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 OF THE 
 
 UNIVERSITY OF CALIFORNIA. 
 
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eflection and Transmission of Electric Waves by 
 a Metallic Grating. ^^\ o *** y~ 
 
 BY PKOF. HORACE LAMB, F.R.S. 
 
 [Extracted from the Proceedings of the London Mathciirtotjfaf Sonch/, 
 Vol. XXIX., Nos. 644, 645.] 
 
 The main problem of this paper consists in the calculation of the 
 disturbance produced in a train of electric waves by a plane grating 
 composed of parallel, equal, and equidistant metallic strips. The 
 treatment is approximate, and involves the assumption that the 
 wave-length is large compared with the distance between the centres 
 of consecutive strips ; the application is, therefore, rather to Hertzian 
 waves than to phenomena of ordinary Optics. The previous mathe- 
 matical literature bearing directly on the subject consists of an 
 investigation by Prof. J. J. Thomson,* and of two papers by Lord 
 Rayleigh.f It will be seen that the author is under great obligations 
 to each of these sources. 
 
 The work of 1, 2, 3 is preliminary, and consists in the solution 
 of several electrostatical problems. With modern methods these 
 present no difficulty ; but the results appeared to possess sufficient 
 interest of their own to warrant the expenditure of a little trouble in 
 the way of graphical representation. The diagrams obtained serve, 
 moreover, to illustrate very clearly some of the conclusions arrived at 
 afterwards in the discussion of the wave problem. 
 
 The problem of 4 is introduced on account of its close connexion 
 with what precedes, although it has no bearing on the main subject 
 of the paper. 
 
 1. Let it be required to find the disturbance produced in a uniform 
 electric field by a grating (of the kind above described) placed with 
 its plane perpendicular to the lines of force. We will suppose that 
 the axis of x is normal to the plane of the grating, and the axis of y 
 in this plane, at right angles to the lengths of the strips. 
 
 In the case of a field symmetrical on the two sides of the grating, 
 it will be found that (with a special choice of units) the potential- 
 and stream-functions are given by the formula 
 
 cosh w = fji cosh z, (1)J 
 
 * Recent Researches on Electricity and Magnetism, 359. The problem attacked is 
 that of reflection by a grating of cylindrical wires. Some notes on this question 
 are included in the present paper, 7. 
 
 t Phil. Mag., April and July, 1897. See also Theory of Sound, 292. 
 
 t [y^/A 1898. This is not new, as the author had supposed. It was set in the 
 Mathematical Tripos, Part II., 1895, and is due to Mr. Lannor.] 
 
524 Prof. H. Lamb on Reflection and Transmission of [May 12, 
 
 where w <f> + iij/, z = x + iy, (2) 
 
 and the constant /x is supposed greater than unity. This result was 
 obtained originally by a modification of Schwarz's method, but it will 
 be sufficient here to verify it a posteriori. The formula (1) gives 
 
 cosh <f> cos \l/ = /A cosh x cos y\ 
 sinh < sin \f/ = p sinh x sin y ) 
 
 The locus <f> = consists of those portions of the axis of y for which 
 
 1 > /x, cos y > 1 ; 
 
 these represent the breadths of the metallic strips, so that (on the 
 scale of our formulae) the half -breadth of a strip is sin' 1 (I//*). For 
 other portions of the plane, < is, so far, indeterminate as to sign ; we 
 will take it to be everywhere positive. Then ;// is indeterminate to 
 the extent of a term 2s7r, but we may suppose that the lines \J/ 0, 
 ^ = TT, *\t = db27r, ... consist partly of the lines y = 0, y = TT, 
 y = 27T, . . . , respectively, and partly of those portions of the axis of 
 y for which 
 
 these correspond to the widths of the apertures between the 
 successive strips. The half-breadth of an aperture, on the present 
 scale, is cos'^l//*). At the centres of the apertures for example, at 
 the origin we have <f> = cosh' 1 //,; these are points of zero force, and 
 the equipotential- and force-lines there meet at angles of 45. 
 I have thought it worth while to trace the curves 
 
 < = const., ty = const., 
 using the formula 
 
 where 
 
 p = -^ (cosh 2< + cos 2i/0 , q = ~ (1 + cosh 2<j> cos 2^) ; (5) 
 
 L./A '-/A 
 
 this is easily deduced from (3). The value of fj. adopted for con- 
 venience of calculation was 
 
 ya = cosh > = 1/2040, 
 
 whence sin" 1 I* = '312^, cos" 1 (l/x) = -18&r ; 
 
 this gives the relative breadths of the strips and apertures. The 
 curves are shewn in Fig. 1. 
 
Or THE 
 
 1898.] 
 
 Electric Wavde by a Metallic Grating. 
 
 525 
 
 FIG. 1. 
 
526 Prof. H. Lamb on Reflection and Transmission of [May 1 2, 
 
 The formulae (3), and the diagram, admit of the usual variety of 
 interpretations in Electrostatics, Electric Conduction, Hydrodynamics, 
 &c. If we introduce a more general linear unit, and denote the 
 breadth of each aperture by a, that of each strip by fr, we may write 
 
 cosh d> cos il/ = u cosh = cos ^7 
 a+b a+6 
 
 K (6) 
 
 . , , . . -i TTX . irii I 
 
 smb. d> sin \1/ = a sinh sin *- 
 
 J 
 
 TTtt Tr /i-7\ 
 
 where /u, = sec = cosec . (7) 
 
 z(a-fo) 2 (a + 6) 
 
 A first application of (6) is to find the capacity of a condenser 
 formed by two metal plates at the same potential with an insulated 
 grating half way between them, the distance between the two external 
 plates being large compared with the interval a + 6. For large 
 positive values of #, the formulae (6) give 
 
 . (8) 
 
 The charge per unit area of one of the plates is 1/4 (a + 6) in the 
 present units, and the capacity of the system per unit area is, therefore 
 
 (9) 
 
 where a? is the half -distance between the external plates. If the 
 central plate had been entire, the capacity would have been l/27nc ; 
 the effect of the apertures is therefore equivalent to increasing the 
 distance between the central plate and each external plate by an 
 amount c, given by 
 
 (10) 
 
 2 (a 
 
 The numerical values subjoined shew what a very large proportion 
 of the central plate can be cut away without sensibly altering the 
 capacity, so long as the period (a + 6) of the residual structure is 
 fej^ge compared with the distance between the plates : 
 
 -^- = 0, 1, -2, -3, -4, -5, -6, -7, -8, -9, I'O 
 
 -~ = 0, -004, -016, -037, -067, 110, 169, -251, -374, "590, oo. 
 a+b 
 
1898.] Electric Waves by a Metallic Grating. 527 
 
 For another interpretation of (6) we may imagine that a current is 
 flowing along a metal strip, of breadth a + &, which is interrupted by 
 a central transverse non-conducting partition (e.gr., a narrow cut in 
 the metal), of breadth a, or by two lateral transverse cuts, opposite 
 to one another, each of breadth \a. It appears that the additional 
 resistance thus introduced is equal to that of a length 2c of the strip. 
 
 2. The function < in (3) is a periodic even function of ?/, the period 
 being TT. It can therefore be expanded by Fourier's Theorem in a series 
 of cosines of multiples of 2y, the coefficients being functions of x 
 whose general form is to be determined by substitution in the equation 
 
 ct <o d <p p. /1 1 \ 
 
 ~r^ + ~jsv- (**) 
 
 Thus, for x positive we find, having regard to the conditions to be 
 satisfied for large values of x, 
 
 <f> = log fjL+x + 2," C m e~ 2mx cos2my. (12) 
 
 The precise values of the coefficients C m hardly concern us, but it 
 may be worth while to record them. The expansion of the function 
 
 w = cosh' 1 (/A cosh z) (13) 
 
 can be effected directly, with the assistance of Lagrange's Theorem, 
 and in this way we obtain the form (12), with 
 
 r -=r V-'Ml m * X(m 2 - 12 ) m 2 (m 2 -l 2 )(m 2 -2 2 ) 1 
 
 ml I 2 ./* 2 l 8 .2./n* P.2 2 .3 2 ./x 6 "/ 
 
 (14) 
 It may be noted that this contains 1 I//* 2 as a factor.* 
 
 * {July, 1898. We have 
 
 c m = (- m ' l -r( m ,- m , 1,1) 
 
 m \ .- 1 
 
 in the hypergeometric notation. 
 
 At the suggestion of one of the referees the proof of (14) is indicated. If we put 
 y = in (13), we have 
 
 <p = cosh- 1 (/j. cosh#) = log |,u cosh#+ V(yu 2 cosh 2 ic 1) j 
 
 where t = e~ 2 *. If we write 
 
528 Prof. H. Lamb on Reflection and Transmission of [May 12, 
 
 The comparison of (12) with (3) leads to a number of results in 
 definite integrals. For example, using square brackets to denote that 
 x has been put = in the values of the enclosed functions, we have 
 
 r 
 
 Jo 
 
 (15) 
 
 "Now it appears from (3) that from y = to y = cos" 1 (!//*) we have 
 
 O] = 0, [<] = cosh" 1 (/x cos y), 
 whilst, from y = cos' 1 (!//*) to y = |TT, 
 
 [</>] = 0, [>] = cos' 1 (/x cos y). 
 Hence (15) gives the theorem 
 
 (17) 
 
 This is easily verified. If we denote the value of the definite integral 
 on the left hand by / (/?), we find 
 
 whence (17) follows by integration, since / (0) = 0.* The verification 
 of (16) is still simpler ; we have only to put d<j>jdx = d\j//dy. 
 
 we find C= l 
 
 say. If x be positive, t will be less than unity, and log can be expanded in 
 powers of t by Lagrange's Theorem, thus 
 
 where u is to be put = 1 , finally. Working this out, we find 
 
 logC=:srcw' H , 
 where C m has the value given above. Hence the value of $ f or y = and x > is 
 
 It easily follows that (12) gives the general value of </> f or x > 0.] 
 
 * The theorem (17) is, substantially, one of a number given by Lobatschewsky, 
 Kasan Memoirs, 1836 (Part i., p. 149), but the proof is quite different. The formula 
 
 i og sec 0, 
 
 
 
 which is immediately deduced from (17) by a partial integration, was obtained by 
 Legendre, Exercices de Calcul Integral, 1. 1., Supplement, Caseix. (1811), as a particular 
 case of a general theorem involving elliptic integrals. 
 
1898.] Electric Waves by a Metallic Grating. 529 
 
 3. When the field of force is not symmetrical with respect to the 
 plane of the grating, the requisite formulas are obtained by the 
 addition of a term Az to w in (1), so that 
 
 cosh (w -{-Az) = /x. cosh z. (19) 
 
 The most interesting special case is where the constant A is so chosen 
 as to annul the force at a distance on one side, say that for which x is 
 negative. We then have A = 1, and the equation (19) reduces 
 to the form 
 
 i coshw 
 
 tanh z = 
 
 , 
 smh w 
 
 whence z = J log - . (21) 
 
 JJL e~ 
 
 If we write //. = e, this can be put into the shape 
 
 ! log =, (22) 
 
 sinh 
 
 or 
 
 11 sinh ^(<j> a) cos 1^ + 2 cosh ^(6 a) sin ii 
 
 (23) 
 This gives, after a little manipulation, 
 
 i 
 
 . , . 
 smh a sin 
 
 cosh 9 cosh a cos 
 By means of these equations the curves 
 
 </> = const., \js = const. 
 
 can be traced without much difficulty. Their forms are shown in 
 Fig. 2 (p. 530) for the case of a = TT. The diagram affords the best 
 verification of the formulae, but the following points may be noticed. 
 In the first place, the line ^ = consists of distinct portions, 
 
 according as <j> a. For < > a we may take 
 
 ' (25) 
 
 the value of x ranging from GO to -fco as < increases from a to oo. 
 This gives the line of force through the origin, which is the centre of 
 
 * The solution for the present case was originally obtained in this form by 
 Schwarz's method. 
 
530 Prof. H. Lamb on Reflection and Transmission of [May 12 
 
-1898.] Electric Waves ly a Metallic Grating. 531 
 
 an aperture, and the lines of force through the centres of consecutive 
 apertures are given in like manner by 
 
 y= TT, 2;r, 37r, ..., 
 corresponding to 
 
 If, on the other hand, if/ and < < a, we have 
 
 (26) 
 
 The value of x ranges from to GO as < increases from to a. The 
 value to be attributed to tan" 1 is determined by considerations of 
 continuity. Thus, if we assign a very small fixed positive value to \l/, 
 and then imagine (j> to decrease from oo through a to 0, the fraction 
 
 sinh a sin if/ 
 cosh (f> cosh u cos if/ 
 
 will pass from a small positive value through infinity to a small 
 negative value. Hence, if if/ approach zero from the positive side, the 
 value of y in (26) will be ^TT. If if/ were to approach zero from the 
 negative side, we should have y = ^TT. The portions of the lines 
 
 y = i^Tr, |7T, |TT, ... 
 
 for which x is negative are lines of force abutting on the backs of the 
 metal strips. 
 
 Again, for if/ = TT, we have 
 
 this value of x ranges from to oo as <f> increases from to oo . And, 
 generally, the portions of the lines 
 
 y = |TT, ifTT, |ir, ... 
 
 for which x is positive are lines of force abutting on the fronts of the 
 metal strips. 
 
 If in (24) we make </> = 0, we get 
 
 = 0, y = fr + l tan-' . ( 28 ) 
 
 I cosh a cos if/ 
 
 This gives a series of isolated portions of the axis of y, representing 
 the breadths of the metal strips. These are bounded by the points 
 for which 
 
 cos if/ = e~ a = l//z, 
 
 and therefore sinh a sin if/ _ = ^ 
 
 I cosh a cos if/ 
 
532 Prof. H. Lamb on Reflection and Transmission of [May 12, 
 
 For example, if we make \J/ increase from to IT, y will diminish from 
 JTT to the minimum value cos"^"", and then increase again up to the 
 value I-TT. 
 
 It is further to be noticed that the lines <f> = const, fall into two 
 systems separated by the curves < = a, which have the lines 
 
 y = 0, 7T, 27T, ... 
 
 as asymptotes. For <<a, the equipotential lines are closed. This is 
 indicated clearly in the figure, which may be compared with the corre- 
 sponding diagram for a grating of cylindrical wires given by Maxwell.* 
 The formulae (24), when generalized as to the linear unit involved, 
 lead to a determination of the capacity of a condenser formed of two 
 parallel plates, one of which is entire, whilst the other is a grating. 
 If the distance x between these plates be large compared with the 
 interval a + 6, where a and b have the same meanings as in 1, we find 
 for the capacity in question 
 
 f ^a + 6, ~^a \ ' (30) 
 
 ** A x + -^- log sec L 
 
 (. 2;r 2(a + 6) ) 
 
 It is also easy to shew that the charge on the grating is divided 
 between its front and back surfaces in the ratio a -f 26 : a. 
 
 4. Although the matter is not very closely connected with the 
 main object of this paper, it may be worth while to notice the inter- 
 pretation of the formulae (3) when the constant p. is less than unity. 
 They give the distribution of electric force in a condenser one of 
 whose plates has a series of parallel equidistant ribs projecting at 
 right angles to its surface. Thus, if, altering the linear unit, we write 
 
 cosh w = i*. cosh , (31) 
 
 a 
 
 or cosh <J> cos \L r = fj. cosh cos | 
 
 (32) 
 
 siiih < sin ^ = p. sinh - sin I 
 a a J 
 
 we find that the line <f> = now consists of the whole of the axis of y, 
 together with those portions of the lines 
 
 y = 0, a, 2a, 3a, ... 
 which extend from the axis of y to the points determined by 
 
 - = . (33) 
 
 a fjL 
 
 Electricity and Magnetism, Vol. I., Fig. xiii. 
 
1898.] Electric Waves by a Metallic Grating. 533 
 
 Fin. 3. 
 
534 Prof. H. Lamb on Reflection and Transmission of [May 12, 
 Hence, if b be the depth of a rib, we have 
 
 /i = sect. (34) 
 
 The curves < = const., if/ == const, are shown in Fig. 3, for the case of 
 
 Proceeding as in 1, we find that the capacity (per unit area) of a 
 condenser of the above type is 
 
 (35) 
 
 4K[x log; cosh J 
 \ TT a I 
 
 x denoting the distance between the plates, which is assumed to 
 be large compared with b. The effect of the ribs in increasing the 
 capacity is therefore equivalent to diminishing the distance between 
 the plates by the amount 
 
 log cosh . (36) 
 
 7T CL 
 
 Again, if we take the curves </> = const, to be lines of electric flow, 
 we get the case of a current flowing along a relatively broad strip 
 which is interrupted along one edge, or along the medial line, by a 
 series of parallel, equal, and equidistant transverse cuts, one behind 
 the other. If b be the length of the cuts, their effect (when at the 
 edge) is equivalent to diminishing the breadth of the strips by the 
 amount (36). As numerical examples, we find, for 
 
 &/= -1, -2, -3, -4, -5, -6, -7, -8, -9, 1-0, 
 
 log cosh *&/._ . 156> -296, -415, -511, '586, -645, '691, 729, -756, -780. 
 Trb/a 
 
 The latter numbers give the ratio which the aforesaid diminution 
 bears to the breadth b of the cuts. For increasing values of b/a this 
 ratio tends obviously to the value unity. 
 
 In the case of cuts across the medial line the quantity (36) must 
 be multiplied by 2, b now denoting the half-length of a cut. 
 
 5. Proceeding to the problem of electric waves, and assuming that 
 all our functions involve the two dimensions x and y only, we have 
 to satisfy an equation of the form 
 
 1 = 0, (37) 
 
 dx z dy* 
 
 subject to the proper condition at the surfaces of the metal. If the 
 
1898.] Electric Waves by a Metallic Grating. 535 
 
 electrical vibrations be sufficiently rapid, the condition is that 
 the component of electric force tangential to the surface, just outside 
 the metal, must be zero. 
 
 We take separately the cases where the direction of the electric 
 force is perpendicular and parallel, respectively, to the plane xy. In 
 the former case, the equation (37) is satisfied by the electric force 
 (R), and the condition which is to hold at the metal surfaces is 
 R = 0, simply. 
 
 Let us now suppose that a train of waves of length 2ir/k and 
 amplitude unity is incident directly on the grating from the side on 
 which x is positive. Omitting the time-factor e'* n , where V is the 
 wave-velocity, this train may be represented symbolically by 
 
 E = e ikx . (38) 
 
 If in place of the grating we had an uninterrupted sheet of metal, the 
 reflected wave would be represented by e~'* x , and the total disturb- 
 ance on the side of x positive would be given by 
 
 E = e--e- te , (39) 
 
 for this satisfies (37) and makes R = for x = 0. 
 
 In the actual circumstances we assume that, on the positive side 
 of the grating, R _ e *._ e - + Xj (40) 
 
 and, on the negative side, R = x'? (41) 
 
 where x and x' must, of course, satisfy (37). Over the portions of the 
 plane x = which are occupied by the metal strips, we must have 
 
 X = 0, X' = 0; (42) 
 
 whilst the continuity of R and dR/dx demands that 
 
 *=*' 1*2 ' <> 
 
 over the apertures. The latter condition will be satisfied provided 
 
 w e make d dv' , AA ^ 
 
 23 _^, -A. ifa (44) 
 
 dx dx 
 
 If the differential equation to be satisfied by x had been 
 
 ffu <Pu = (45) 
 
 dx 2 dy* 
 
 in place of (37), the conditions relating to this function would 
 (save as to the factor ik) have been the same as in the hydro- 
 dynamical problem where a system of equal, parallel, and equidistant 
 
536 Prof. H. Lamb on Reflection and Transmission of [May 12, 
 
 blades, all in the same plane, is moved perpendicular to itself 
 in a frictionless liquid, which may have an independent general 
 motion in the same direction. The function x/.ik would, in fact, give 
 the value of the velocity-potential on the positive side of the plane 
 x = 0. It must be borne in mind in this analogy that the blades 
 correspond to the apertures of our grating, and the intervals between 
 them to the metallic strips.* Xow the relative motion of the fluid in 
 such a case is shown in Fig. 1, to which the formulae (6) apply ; and 
 a value of x satisfying the conditions (42) and (44) would therefore be 
 
 x = -ikx + C<f>, (46) 
 
 where <j> is the positive function determined by (6). 
 
 This value of x is necessarily periodic with respect to y, the period 
 being a + b ; and the line y = is a line of symmetry. Hence, 
 expanding by Fourier's Theorem in a series of cosines of multiples 
 of 27r?//(a + 6), we find 
 
 x = C log n + ( -^ - tfc) x + CSr O m e-*4* cos ^ , (47) 
 \a + o / ft-f-o 
 
 where K H , = , (48) 
 
 a+o 
 
 and the first two terms have been determined from a comparison of 
 (46) and (8) . The terms involving e** are excluded by the condition 
 of finiteness for x = oo. 
 
 The true value of x can ^ e similarly expanded in a Fourier's series, 
 the forms of the coefficients as functions of x being* determined by 
 means of the differential equation (37). The resulting form is 
 
 x = jV-'** + 2r^-V*cos, (49) 
 
 provided 
 
 Since, by hypothesis, a + ~b is small compared with the wave-length 
 27T/&, the right-hand side of (50) is positive. Hence the quantities \ m 
 are real, and, moreover, differ respectively very little from *,. Terms 
 involving e '" x are excluded as before by the condition of finiteness for 
 x = oo, so that the waves represented by x are ultimately plane. The 
 fact that they must travel outwards from the grating justifies the 
 omission of the term in e ikx . 
 
 * The foregoing method of analysing the conditions to be satisfied is taken from 
 Lord Rayleigh's paper, Phit. Mag., April, 1897- 
 
'1898.] Electric Waves by a Metallic Grating. 537 
 
 For small values of x, the expansion (49) reduces to the form 
 
 a-\- b 
 
 approximately; and it appears, on comparison with (47), that the 
 conditions incumbent on x w ^ a ll ^ e satisfied, subject to an error 
 of the order tf (a + 6) 2 /47r 2 , if 
 
 (52) 
 
 Substituting the value of ft from (7), we find 
 
 where c is the linear magnitude defined by (10). 
 
 The conditions for x are all fulfilled if we suppose that its value at 
 any point P on the negative side of the grating is equal, both in 
 magnitude arid in sign, to the value of x a ^ the image P of P' on the 
 positive side. Hence the Fourier expansion of x i g 
 
 ' 7~> ikr ^ T> -m SK. A\ 
 
 x = J3 e +2i B m e '" cos *. (54) 
 
 In the above investigation, the coefficient of the primary wave has 
 been taken to be unity. On the same scale, the coefficients of the 
 reflected and transmitted waves are 1 + # and _B , or 
 
 1 ike 
 
 and 
 
 I + ikc 1+t'fo' 
 
 respectively. Hence the intensities J, T (say) of these waves, in 
 terms of that of the primary wave as unit, are given by 
 
 It appears from the numerical values given near the end of 1 that 
 the value of c/(a + 6) may be very moderate even when the breadth (6) 
 of the metal strips is relatively small. Hence even a very open grating 
 (i.e., one in which a very small proportion of the area is occupied by 
 metal) will produce almost total reflection of waves in which the 
 electric displacement is parallel to the bars, provided the wave-length 
 (2ir/k) be moderately large compared with the period a-\-b of the 
 structure. 
 
 * Cf. J. J. Thomson, loc. tit. 
 
538 Prof. H. Lamb on Reflection and Transmission of [May 12, 
 
 The circumstances when the reflection is practically total are 
 clearly illustrated in Fig. 2, where the lines <f> = const, represent the 
 lines of (alternating) magnetic force in the immediate neighbourhood 
 of the grating. A similar interpretation applies also to Maxwell's 
 diagram already referred to. 
 
 6. When the electric force is everywhere parallel to the plane xy, 
 the magnetic force (y) is everywhere perpendicular to this plane, and 
 satisfies an equation of the form (37). The components of displace- 
 ment-current in the dielectric are then 
 
 _1_ dv __ 1_ dy Q 
 4<7r dy 4tTr dx 
 
 and the condition to be satisfied at the surfaces of the metal strips is 
 dy/dn = 0, where dn denotes an element of the normal. 
 Considering a train of waves represented by 
 
 7 = e to , (56) 
 
 and therefore incident directly on the grating from the side of 
 x positive, we should have, if the apertures were filled up, 
 
 y = e ik * + e- ikx (57) 
 
 on the positive side, and y = on the negative side. In the actual 
 case we assume, for the positive side, 
 
 y^e^ + e-'^ + x, (58) 
 
 and, for the negative side, y = \-> (59) 
 
 where x and x' must satisfy (37) ; and the conditions to be fulfilled 
 over those portions of the line x = which correspond to the metal 
 strips are 
 
 fv = 0, &=o, (60) 
 
 dx dx 
 
 whilst over the apertures we must have 
 
 on account of the continuity of y and dy/dx. The former of the con- 
 ditions (61) will be satisfied provided 
 
 X=-l, x' = l. (62) 
 
 If the differential equation (37) were replaced by (45), the con- 
 ditions applicable to x would be similar to those which govern the 
 
UNIVERS/TY 
 1898.] Electric Waves b^J^^^JSHrating. 539 
 
 electric potential in the case of a grating placed transverse to the 
 lines of force in a uniform field ; but it must be remembered in this 
 analogy that the bars in one problem correspond to the apertures in 
 the other, and vice versa* We should therefore have 
 
 where < is determined by (6) ; but, if we still continue to denote by a 
 and 6 the breadths of the apertures and of the strips, respectively, in 
 our grating, the formula (7), giving the value of /x, must be replaced 
 
 
 
 7TO 7TCI /aA\ 
 
 p = sec - = cosec - . (64) 
 
 2(a + &) 2(a+6) 
 
 The Fourier expansion of (63) is 
 
 (65) 
 
 whilst that of the true function x nas the form 
 
 where K, n and \ m are defined by (48) and (50) . For small values of as 
 the latter expansion takes the form 
 
 x = C (1 -i fee) + Si" B m e ~ Km * cos %22t . (67) 
 
 a ~T~ 
 
 By comparison we find that (66) will satisfy approximately all the 
 conditions of the problem, provided 
 
 (68) 
 
 Hence " 0.= -, (69) 
 
 provided c x = - log sec - . (70) 
 
 As regards x'? all the conditions are satisfied if we suppose that its 
 value at any point P* on the negative side of the grating is equal in 
 absolute magnitude, but opposite in sign, to that of x at the image 
 P of P f on the positive side. Hence the Fourier expansion of x' is 
 
 7T f *-i ^ 
 
 cos --. (71) 
 
 * See the footnote on p. 536. 
 
540 Prof. H. Lamb on Reflection and Transmission of [May 12, 
 The coefficients of the reflected and transmitted waves are 1 + C 
 
 and C , or 
 
 ike, -, 1 
 ^- and 
 
 1+ikc^ 
 
 respectively, that of the primary being taken as unit. Hence the 
 intensities J, /' of these waves are given by 
 
 (72) 
 
 For sufficiently great wave-lengths there is very little reflection, even 
 when the intervals a between the strips are very narrow compared 
 with the breadths of the strips themselves. 
 
 * [These results, combined with those of 5, serve to explain the 
 polarizing efficacy of a grating, as employed originally by Hertz. As 
 a numerical example, let us suppose that the wave-length is ten times 
 the interval a + b. Then, even if one-tenth only of the area of the 
 grating be occupied by the metal strips, the coefficients of reflection 
 and transmission are, for electrical vibrations parallel to the lengths, 
 
 I = -879, r = -121. 
 
 The same numbers in reverse order give the coefficients of reflection 
 and transmission for electrical vibrations transverse to the length, 
 when the apertures occupy only one-tenth of the area ; viz., 
 
 /=-121, J"=-879. 
 
 The coefficient of reflection in the one case, and of transmission in 
 the other, increases rapidly with the wave-length.] 
 
 In the immediate neighbourhood of the grating the lines y = const. 
 coincide with the curves < = const, of Fig. 1. These give the direc- 
 tions of the electric oscillation ; but it must be remembered that the 
 portions of the vertical line of symmetry which represented the metal 
 strips in the electrostatic problem of 1 now correspond to apertures, 
 and vice versa. 
 
 The mathematical conditions to which the magnetic force y is 
 subject in the problem of this section are exactly those which apply 
 to the velocity-potential in the case of sound-waves incident directly 
 on a grating of the kind considered. f It appears that a series of 
 narrow slits, with relatively broad opaque intervals, may transmit a 
 very large proportion of the original sound, provided the wave-length 
 be sufficiently great. 
 
 * Paragraph inserted July, 1898. 
 
 t Of. Rayleigh, Phil. Mag., April, 1897, p. 272. 
 
1898.] 
 
 Electric Waves by a Metallic Grating. 
 
 541 
 
 7. As mentioned in the introduction, the problem of reflection by a 
 grating composed of parallel cylindrical bars has been discussed by 
 Prof. J. J. Thomson in the case where the electric displacement is 
 everywhere in the direction of the length. The result arrived at is 
 that the reflection is total, without any limitation as to the thinness 
 of the bars. It may be worth while to examine this point, and to 
 complete the investigation by considering also the case where the 
 electric displacement is transverse. 
 
 If, as in 1, we write 
 
 w 
 
 (73) 
 
 the potential- and stream -functions corresponding to a system of equal 
 and equidistant line-sources cutting the plane xy perpendicularly at 
 the points (0, 0), (0, =fca), (0, 2a), ..., are given by the formula 
 
 w a log z + log (zid) + log (z + id) + log (z 2id) + log (z -f 2m) + . . . , 
 
 (74) 
 
 or, say, w = log sinh . (75) 
 
 This makes < = 1 log^ (cosh - cos 
 
 2 2 \ c 
 
 = tan-'n 
 
 , , TfX 
 
 tanh 
 
 (76) 
 
 in agreement with Maxwell, Electricity and Magnetism, 203. 
 
 The case of a row of double sources having their axes parallel to x 
 is obtained by differentiating (75) with respect to z ; thus we find 
 
 7T -i , -I TTZ 
 
 w = log coth 
 a a 
 
 (77) 
 
 sinh 
 
 2-rrx 
 
 i 2irx 2-Try 
 
 cosh cos 
 
 sin 
 
 cosh 'cos 
 
 a a 
 
 (78) 
 
 If the double sources have their axes parallel to y, we have merely to 
 interchange the values of (f> and fy in these formulae. 
 
542 Prof. H. Lamb on Reflection and Transmission of [May 1 2, 
 
 These results enable us to solve very simply a number of problems 
 in Hydrodynamics, Magnetic Induction, and so on. For example, the 
 stream -function for a liquid flowing through a grating of parallel 
 cylindrical bars is 
 
 sin 
 
 a , 2-jrx 
 
 cosh cos - 
 
 (79)* 
 
 where V is the velocity at infinity. This formula is easily verified, 
 since, for small values of x and y, it reduces to the form 
 
 (80) 
 
 It is here assumed that the radius b of a bar is small compared with 
 the distance a between the axes of consecutive bars. 
 
 Again, in the case of a liquid flowing parallel to a row of cylindrical 
 obstacles, we have 
 
 sinh 
 
 cosh cos 
 
 a a 
 
 (81) 
 
 Returning to the formula (75), we find, if the real part of z be 
 positive, w __ 
 
 (82) 
 (83) 
 
 (84) 
 (85) 
 
 TTZ -I Q ^oo 1 2wirZ 
 
 " ~a~ ' l m 6 
 
 TTX o 1 
 
 whence < = log 2 Sj e -** cos 
 
 a m a 
 
 Since < is an even function of a/, we have, for x negative, 
 
 2miry 
 
 = __]og2-3Tl 
 
 a m 
 
 la cos 
 
 For small values of a?, we have, from (76), 
 
 .; If : * = 1 gf> 
 
 where r = V(a? + y' 
 
 When the electric force (JK) is everywhere parallel to the bars of 
 the grating, we assume, for x positive, 
 
 * The curves vj/ = const, have been traced from this formula by Prof. Hele Shaw, 
 Trans. InsL Nov. Arch.. 1898. 
 
1898.] Electric Waves by a Metallic Grating. 543 
 
 ^ ^TTZ'TTT/ xr\>^x 
 
 -*m*COS f, (86) 
 
 and, for x negative, 
 
 R = 5e tA;a; 4-^D m e A -m a; cos ^^, (87) 
 
 where X m is determined by 
 
 \ 2 4m 7T 79 fQO\ 
 
 A w - K . (OO) 
 
 For small values of x, we may ignore the difference between X m 
 and 2ra7r/a ; and, if we further put D m = 0/m, the formulae (86) 
 and (87) may be put into the shapes 
 
 / 7TC \ 
 
 \ a I ' 
 
 / TTfcC r\ \ s/\r\\ 
 
 and B = Jo -f ikBx + ( <^> H h log 2 1 , (90) 
 
 \ d / 
 
 respectively, where <j> is defined by (83) and (84) . The continuity of 
 B and dB/dx requires that 
 
 a a 
 
 At the surface of the metal wires, where r = b (say), we must have 
 B = 0. This cannot be satisfied exactly ; but, if the ratio 6/a be 
 sufficiently small, it is satisfied approximately if we make 
 
 (92)* 
 a 
 
 If we now write c = log -^- , (93) 
 
 7T 27T& 
 
 we find, from (91) and (92), that 
 
 (94) 
 
 , 
 l<+iko 
 
 The intensities of the reflected and transmitted waves, in terms of 
 that of the primary wave, are therefore 
 
 I f the wave-length is at all large compared with c, kc is small, and 
 the reflection is almost total. But, for any given wave-length (large 
 compared with a) , kc may be made as great as we please by sufficiently 
 diminishing the radius 6 of the wires. In this way we can pass to 
 the case of free transmission. 
 
 * \_July, 1898. Viz., this makes R = for x = 0, r = .] 
 
544 On Reflection and Transmission of Electric Waves. 
 
 When the electric force is everywhere parallel to the plane xy, it 
 is convenient (as in 6) to adopt as dependent variable the magnetic 
 force (y), which is everywhere perpendicular to this plane. We 
 assume, for x positive, 
 
 y = e <* + Ae ik * + S E M e " x "* cos ^^ , (96) 
 
 and, for x negative, 
 
 y = Be^-ZE,^ cos ^^- , (97) 
 
 CL 
 
 where \ m is denned by (88). If we put E M = SvC/a, and ignore the 
 difference between \ m x and 2m7rx/a, this gives, for small values of x, 
 
 k (l-A)x+C &-) , (98) 
 
 a I 
 
 and y= B + ikBx + C f ^ + ), (99) 
 
 respectively. The continuity of y and dy/dx on the two sides of the 
 plane x = requires that 
 
 I + A- C = B + C, l-A = B. (100) 
 
 a a 
 
 The surface-condition is that dy/dr = for r = b. If we put 
 x = r cos 0, d<j>/dx = cos 0/r, this gives 
 
 |r = o. (101) 
 
 7T& 2 
 
 Hence, if we write < 
 
 we get A = ^ . B = ; . (103) 
 
 The intensities of the reflected and transmitted waves are now 
 
 (104) 
 
 If the half wave-length be large compared with tf/a, we have free 
 transmission, with hardly any reflection. 
 
 As in the problem of 6, this last calculation has also an acoustical 
 interpretation, and serves to further illustrate the " extreme smallness 
 of the obstruction offered by fine wires or fibres to the passage of 
 sound." * 
 
 1. ii., 343. 
 
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