Study on Arrival Times of Transient Creeping Wave and Transient Whispering-Gallery Mode

Keiji Goto*, Kojiro Mori, Yuki Horii, and Mizuki Sawada Department of Communication Engineering

National Defense Academy Yokosuka, Kanagawa, Japan

Abstract—We study the arrival times of a transient creeping wave (CW) and a transient whispering-gallery mode (WG) when a pulse wave is incident on one of the edges of a cylindrically curved conducting open sheet. The time-domain asymptotic solu-tions for the transient CW and the transient WG are derived asymptotically by applying the Fourier transform method. We show the interesting phenomenon that the transient CW propagates along the curved sheet later than the transient WG.

I. INTRODUCTION By technological advances in the area of the analysis of

radar cross section, high resolution radar, and target identification, it is becoming important to study the asymptotic analysis methods for the frequency-domain (FD) and the time-domain (TD) electromagnetic scattered field by various kinds of curved structures with edge or wedge [1]---[4].

In the previous study [5], we have derived the TD asymptotic solution for the transient scattered field when a pulse wave is incident on a cylindrically curved conducting surface with edges. The TD asymptotic solution has been represented by a combination of the geometric optical ray (GO), the surface diffracted ray (SD) converted from the creeping wave (CW), the edge diffracted ray (ED), the edge diffracted and reflected ray (EDR), and the whispering-gallery mode (WG) radiation field. We have confirmed the validity of the TD asymptotic solution by comparing with the reference solution.

In the present study, we consider the arrival times of a transient CW and a transient WG when a pulse wave is incident on one of the edges of a cylindrically curved conducting open sheet. We derive the TD asymptotic solution for the nth-order CW incidence and edge diffracted ray (CWn−ED) and the one for the mth-order WG incidence and edge diffracted ray (WGm−ED) by applying only the Fourier transform method. From the propagation path difference between the CWn−ED and the WGm−ED and from the group velocity difference between the CWn and the WGm, we show the interesting phenomenon that the CW propagates along the curved sheet later than the WG. The time factor )exp( tiω− is adopted and suppressed in this paper.

II. FORMULATION AND ARRIVAL TIMES OF TRANSIENT CWn AND TRANSIENT WGm

Fig. 1 shows a cylindrically curved conducting open sheet with edges A and B defined by ,0, ABφφρ ≤≤= a the coordi-nate systems ),,( φρ ),,,( zyx and ).,( ψr We assume that the

radius of curvature a is sufficiently large compared with a wavelength λ (i.e., :,1 ωω >>ca angular frequency, :c speed of light) and that the modulated plane pulse wave which has a magnetic field component in the z − direction propagates x − positive direction. Fig. 1 shows the propagation paths of the transient CWn−ED ( ) and the transient WGm−ED ( ) excited by the equivalent source at the edge A after the magnetic type (H-type) pulse wave is incident on the edge A.

We assume the following truncated pulse source )(ts defined by the product of the modulated wave )(0 ts and the carrier wave )](exp[ 00 tti −− ω whose central angular frequency is 0ω :

,2,0for0

20for)](exp[)()(

0

0000

⎩⎨⎧

><≤≤−−

=ttt

ttttitsts

ω (1)

where 0t denotes the constant parameter.

A. Arrival Time of Transient CWn− ED The TD asymptotic solution for the transient CWn−ED with

the path PBA0 →→Q (see in Fig. 1) may be given by [5]:

.])(

(exp[

)()(~)(

BP0CW,CW,

A00

EDCW,00EDCWEDCW

AB

0

cLL

cLtti

ttsAty

n

nnn

pn

p

−−

−−−

−•

−−−

ωνω

ω

Q (2)

Fig. 1. Propagtion phenomena of transient CWn−ED ( ) and transient WGm−ED ( ) excited by the edge A of a cylindrical-ly curved conducting open sheet ).,( ABφa

edge B

edge A

Q2

y

O′

P

Q0

),( ψr

Q1

modal caustic )( 0ωρm

Oz

x

plane (pulse) waveH-type incident

ABφa

r

ψ•

CWn WGm

CWn-ED

WGm-ED1

2

1

2

1

1 2

1 2

2214978-1-4799-3540-6/14/$31.00 ©2014 IEEE AP-S 2014

.)()( 0CW,CW,BPAEDCW, AB0ων

nn gnp LcLLt ++=− Q (3)

Here )( 0EDCW ω−nA is the amplitude term of the CWn−ED and

notations A0QL , ABCW,nL , and BPL denote respectively the

distances along the paths A,0 →Q B,A and P.B → While, )( 0CW, ων

np in (2) and )( 0CW, ωνng in (3) denote the

phase and the group velocities of the transient CWn defined by

,))(2)(4(1)(,)()( 32000CW, acccc nnnp n

ωσωωων +== (4)

.)()()(1

)()(

0CW,0CW,0

0CW,0CW, c

nn

nn

pp

pg <

′−=

ωνωνωων

ων (5)

Here nσ in (4) denotes the nth zero of the Airy function derivative (i.e., 0)( =′ niA σ ). It is interesting to note that the envelope shape of the )(EDCW ty

n − in (2) coincides with that of the modulated wave )(0 ts in (1) and takes the maximum value at the arrival time EDCW, −npt in (3).

B. Arrival Time of Transient WGm− ED The TD asymptotic solution for the transient WGm−ED

with the path PBA 210 →→→→ QQQ (see in Fig. 1) may be given by [5]:

,])(

(exp[

)()(~)(

BP0WG,

A00

EDWG,00EDWGEDWG

AB

0

cLcL

cLtti

ttsAty

m

p mmm

−−

−−−

−•

−−−

ωω

ω

Q (6)

.)()()( 0WG,0WG,BPAEDWG, AB0ωνω

mm gmp LcLLt ++=− Q (7)

Here )( 0EDWG ω−mA is the amplitude term of the WGm−ED and

notations A0QL , )( 0WG, ABωmL , and BPL denote respectively

the distances along the paths A,0 →Q B,A 21 →→ QQ and P.B → While, )( 0WG, ων

mg in (7) denotes the group velocity of the transient WGm defined by

.)()()(1

)(0WG,0WG,0

0WG,ABAB

cLL

c

mmg m

<′+

=ωωω

ων (8)

Please note that the envelope shape of the )(EDWG tym − in

(6) coincides with that of the modulated wave )(0 ts in (1) and takes the maximum value at the arrival time EDWG, −mpt in (7).

C. Arrival Times of Transient CWn and Transient WGm The arrival times of the transient CWn propagating along

the path BA in and the transient WGm propagating along the path BA 21 →→ QQ in are given by the second term of the right-hand side in (3) and (7), respectively. While, between the distances of the CWn and the WGm , and the group velocities of the CWn and the WGm , the following inequalities are realized, respectively.

.)()(),( 0WG,0CW,0WG,CW, ABABcLL

mn ggmn <<> ωνωνω (9)

By applying inequalities in (9) to the second terms of the right-hand side in (3) and (7), one may confirm that the transient CWn propagates along the curved conducting sheet from the edge A to the edge B later than the transient WGm.

III. NUMERICAL RESULTS AND DISCUSSIONS Figs. 2(a) and 2(b) show the waveform and the envelope

shape of the lowest-order )1( =n transient CW1−ED and those

Fig. 2. Envelope shape and response waveform of transient CW1−ED (Fig. 2(a)) and transient WG1−ED (Fig. 2(b)) excit-ed by the edge A of a curved conducting open sheet (see Fig. 1). Numerical parameters: ),0.120,m0.1(),( AB =φa ),(P ψr

).8.30,m0.100(= Pulse source )(ts defined by (1) with ])4()(exp[)( 22

00 dttts −−= used in the calculation =0(t ).srad100.3,s101.4,s103.2 9

0109 ×=×=× −− ωd

of the lowest-order )1( =m transient WG1−ED, respectively. It is clarified that, each pulse element ( ) calculated from the TD asymptotic solutions in (2) and (6) agrees excellently with the reference solution ( ) by using the FD asymptotic solution [4] and the fast Fourier transform (FFT) numerical code. It is observed that the transient CW1−ED arrives at the observation point later than the transient WG1−ED. From the path difference between the CW1−ED and the WG1−ED and from the relations in (9), it is very interesting to confirm that the transient CW1 propagates along the convex sheet from the edge A to the other edge B later than the transient WG1.

IV. CONCLUSION We have studied the arrival times of the transient CWn−ED

and the transient WGm−ED excited by one of the edges of a cylindrically curved conducting open sheet. We have con-firmed the interesting phenomenon that the transient CW propagates along the curved sheet later than the transient WG.

ACKNOWLEDGEMENT The work was supported in part by the Grant-in-Aid for

Scientific Research (C) (24560492) from Japan Society for the Promotion of Science (JSPS).

REFERENCES

[1] P. R. Rousseau and P. H. Pathak, “Time-domain uniform geometrical

theory of diffraction for a curved edge,” IEEE Trans. Antennas Propagat., vol. 43, no. 12, pp. 1375---1382, Dec. 1995.

[2] K. Goto, T. Ishihara, and L. B. Felsen, “High-frequency (whispering-gallery mode)-to-beam conversion on a perfectly conducting concave-convex boundary,” IEEE Trans. Antennas Propagat. Mag., vol. 50, no. 8, pp. 1109---1119, Aug. 2002.

[3] F. A. Molinet, “Edge-excited rays on convex and concave structures: A review,” IEEE Trans. Antennas Propagat. Mag., vol. 47, no. 5, pp. 34---46, Oct. 2005.

[4] K. Goto, T. Kawano, and T. Ishihara, “High-frequency analyses for scattered fields by a cylindrically curved conducting surface,” IEICE Trans. Electron., vol. E92-C, no. 1, pp. 25---32, Jan. 2009.

[5] K. Goto, T. Kawano, and T. Ishihara, “Time-domain asymptotic solution for transient scattered field by a cylindrically curved conducting surface,” IEICE Electron. Express, vol. 6, no. 6, pp. 354---360, March 2009.

1

2

2

-6

0

6[×10-7]

-0.0003

0

0.0003

(a)

6

-6

3

-3resp

onse

wav

efor

m

WG1-ED

0

0

335 340 345

(b)time [ns]

time difference

X 1

0-4

CW1-ED

X 10

-7 1

2

2215