is 

2 dB/100 ft. The corrected value is therefore 

which is independent of frequency. 

The loss rate Lhd is shown in Table I as a function of frequency for the 14ft X 7-ft tunnel. The values are the 

sum of the roughness and tilt losses calculated by (5) and 

(6) for h = 4 in rms and 0 = lo rrns. The diffuse component level relative to the Eh mode, calculated by (7), 

is given in the fourth column of Table I. The diffuse component is larger at high frequencies oning to the increased 

scattering of the Eh mode by mall t.ilt. 

TABLE I 

DIFFUSE RADIATION COMPOSENT IX R~IS TUNNEL ANDAT 

BEGINKING OF CROSS TUNNEL 

f k Lhd Id& 

'h, main 

(MHz) (Ft.) (dB/lM) ft.) (dB) 

4,000 245 5.4 -138 

3,000 327 4.1 -14.7 

2,000 .49 2.8 -16.4 

1,000 98 1.5 -19.0 

415 2.37 1.1 -20.6 

200 4.92 1.3 - 19.7 

Ide 

lh, main 

(dB) 

- 21.7 

- 229 

- 24.6 

- 27.2 

- 28.8 

- 27.9 

PROPAGATION AROUND A CORNER 

From solid angle considerations (see Appendix F) one 

finds that the fraction of the diffuse con~ponent in the 

main tunnel that enters the 14ft X 7-ft aperture of a 

cross tunnel is 15 percent or -8.2 dB. The diffuse level 

just inside the 'perture of the cross tunnel, relative to the 

Eh mode level in the main tunnel is therefore obtained by 

subtracting 8.2 dB from t,he values in column 4 of Table I. 

The results are shona in column 5 of the table. A dipole 

antenna with either horizontal or vertical orientation 

pla.ced at this point responds to one half of the diffuse 

radiation, and therefore gives a signa.1 that is 3 dB less 

than the va.lues in column 5 of Table I, relative to a 

horizontal antenna in the main tunnel. 

If a horizontal antenna is moved down the cross tunnel 

the loss ra.te is initially 119 dB/100 ft (the value calculated 

here nit,hout correction for tunnels branching from the 

cross tunnel). Ultimately, however, the loss rate becomes 

that of the Eh mode excited in the cross tunnel by the 

diffuse radiat$ion in the main tunnel. We determine the 

Eh level at the beginning of the cross tunnel by calculat,ing 

the fraction of the diffuse radiation leaving the exit 

aperture of the main tunnel, which lies within the solid 

angle of acceptance of the Eh mode in the cross tunnel 

(see Appendix F) . The result is 

This ratio, in dB, is given in column 2 of Table 11. 

Column 3 of Table I1 is the Eh level at the beginning of 

the cross tunnel relative to the Eh level in the main tunnel 

found by adding column 2 of Table I1 and column 4 of 

Table I. We find the corresponding ratio at 100 ft down 

the cross tunnel by adding the Eh propagation loss rates 

given in Fig. 3 for 0 = lo. The results are shown in the 

last column of Table 11. 

The foregoing theoretical results for the diffuse and Eh 

components in the cross tunnel allon, us to plot straight 

lines showing the initial and final trends in signal level in 

the cross tunnel. These asymptotic lines are shorn in 

Figs. 6 and 7 for 415 MHz and 1000 ~IHz, in comparison 

with the cross tunnel measurements. The agreement both 

in absolute level and distance dependence gives good 

support to the theoretical model. 
0 

10 I 

I Frequency - 415 MHz 

20 

30 

Corner Loss 40 

in dB 50 

60 

70 

80 

Ih,crovj 

Asymptotes 

I 

I I 1 I I I I I I 0 20 40 60 80 100 120 

Distance Around Corner in Feet 

Fig. 6. Grner loss in high-wd (frequency 415 MHz). 

80 Asymptotes 

90 I 

0 

10 

I I t t I I I I 

0 20 40 60 80 100 120 140 

Distance Around Corner in Feet 

- I 

I Frequency - 1000 MHz 

- I Antenna Orientation (HH) I 

Fig. 7. Corner loss in high-wal (frequency 1000 MHz). 

-Theory 

Exp. Data 14' 

Corner Loss 

in dB 40 - 

50 .- 

EFFECT OF ANTENNA ORIENTATION 

The theoretical model also allows us to predict the 

effect of antenna orientation when the transmitting and 

receiving antennas are far enough apa.rt so that dyna.mica.1 

equilibrium between the Eh mode and the diffuse component is established. M7e start with both antennas horizontal (HH codguration) and consider this as the 0-dB 

reference. Then if the receiving antenna is rotated to the 

vertical (HV coniig~r~tion) this a,ntenna is now orthogonal 

to the Eh mode, and therefore responds only t,o one ha,lf of 

the diffuse component, so that the loss is 3 dB more than 

the values in Table I, column 4. The result is shown in 

Table 111, column 2. Now, by the principle of reciprocity, 

the tra.nsmission for VH is the same as for HV as shown 

in column 3 of Table 111. We now rotate the receiving 

antenna to get the codgura.tion W. Again we incur an 

additional transmission loss of 3 dB more than the values 

TABLE I11 

EFFECT OF BNTICKYA ORIENTATIOX 

f HH HV VH W 

(MHz) (dB) (dB) (dB) (dB) 

TABLE IV 

INSERT~OX LOSS (Li) (FOR HALF-WAVE A~TENN.~) 

F X L: 

(MHz) (Feet) (dl~) 

4000 0245 35.0 

3000 0.327 32.4 

2000 0.49 28.9 

loo0 0.98 22.9 

415 2.37 152 

200 4.92 8.9 

in Table I, column 4. The W values are shown in Table 

111, column 4. 

AiiTENNA INSERTION LOSS 

Dipole or whip antennas are the most convenient for 

portable radio communications between individuals. However, a considerable loss of signal power occurs at both the . 

transmitter and receiver when simple dipole a.ntennas are 

used because of the inefficient coupling of these ante~as 

to the waveguide mode. The insertion loss of each dipole 

antenna can be calculated by a standard microwave circuit 

technique for computing the amount of power coupled 

into a wa,veguide mode by a probe, whereby the dipole 

antenna is represented as a surface urrent iilament having F 

a sinusoidal current distribution along its length (see 

Appendix G). For the Eh (1,l) mode the redt for a 

dipole placed at the point (zo,yo) in the tunnel cross section 

is 

ZO,I,I is the characteristic impedance of the Eh (1,l) mode 

and R, is the radiation resistance of the antenna, which are 

approximately the free space values 377 D and 73 Q, respectively, provided that X is small compared with dl and dz. 

For the case of a.ntennas placed at the center of the 

tunnel (XO = 0,yo = 0) , (11) gives the results shown in 

Table IV, where the insertion loss Li in dB is equal to 

- 10 loglo C. It is seen that the insertion loss decreases 

rapidly with increasing wavelength, as one would expect, 

since the antenna size occupies a larger fraction of the 

width of the waveguide. The overall insertion loss, for 

both antennas, is twice t,he value given in the table. A 

considerable reduction in loss would result if high gain 

a,ntenna systems were used. 

OVERALL LOSS IN A STRAIGHT TUNNEL 

The overall loss in signal strength in a straight tunnel is 

the sum of the propagation loss and the insertion losses of 

the transmitting and receiving antennas. Table V lists the 

component loss rates for the (1,l) Eh mode due to direct 
TABLE V 

CALCULATION OF OVERALL LOSS FOR Eh MODE -WITH TWO HALF-WAVE DIPOLE ANTENNAS (h = 5 ft, 0 = lo, K1 = K. = 10. dl = 14 ft, dn = 7 ft) , - . - 

Lrefraction Lroughness hilt Lpropagation Linrwtion 7 L~~era~~ (dB)-------\ 

(MHz) ldB/100') (dB/100') (dB/100') (dBl100') (dB) 100' 500' 1000' 1500' 2000' . . 

QOOO .06 .05 5.33 5.44 69.90 75 97 124 152 179 

3000 .10 .07 3.99 4.16 6488 69 86 107 127 148 

moo 23 .lo 2.66 2.99 57.86 61 73 88 103 118 

1000 .9 1 21 1.33 2.45 45.82 48 58 70 81 93 

415 5.34 50 0.55 6.39 30.48 37 62 94 126 158 

200 23.00 1.04 027 24.31 17.80 42 139 261 383 504 

100 92.00 2.08 0.14 94.20 5.80 100 477 948 1419 1890 

150 

Total Loss in dB 

0 

0 500 1000 1500 2000 2500 3000 3500 

Frequency (MHz) 

0 rms = lo 

h rms = 113 Ft. 

K, = K,= 10 

d, = 14 Ft.. d, = 7 Ft. 

Orientation (HH) 

I I I I I I I 

Fig. 8. Overall lorn for various dist.ances along a straight tunnel (for halfwave antennas). 

refraction, roughness, and tilt; the total propagation loss 

rate; the insertion loss for two half-wave antennas; and 

the overall loss for five different distances. The overall loss 

for the HH orientation is also shown in Fig. 8, where it is 

seen that the optimum frequency for minimum overall 

loss is in the range 500-1000 MHz, depending on the 

desired communication distance. 

It is also of interest to combine the results in Table V 

with those in Table I11 to obtain the overall loss versus 

distance for the HH, HTT (or VH), and W antenna 

orientations. In order to compare the theoretical values 

RTith the experimental data, which are expressed with 

refercncc to isotropic antennas, we add 4.3 dB to the 

overall loss calculated for half-wave dipoles. The theoretical results for the three different antenna orientations for 

frequencies of 415 MHz and 1000 MHz are compared 

with the experimental data in Figs. 9 and 10. It is seen 

that the theory agrees quite well with the general trend of 

the data. 

OTTERALL LOSS ALONG A PATH WITH ONE 

CORNER 

Table TrI gives the overall Eh mode loss for a path from 

one tunnel to another, including the corner loss involved 

in reestablishing the Eh mode in the second tunnel. The 

loss is the sum of the corner loss, given in Column 3 of 

Table I1 and repeated in Table VI, and the straight tunnel 

loss given in Table V for various total distances. The results 

in Table VI are for the case of half-wave dipole transmitting and receiving antennas and are valid when neither 

antenna is within about 100 ft of the corner. The overall 

loss is less than the values in Table VI if the receiving 

antenna is within this distance, oning to the presence of 

the rapidly attenuating diffuse component that passes 

around the corner. From the principle of reciprocity, the 

same is true if the h-ansmitting antenna is within 100 

ft of the corner. 

The results indicate that the optimum frequency lies 
Overall Loss 

in dB 

Frequency415 MHz 

Thwry - 

Experiment 14' Antenna 

Orientation 

150 - 

160 - 

170 - 

Distance in Feet 

Fig. 9. Overall loss in straight tunnel in high-cod (for isotropic 

antennas, frequency 415 MHz). 

Overall Loss 

in dB 

30 

40- Frequency - 1000 MHz 

Theory - 

Experiment 14' Antenna 

Orientation 

0 HH 

130 - 

140 - 

150 - 

160 - 

170 - 

180 b I I I I I 

500 1000 1500 2000 2500 300 

Distance in Feet 

Fig. 10. Overall loss in straight tunnel in high-coal (for isotropic 

antennas, frequency 1000 MHz). 

in t,he range 400-1000 MHz. However, if one installs 

horizontal half-wave resonant scattering dipoles with 45' 

azimuth in the important tunnel intersections, in order 

to guide the Eh mode around the corner, the optimunl 

ma,y shift to somewhat lower frequencies since a greater 

fract,ion of the incident Eh wave will be deflected by the 

longer low-frequency dipoles. 

EXPECTED COMMUNICATION RANGE 

BETWEEN TWO ROVING I\IINERS 

Communication can be maintained between two separated individuals until the separation distance increases 

to a point where the signal strength is not sufEcient to 

overcome the background electrical noise. To obtain 

estimates of this communication range for a mobile 

application involving roving miners equipped with portable personal radio transceivers, three frequency independent loss factors should be added to the values of 

overall loss presented in Tables V and VI. These factors 

are: polarization loss-to account for likely misalignment 

of transmit and receive antennas; ante~a efficiency 

TABLE VI 

OVERALL LOSS ALOWG PATH IKCLCDING ONE CORNER (Eh MODE WITH HALF-WAVE DIPOLE ANTENXAS) 

Eh Loa 

per Corner 

(dB) 

802 

77.6 

74.1 

67.6 

57.7 

47.3 

Overall Loss (dB) 

500' 1000' 1500' 2000' 

177 205 232 259 

163 184 205 226 

147 162 177 192 

126 138 148 161 

120 152 184 216 

187 308 43 551 

loss-to account for the nonideal antenna installation on the 

portable units; and fade margin-to account for signal 

cancellation effects due to destructive interference. Nominal values that appear reasonable for these factors are 

12 dB, 4 dB, and 12 dB, respectively, resulting in a total 

of 28 dB to be added to the aforementioned values of 

overall loss. By exercising care in the orientation and 

position of the portable radios in the mine tunnel cross 

section while communicating, these polarization and signal 

fading losses can of course be reduced, thereby producing 

a corresponding increase in range. 

Representative values of receiver sensitivity and transmitter power for FM portable radios in the UHF 450- 

MHz band are 0.5 pV for 20 dB of quieting (- 113 dBm 

into a 50-9 input resistance) and 2 W (33 dBm), respectively, resulting in a total allowable loss of 146 dB. In 

this frequency band, measurements in mines have shorn 

that the intrinsic electrical noise of the UHF receiver mill 

predominate over externally generated nine electrical 

noise. Using the aforementioned parameter values in conjunction with the 415-MHz overall loss values presented 

in Tables V and VI for straight line transmission paths 

and paths including one corner, predictions of comrnunication range along haulageways and in working sections of 

mines can be obtained, as shown in Fig. 11. Fig. 11 illustrates the coverage expected in a high-coal mine between 

a centrally located miner with a portable radio and a 

second miner roving throughout a typical 600- X 600-ft 

room and pillar mine section with another portable unit 

at an operating frequency of 415 MHz. 

When the signal must go around only one corner, satisfactory communication can be expected over a linear distance of approximately 500 ft down an entry and cross-cut. 

When no corners are encountered, as in a haulageway 

transmission path, satisfactory straight line communication can be expected over distances in excess of 1500 ft. 

These range limits are somenrhat conservative estimates, 

and as mentioned, can usually be somewhat extended if 

the portables are rotated into the horizontal plane, 

pointed across the tunnel, and translated a little to a more 

favorable signal strength position, thereby taking full 

advantage of the dominant horizontal field component 

and minimizing destructive interference effects. 

Wireless coverage has been estimated using the 415- 

MHz frequency results because the 450-MHz UHF frequency band is the present upper limit for commercially 

available portable radio transceivers, and because operating frequencies near 400 MHz are most favorable for 
0 50 100 ft. 

L_ 

Scale 

[3 coal 

0 Satisfacton, Coverage 

Marginal Coverage 

No Coverage 

Fig. 11. Predicted UHF wireless radio coverage for 2-W portable radios in high-(7-ft) cod mine section, frequency 41.5 NHz. 

U.S. high-coal room and pillar section applications where 

transmission paths typically include one c~rner.~ 

CONCLUSIONS 

The kind of propagation model developed in this paper, 

involving the (1,l) Eh waveguide mode accompanied by 

a "diffuse" component in dynamical equilibrium with it 

seems to be necesmry too account for the many effects 

observed in the measurements of Goddard [4]: the exponential decay of the wave; the marked polarization effects 

in a straight tunnel; the independence of decay rate on 

antenna orientation; the absence of polarization at the 

beginning of a cross tunnel; the two-slope decay characteristic in a cross tunnel; and overall frequency dependence. All of these effects are moderately well accounted 

for by the theoretical model. However, considerable 

refinement of the theory could be made by removing 

some of the present oversimplifications, such as the 

assumption of perfectly diffuse scattering both in the 

main tunnel and immediately around a corner in a cross 

tunnel, the use of the "average ray" approximation, and 

the description of the propagation around a corner in 

terms of two asymptotes only. More dat.a at greater distances down a cross tunnel and data covering a wider 

frequency range in both main and cross tunnels nrould 

allow a more stringent test of the theory. 

ACKI\TOWLEDGMENT 

The authors wish to thank H. Parkinson of the U. S. 

Bureau of Mines, Pittsburgh Mining and Safety Research 

Center for his encouragement and support of this work, 

and A. Goddard and D. Anderson of Collins Radio Co., 

Cedar Rapids, Io+a, for their cooperation in supplying 

and discussing their experimental data prior t,o its publication. 

APPENDIX A 

MODE CALCULATIONS 

A. (1,l) Eh and E, lModes 

The fundamental (1,l) Eh mode is a.pproximately a 

TEM wa.ve given by (1) and (2). The complete field of 

this form in the tunnel (-d1/2 I x < d1/2, -d2/2 I y I 

&/2), satisfying Maxwell's equations, is 

E, = Eo cos klx cos by exp (-ihz) (Al) 

ikl E', = - Eo sin klx cos by exp ( -ib) 

k3 (A3 

Hz = - "' Eo sin klz by exp ( -ikj.z) wmk3 (A41 

"dditional information regarding the practical application of Hu = 

(k12 + k32) EO cos klx cos kpy exp (-ikg) (A5) UHF wireless radio systems to mine haulageways and sections is WCLO~~ 

given in a paper coauthored by R. Lagace of hrthur D. Little, Inc., 

and H. Parkinson of the U. S. Bureau of Mines. Pittsburgh Mining 

and Safety Research Center. entitled "Two-way communications ih H~ = - E, cos klx sin lczy exp ( -ik3z) with roving miners," and published in U. S. Bureau of Mines In- WPO 

(A61 

formation Circular 8635, "Mine Communications" 151. 
where 

Since the wavelengths of interest are small compared with 

the tunnel dimensions, the wave vector components kl 

and kz are small compared with ka, which is close to 

ko = 27r/X. Therefore H, reduces to the expression given 

in (2) and E,, Hz, and Hz a.re very small. 

In the roof (y 2 dz/2) of dielectric constant Kz, the 

field must represent an outgoing wave in the y direction 

and therefore has t,he form 

E, = B cos klx exp ( -ikzfy) exp ( -ikS) (-48) 

E, = 0 (A91 

ikl E, = - B sin klx exp ( --ikz'y) exp ( -ik3z) kt (A10) 

H, = - B sin klx exp ( -ihty) exp ( -ik~) ~d3 

This equation could readily be solved numerically for the 

eigenvalues of kz, for given values of the other quantities. 

However, we prefer here to obtain an approximate closed 

form expression that provides a better insight into the 

prgblem. 

Since kzdz/2 << 1 w; 6nd for the lowest Eh mode 

This result shows that, except for a small imaginary part, 

kz has the same value as for a metal waveguide. The 

imaginary pa.rt arises from the power loss due to the outgoing refracted wave. The a.pproximation involved in 

(A20) is va.lid provided that X is small compa.red with 

.~r&(Kz - 1)'12. This criterion is very well satisfied, for 

exanlple, for X = 1 ft, dz = 7 ft, Kg = 10. 

In the side mall (x 2 d1/2), of dielectric constant K1, 

the field has the form 

Ez = A exp (-ikl'x) cos kzy exp (-ikg) (A211 

(k32 + k12) )js = - kl' H, = B cos klx exp ( -i&y) exp ( -ik3z) - exp ( -ikl'x) cos by exp ( -ikaz) 

LJPO~ k3 (A231 

which satisfies Maxwell's equations. The wave number H, = + k32) A exp ( -ik{z) cos kly exp ( -ikg) 

component hLf in the dielectric is given by the relation wcraks 

The ib boundary conditions at y = &/2 are that the t,an- H~ = -A (-ikiX) by exp (-ikaz) (A26) 

gential components of E and H are continuous. These w~o 

conditions require that where 

-iK& Eocos~))=Bexp(T) (A151 k1'2 + kz2 + k3' = K1lio2. (A271 

Continuity of the tangential E-field gives the condition 

and 

rczdz -ilczfdz ,%ao sin = ikiB exp (T) (A16) - ik:dl kJ30 sin r*) = ikl'A exp (T) . (A28) 

from which we obt.a.in the condition Continuity of H, and H, requires that 

Ic2dz kz tan - = ilez'. 2 (A171 (kr2 + k12)Eo cos ($) = (ksZ 4- kI")A exp -- 

(-?'dl) 

Since kl and kz are small compared with ko we hd from 

(A7) and (A14) that kZf is given approximately by and 

kpf = kO(K2 - 1)lI2. @la) - ikidl 

Therefore, from (A17) and (A18) we obtain the following Wo cos (*) = k4 exp (T) . (A30) 

mode condition for X-2, for modes that are even functions 

of y: Since (A29) a.nd (A30) are inconsistent we can only 

satisfy the H boundary condition approximately. We note 

kzd2 b tan - = iko(Kz - (A19) that since K1 >> 1, I kl' 1 is of the same order as ko, whereas 

2 I k2 I is much smaller. Therefore we may ignore (A30) and 
TABLE VII 

Loss RATES FOR VARIOUS MODES 

dso, from (A7) and (A27), write 

k12 + k32 NN b2 (A311 

Then from (A28), (A29), (A31), and (A32) me obtain, 

approximately , 

Again taking advantage of the smallness of kl and kn 

relative to ko, we find for the lowest Eh mode that 

which shows that the mode shape in the x direction is also 

the same as for a metal waveguide, except for a small 

imaginary part. The approximation is valid if X is small 

compared with lrdl(K1 - 1/K1)1f2. The criterion is well 

satisfied for h = 1 ft, dl = 14 ft, K1 = 10. 

On ~ubst~ituting for B and k2 from (A34) and (A20) 

into (A7) we hd, on neglecting second-order terms, that 

the propagation constant in the z direction is 

3. Higher Modes 

One can readily generalize (A36) and (A37) to the 

case of a higher mode (n.l,nz) with approximately nl halfwave loops in the x direction and n2 in the y direction. 

The results are 

Table VII shows the loss rates for a ;umber of modes for 

f = 1000 MHz, X = 0.98 ft, dl = 14 ft, & = 7 ft., K1 = 

Kz = 10, z = 100 ft. 

The validity criteria for the approximate solutions of 

the mode conditions (A19) and (A20) are the same as for 

the (1,l) modes, i.e., they do not depend on nz or 121. 

APPENDIX B 

EFFECT OF CONDUCTIVITY ON 

PROPAGATION LOSS 

To hd the effect of the finite conductivity of the walls 

of the tunnel on the propagation loss we now regard 

K1 and K2 as complex numbers in the approximate solutions for the eigenvalues of kl and k2, which, for t,he general 

case of an Eh mode, take the form 

The power loss in dB for the (1,l) En mode for a distance The propagation consta'nt' k3 is given by 

z is therefore k32 = k02 - k12 - b2. (B3) 

LEh = -8.686 Im (k3) Since kl and kz are assumed to be small compared with 

KI ko, we can express the power ab~orpt~ion coefficient ah for 

Eh modes in the form 

(A361 ah = - 2 Im (kg) - 1 Im (klZ + M) 

We obtain the loss for the (1,l) E, mode by interchanging ko 

the subscripts 1 and 2 in (A36) X 

1 = - Im (b2 + h2). 2lr 034) 

Lg* = 4.343122 (d13(Kl - I) 112 + On substituting for kl and kz from (Bl) and (B2) into 

(A37) (B4) we obtain 

As a check on these formulas we find that exactly the n12K1 7.~2~ 

same results are obtained if one adds the losses for tvo = X2 Re 

Snite slot waveguides of slot widths dl, dz and dielectric (dl3 (& - 1) &a (KZ - 1) 

constants K1, K2, respect.ively. The numerical results given h3 nl2K12 by (836) and (A37) a.lso agree well wit,h t,hose given by 2a Q2 ) * (BS) 

the ray method (see Appendix C) . - - Im (d+(Kl - 1) U(4 - 1) 
We now n~ite K1 and K2 in the form 

K1 = KI' - iKll' (B6) 

K2 = K2' - iKil (B7) 

where Ky and K2" axe assumed to be small compared 

with K11 and Kz'. Under these conditions (B5) becomes, 

up to first-order terms in Kit and Kc 

If a1 and n2 are the conductivities of the vertical and 

horizontal walls of the tunnel, and f is the frequency of the 

radiation, t,hen 

Let ul = = 1W2 ~nho/m,~ f = 1000 MHz, €0 = 8.85 X 

10-l2 F/m. Then Kp = K2" = 0.18. For Kt = Ks' = 10, 

dl = 14 ft, dz = 7 ft, h = 1 ft, nl = nz = 1, the h2 and X3 

terms in (B8) are then 2.19 X ft-I and 5.89 X lo-" 

ft-1, respectively. Thus the effect of conductivity is completely negligible in the frequency range of interest. It is 

to be noted that tan 6 = 0.018 for the case chosen. 

The very small contribution of conductivity to the 

propagation loss for small values of the loss tangent of t-he 

dielectric agrees with the results obtained by Glaser [3] 

for a hollow cylindrical channel in a lossy dielectric. 

APPENDIX C 

RAY METHOD 

The allowed modes in a rectangular tunnel in a dielectric 

can also be determined approximately by a ray theory 

approach. In this method we consider a ray of the radiation t,ha.t bounces from wall to wall of the tunnel ma,king 

a gra.zing angle 41 with the side walls a.nd 42 with the floor 

and roof. The Eh and E, (1,1) modes are both defined in 

the ray picture by the phase relations 

A 

sin 41 = - 

2dl 

phase sbift undergone by the ray is exactly 360" after 

successive reflections from the two side walls or from the 

floor and roof. 

For frequencies around 1000 MECz, h is small compared 

with dl and d2. Therefore we can use the approximate 

relations 

The numbers of reflections N1 and Nz experienced by a ray 

at the vertical and horizontal mils of the tunnel, while 

traveling a distance z along the tunnel, are given by 

The attenuation factor for the ra.y intensity for this 

distance is 

where Rl and R2 are the power reflectances of the vertical 

and horizontal surfaces at the grazing angles 41 and &, 

respectively. 

On combining (C3)-(C7) we find for the loss L in 

decibels 

For the Eh mode the reflectances R1 and Rz of the vertical and horizont,al walls, respectively, are given by the 

Fresnel formulas 

RLEh = 

For the E mode the corresponding formulas are 

K1 sin & - (sin2 ~#q + K1 - 1)'12 

K1 sin & + (sin2 41 + KI - 1) 'I2 

E2~h = 

A 

sin&=-. (C2) Table VIII gives calcula,ted values of grazing angle, 2d2 number of reflections, reflectance, and loss for the Eh 

Equations (Cl) and (C2) are the conditions that the mode in a 14ft X 7-ft high-coal tunnel with K1 = K2 = 

10. L and Lz a.re the loss rates at the side walls and at 

sin & - (<in2 & + K2 - 1) 

sin & + (sin2 42 + K2 - 1)'12 

-4 representat,ive nominal value for tunnel walls in bituminous the roof and floor, respectively. The total loss L agrees 

coal seams. very well with the Eh mode theory results shonm in Fig. 2. 
TABLE J'III 

RAY METHOD CALCULATIOXS (FOR En MODE) 

N. N, L, L, L 

APPENDIX D 

a vertical axis. Then the reflected beam is rotated through 

an angle 20. This means tha.t t,he electric field is changed 

from 

LOSS DUE TO SURFACE ROUGHNESS 

Ex = F(x,y) exp ( -ik3z) When a parallel beam of radiation of intensity I0 st,rikes (El) 

a rough surface a.t normal incidence the reflected radia,tion to 

consists of a parallel beam of reduced intensity I together = F(z,y) c-(L,(~ 20 + sin 20)~. (E~) 

with a diffuse component. If the surface is a. perfect 

reflector The power coupling factor gl of the disturbed field (E2) 

where is the wa.velength and h is the root-mean-square 

roughness. For incidence a,t a, grazing a.ngle 4 one may 

assume tha.t the effective roughness is now IL sin #I, so the 

loss factor becomes [6] 

In the case of the dominant mode in a dielectric waveguide 

we can, from (Cl) and (C2), write for the roughness loss 

factors per reflection for the vertical and horizontal walls 

back into the mode (El) is given by 

where the integrations are over the cross section of the 

tunnel. The bar over E,' indicates complex conjugate. 

Since 0 is small me can replace cos 20 by 1 and sin 20 by 

20. Then (E3) becomes 

I S$ I F l2 exp (2ik3xe) dx dy l2 

g1 = ($I I F l2 dx dy)2 (E4) 

Instead of using the actual function cos klx cos kzy for F, 

we find it more convenient to use an equivalent Gaussian 

function 

- 

(D4) and integrate over idnite limits. The result is 

g1 = exp ( - $k32aW) . The loss factor for a dista.nce z is therefore, from C(3)- (E6) 

c (6) Next we assume that F2 falls to l/e at the point x = d1/2, 

y = 0, which is at the surface of the waveguide. Then 

a2 = 1/2d12 and 

g1 = exp ( - $k32d1202). (E7) 

(D5) Likewise, tilt:mg of the floor or roof gives a coupling factor 

The loss in dB is then g2 = exp ( - $k3"2262) . (E8) 

The loss factor for a dist.a.nce z is 

.Lmugaons = 4.3439h2h ($ + $) Z. (D6) 

g = g1N~g2hT* = exp (- y) (E9) \ .., APPENDIX E 

where we have replaced ks by ko. 

LOSS DUE TO TILT OF TUNNEL WALLS The loss in dB is therefore 

The effect of wall tilt can be est.imated a.s follows. 

Suppose that a ray of the Eh mode encountem a portion 

of a side wall that is tilted through a small angle 0 about 
APPENDIX F / 

PROPAGATION AROUND A CORNER 

Maun Tunnel 

A. Tranmnission of Dime Component Around a Corner 

To calculate the fraction of the diffuse component in 

the main tunnel that goes around a corner into a cross 

tunnel, me use results given in graphical form by Sparrow 

and Cess 171 for the angle factors for diffuse radiation 

transfer between rectangular areas, on the assumption 

that the radiation in the main tupnel is perfectly diffuse. Cross Tunnel In the high-coal case of intersecting tunnels each of dimensions 14 ft by 7 ft, the angle factor between the main 

tunnel aperture 1 &nd the cross tunnel aperture 2 is Fig. 12. Geometry for coupling to cross tunnels. 

Fl,z(high coal) = 0.15 = -8.2 dB. (F1) dl and d2 being the horizontal and vertical dimensions of 

In the case of low-coal tunnels of dimensions 14 ft x 3.5 ft, the tunnel. Therefore 

the result is 

(low coal) = 0.10 = - 10 dB. (F2) 

Since 41 and & are small, the contributions of various 

B- l3xcitdion of Eh Mode in cross Tunnel by B.fuse elements dA are approximately equal since shadowing 

Component in Main Tunnel effects can be neglected. Therefore (F4) becomes 

Fig. 12 depicts the geometry used for computing the 

degree to which the Eh mode in the main tunnel couples 

Po lTi2 

P = (242) - cos e ~ s de 

2~ e=r/z-+~ 

to the Eh mode in a cross tunnel. Diffuse radiation passing 

through a cross section A of the main tunnel has the angular distribution 

Po cos 9 dP = - 

aA dQ dA 

dP power in element of solid angle dB from element of 

area dA, 

6 angle with normal to dA, 

PO total diffuse power, 

A total area of cross section. 

me power entering the E~ in the cross tunnel is The power ratio of the Eh mode in the cross tunnel and 

therefore main tunnel is 

where the intwation is taken over the whole area A of where PO is the diffuse power level in the main tunnel. 

the cross and mer the solid angle 0 of the mode. This result neglects any contribution from scattering by 

me factor 1/2 alloms for the ho~ontal pokation of the floor and roof of the intersection area between the two 

the mode. Q is defined by the angles tunnels. 

APPENDIX G 

ANTENNA INSERTION LOSS 

X 

&=- (F6) A half-wave dipole antenna centered at xo, yo in the 2d2 tunnel and oriented in the x direction is approximated by 
a surface current distribution to the left and to the right of the dipole, is 

where uo and are the unit impulse and step functions. 

For t,he case of an infinite tunnel extending to either 

side of t,he dipole, the tangential field components take 

on the usual form 

E* = C gikVfjk exp (&yjl;z) 

j. k 

((33) 

ijk = hjk cos JC~X cos kzy ((34) 

where Zo,t, 'C"*jr:, ioik, and Grrc are the characteristic impedance, + and - wave voltage coefficients, and normalization constants, respectively, for each waveguide mode. 

By matching the tangential boundary conditions over t,he 

cross section containing the dipole and selecting the 

dominant (1,l) Eh mode contribution by using the mode 

orthogonality properties, -n-e obtain 

V+,I,I = - - [ [ (E, X I?(xly)) -i~.l(x,~) dz dy. (G7) 

Carrying out the integration we get 

The total power coupled into the dominant (1,l) Eh 

waveguide mode propaga,ting in both tunnel directions, 

The power ra.diated by the dipole is 

where R, is the dipole radiation resistance. The desired 

coupling factor C is the fraction of the dipole radiated 

power that is coupled t,o the (1,l) Eh mode propagating in 

one of the tunnel directions for a. half-wave dipole; C is 

given by 

When the wavelenpth - is small commred with the tunnel 

cross-sectional dimensions, the cosine factors are approximately zero at the tunnel walls, so that kl r/dl, JC2 E 

?r/dz and 2/(d1d2) 'I2. Then (G11) becomes 

REFERENCES 

[I] A. G. Emslie, R. L. Lagace, and P. F. Strong, "Theory of the 

propagation of UHF radio waves in coal mine tunnels," in Proc. 

Through-the-Earth Electro~napetics U'orbhop, Colorado School of 

Mines, Golden, Colo, Aug. 1.5517. 1973, available from Nat. 

Tech. Inform. Service, Alexandria, Va. 

[2] E. A. J. Marcatili and It. A. Schrneltzer, "Hollon- metallic and 

dielectric waveg~udes for long distance optical transmission and 

lasers," Bell Syst. Trch. J., vol. 43, p. 1783, 1964. 

[3] J. I. Glaser, "Attenuation and guidance of modes in hollow dielectric m-aveguides," IEEE Trans. :llicrowave Thmy Tech. 

(Corresp.), vol. i\iTT-17, pp. 173-174. Mar. 1969; and M.I.T., 

Cambridge, hIas., Ph.Z). thesis, Dep. Elec. Eng., "Low-loss waves in hollow dielectric tubes," Feb. 1967. 

[4] A. E. Goddard, "Radio propagation measurements in coal mines 

at UHF and VHF," in Proc. Through-the-Earth Electronmgnelics 

Vorkshop, Colorado School of Mines, Golden, Colo., Aug. 1517, 

1973, available from Nat. Tech. Inform. Service, Alexandria, Va. 

[5] Bureau of Mines Information Circular/l974, IC-8635, "Mine 

communications," Proc. Bureau of :lfines Technology Transfer 

Seminar, Bruceton, Pa., hlar. 21-22, 1973. 

[6] P. Beckrnann and A. Spizzichino, The Scattering of Eledr~nagnetic Waves from Rough Surfaces. New York: 3iacmillan, 1963, 

p. 81, eq. (6). 

[7] E. M. Sparrow and R. D. Cess, Radiation Heat Transfer. Belmont,, Calif.: Wadsworth, 1966, App. A.