Forward Wave Amplitude Research Articles

Abstract 062: Aortic Stiffness, Blood Pressure Progression and Incident Hypertension: The Framingham Heart Study

Background: Increased vascular stiffness and excessive blood pressure (BP) pulsatility are important risk factors for age-related morbidity. Vascular stiffness and BP pulsatility are related, with a prevailing view that hypertension antedates and contributes to premature vascular aging and a secondary increase in vascular stiffness. However, temporal relations between comprehensive vascular measures and BP elevation have not been fully delineated in a large community-based sample. Methods: We examined longitudinal relations of BP and 3 measures of vascular stiffness and pressure pulsatility derived from arterial tonometry (carotid-femoral pulse wave velocity [CFPWV], forward wave amplitude and augmentation index) over a 7-year period in 1,898 Framingham Offspring participants (mean age 60 yrs, 1,057 women). We also examined relations between measures of microvascular and endothelial function derived from brachial artery Doppler and future progression of BP or vascular stiffness. Results: In multivariable-adjusted regression models, baseline tonometry measures were separately and jointly associated with higher systolic and pulse pressure and incident hypertension ( Table ). Conversely, higher baseline BP was associated with higher forward wave amplitude and augmentation index (all p<0.05) but not CFPWV at follow-up. Higher baseline resting brachial artery flow and lower flow-mediated dilation were associated with incident hypertension in models that included BP and tonometry measures ( Table ). Conclusion: Higher aortic stiffness (CFPWV), pressure pulsatility (forward wave amplitude), and wave reflection (augmentation index) and lower flow-mediated dilation are associated with blood pressure progression and incident hypertension. Our findings support the notion of aortic stiffness as a precursor of hypertension and further suggest a vicious cycle of increasing pressure pulsatility with advancing age. Table. Correlates of incident hypertension. Predictor Variables (baseline) OR 95% CI P Systolic BP 3.24 (2.17; 4.84) <0.0001 Diastolic BP 1.47 (1.13; 1.92) 0.0042 CFPWV 1.30 (1.02; 1.67) 0.037 Forward wave amplitude 1.66 (1.32; 2.09) <0.0001 Augmentation index 1.78 (1.45; 2.17) <0.0001 Brachial artery baseline flow 1.23 (1.05; 1.45) 0.013 Flow-mediated dilation 0.83 (0.70; 0.98) 0.029 Results of a single multivariable model that further adjusted for age,sex, BMI, height and triglycerides in 1,019 participants free of hypertension at baseline who experienced 337 cases of incident hypertension during follow-up. OR expressed per 1 SD of the independent variable.

Hemodynamic Correlates of Blood Pressure Across the Adult Age Spectrum

Background— Systolic blood pressure and pulse pressure are substantially higher in older adults. The relative contributions of increased forward versus reflected pressure wave amplitude or earlier arrival of the reflected wave to elevated pulse pressure remain controversial. Methods and Results— We measured proximal aortic pressure and flow, forward pressure wave amplitude, global wave reflection, reflected wave timing, and pulse wave velocity noninvasively in 6417 (age range, 19 to 90 years; 53 women) Framingham Heart Study Third Generation and Offspring participants. Variation in forward wave amplitude paralleled pulse pressure throughout adulthood. In contrast, wave reflection and pulse pressure were divergent across adulthood: In younger participants, pulse pressure was lower and wave reflection was higher with advancing age, whereas in older participants, pulse pressure was higher and wave reflection was lower with age. Reflected wave timing differed modestly across age groups despite considerable differences in pulse wave velocity. Forward wave amplitude explained 80 (central) and 66 (peripheral) of the variance in pulse pressure in younger participants (<50 years) and 90 and 84 in the older participants (≥50 years; all P <0.0001). In a stepwise model that evaluated age–pulse pressure relations in the full sample, the late accelerated increases in central and peripheral pulse pressure were markedly attenuated when variation in forward wave amplitude was considered. Conclusions— Higher pulse pressure at any age and higher pulse pressure with advancing age is associated predominantly with a larger forward pressure wave. The influence of wave reflection on age-related differences in pulse pressure was minor.

Clinical Summaries

Clinical Summaries

Wave reflection and arterial stiffness in the prediction of 15-year all-cause and cardiovascular mortalities: a community-based study.

The value of increased arterial wave reflection, usually assessed by the transit time-dependent augmentation index and augmented pressure (Pa), in the prediction of cardiovascular events may have been underestimated. We investigated whether the transit time-independent measures of reflected wave magnitude predict cardiovascular outcomes independent of arterial stiffness indexed by carotid-femoral pulse wave velocity. A total of 1272 participants (47% women; mean age: 52+/-13 years; range: 30 to 79 years) from a community-based survey were studied. Carotid pressure waveforms derived by tonometry were decomposed into their forward wave amplitudes, backward wave amplitudes (Pb), and a reflection index (=[Pb/(forward wave amplitude+Pb)]), in addition to augmentation index, Pa, and reflected wave transit time. During a median follow-up of 15 years, 225 deaths occurred (17.6%), including 64 cardiovascular origins (5%). In univariate Cox proportional hazard regression analysis, pulse wave velocity, Pa, and Pb predicted all-cause and cardiovascular mortality in both men and women, whereas augmentation index, reflected wave transit time, and reflection index were predictive only in men. In multivariate analysis accounting for age, height, and heart rate, Pb predicted cardiovascular mortality in both men and women, whereas Pa was predictive only in men. Per 1-SD increment (6 mm Hg), Pb predicted 15-year cardiovascular mortality independent of brachial but not central pressure, pulse wave velocity, augmentation index, Pa, and conventional cardiovascular risk factors with hazard ratios of approximately 1.60 (all P<0.05). In conclusion, Pb, a transit time-independent measure of reflected wave magnitude, predicted long-term cardiovascular mortality in men and women independent of arterial stiffness.

Abstract 5151: Waveform Reflections and Arterial Stiffness in the Prediction of 15.1-Year All-Cause and Cardiovascular Mortalities: A Community-Based Study

Background- The role of arterial wave reflection, usually assessed by the transit-time dependent augmentation index (AI) and augmented pressure (Pa), in the prediction of cardiovascular events may have been underestimated. We investigated whether the transit-time independent measurements of reflected wave magnitude predict cardiovascular outcomes independently of arterial stiffness indexed by carotid-femoral pulse wave velocity (PWV). Method- A total of 1272 participants (47.0% women) from a community-based survey were studied. Carotid pressure waveforms derived by tonometry were decomposed to generate forward wave amplitude (Pf), backward wave amplitude (Pb), and reflection index (RI, = [Pb/(Pf+Pb)]), in addition to AI, Pa, and reflected wave transit time (RWTT). Results During a median follow-up of 15.1 years, 225 deaths occurred, including 64 cardiovascular origins. In univariate Cox proportional hazards regression analysis, PWV, Pa and Pb predicted all-cause and cardiovascular mortalities in both men and women whereas AI, RWTT and RI were predictive only in men. In multi-variate Cox analysis accounting for age, height, and heart rate, Pb predicted all-cause mortality in women and cardiovascular mortality in both men and women, whereas Pa only predicted cardiovascular mortality in men but not in women. When PWV and Pb were jointly entered into the multivariate model accounting for age, sex, height, heart rate, and brachial mean blood pressure, Pb (hazard ratio 1.066, 95% confidence interval 1.023–1.110) but not PWV (1.086, 0.993–1.189), independently predicted cardiovascular mortality. Conclusion- Pb, a transit-time independent measurement of reflected wave magnitude, predicted long term cardiovascular mortality in men and women independently of PWV.

Central Aortic Stiffness is Increased in Patients With Heart Failure and Preserved Ejection Fraction

Background Hypertension is an important risk factor for the development of heart failure with preserved ejection fraction. Although heart failure in hypertensive patients is usually ascribed to intrinsic myocardial abnormalities, noncardiac factors may contribute. Methods and Results Using arterial tonometry and Doppler echocardiography, we assessed arterial stiffness and cardiac diastolic function in 53 individuals with ejection fraction >=0.50, including 23 with hypertension but no heart failure, 16 with hypertension and heart failure, and 14 healthy, normotensive controls. Relative to healthy controls and hypertensives, subjects with heart failure had higher systolic blood pressure, body mass index, creatinine, and left ventricular mass. Diastolic function, as estimated by myocardial relaxation velocity, was not different among the 3 groups. Peripheral arterial stiffness was similar across all groups, but key measures of central aortic stiffness (carotid-femoral pulse wave velocity, characteristic impedance, forward wave amplitude) steadily increased with progression from healthy to hypertensive to heart failure despite adjustment for body mass index, systolic blood pressure, and renal function and were positively correlated with both left ventricular mass and filling pressure. Conclusions We conclude that patients with heart failure and preserved ejection fraction have increased central aortic stiffness relative to age-matched healthy and hypertensive subjects without heart failure. These changes exceed differences in diastolic function and suggest that abnormal ventricular-vascular coupling may contribute to the pathophysiology of heart failure with preserved ejection fraction.

Association between arterial stiffness and variations in oestrogen-related genes

Increased arterial stiffness and wave reflection have been identified as cardiovascular disease risk factors. In light of significant sex differences and the moderate heritability of vascular function measures, we hypothesized that variation in the genes coding for estrogen receptors alpha (ESR1) and beta (ESR2) and aromatase (CYP19A1) is associated with aortic stiffness and pressure wave reflection as measured by noninvasive arterial tonometry. 1261 unrelated Framingham Offspring Study participants who attended the 7th examination cycle (mean age 62±10 years, 52% women) and had arterial tonometry and genotyping data were included in the study. ANCOVA was used to assess the association of polymorphisms with forward wave amplitude, augmented pressure, augmentation index, carotid-femoral pulse wave velocity, and mean arterial pressure with adjustment for potential confounders. In the sex-pooled analysis, those homozygous for the minor allele at any of four ESR1 variants that were in strong linkage disequilibrium ((TA)n, rs2077647, rs2234693 and rs9340799) had on average 18% higher augmented pressure and 16% greater augmentation index compared to carriers of one or two major alleles (p=0.0002–0.01). A similar magnitude of association was detected in those homozygous for the common allele at two ESR2 SNPs (p=0.007–0.02). Our results are consistent with the hypothesis that variation in ESR1 and ESR2, but not CYP19A1, is associated with increased wave reflection, which may contribute to previously demonstrated associations between these variants and adverse clinical events. Our findings will need to be replicated in additional cohorts.

Stability and instability of nonlinear defect states in the coupled mode equations—Analytical and numerical study

Coupled backward and forward wave amplitudes of an electromagnetic field propagating in a periodic and nonlinear medium at Bragg resonance are governed by the nonlinear coupled mode equations (NLCME). This system of PDEs, similar in structure to the Dirac equations, has gap soliton solutions that travel at any speed between 0 and the speed of light. A recently considered strategy for spatial trapping or capture of gap optical soliton light pulses is based on the appropriate design of localized defects in the periodic structure. Localized defects in the periodic structure give rise to defect modes, which persist as nonlinear defect modes as the amplitude is increased. Soliton trapping is the transfer of incoming soliton energy to nonlinear defect modes. To serve as targets for such energy transfer, nonlinear defect modes must be stable. We therefore investigate the stability of nonlinear defect modes. Resonance among discrete localized modes and radiation modes plays a role in the mechanism for stability and instability, in a manner analogous to the nonlinear Schrödinger/Gross–Pitaevskii (NLS/GP) equation. However, the nature of instabilities and how energy is exchanged among modes is considerably more complicated than for NLS/GP due, in part, to a continuous spectrum of radiation modes which is unbounded above and below. In this paper we (a) establish the instability of branches of nonlinear defect states which, for vanishing amplitude, have a linearization with eigenvalues embedded within the continuous spectrum, (b) numerically compute, using Evans function, the linearized spectrum of nonlinear defect states of an interesting multiparameter family of defects, and (c) perform direct time-dependent numerical simulations in which we observe the exchange of energy among discrete and continuum modes.

Vascular Stiffness and Genetic Variation at the Endothelial Nitric Oxide Synthase Locus

Arterial stiffness is a moderately heritable trait that is affected by alterations in the bioavailability of NO. Previous studies have found associations between variants in the gene for endothelial NO synthase (NOS3) and arterial properties. We previously identified a linkage peak for forward pressure wave amplitude in the immediate vicinity of NOS3. Therefore, we evaluated relations between arterial stiffness measures and common genetic variants at this locus. Eighteen single nucleotide polymorphisms capturing approximately 90% of underlying common variation in NOS3 were genotyped in unrelated Framingham Heart Study participants (N=1157; 52.2% women; mean age: 62 years) with routinely ascertained tonometry data that provided 5 arterial phenotypes (forward and reflected pressure wave amplitude, central pulse pressure, carotid-femoral pulse wave velocity, and mean arterial pressure). In women but not men, the genotype for the common NOS3 missense mutation (Glu298Asp, rs1799983) was related to central pulse pressure (women: GG=53+/-0.9, GT=54+/-0.9, and TT=47+/-2.0 mm Hg, P=0.0047; men: GG=50+/-1.0, GT=49+/-0.9, and TT=47+/-1.8 mm Hg, P=0.30) and forward wave amplitude (women: GG=41+/-0.7, GT=42+/-0.7, and TT=38+/-1.6 mm Hg, P=0.029; men: GG=42+/-0.9, GT=41+/-0.8, and TT=39+/-1.5 mm Hg, P=0.47). The only other nominally significant sex-specific association in men but not women was between an intronic polymorphism (rs1800781) and reflected wave amplitude (women: AA=10.4+/-0.4, AG=11.1+/-0.6, and GG=8.9+/-2.2 mm Hg, P=0.50; men: AA=6.1+/-0.3, AG=7.3+/-0.5, and GG=11.3+/-2.3 mm Hg, P=0.014). After adjusting for multiple testing (18 polymorphisms and 5 phenotypes), these nominal associations were no longer significant. The present study was suggestive of modest relations between common genetic variants at the NOS3 locus and arterial stiffness.

Impedance Progress

Changes in global arterial function and blood pressure pulsatility play an important role in the exponential increase in the incidence of various common afflictions of aging. However, our current approach to hemodynamics, evaluating the peak (systolic blood pressure) and trough (diastolic blood pressure) of the blood pressure waveform in the arm, provides a limited view of pulsatile hemodynamics. As a first refinement, blood pressure can be separated into steady-flow and pulsatile components, mean arterial pressure (MAP) and pulse pressure (PP), respectively. Numerous recent studies have shown that many serious clinical events may be more closely linked to PP, which is related to large artery stiffness, rather than MAP, which is determined by small artery function and cardiac output. However, systolic blood pressure or PP in the arm probably does not tell the full story. Conventional brachial systolic blood pressure represents a variable composite of PP (large arteries) and MAP (small arteries), depends on ventricular ejection characteristics and arterial properties, and, because of transmission delays and wave reflection, may not be fully representative of central aortic pressure, which may be more relevant to cardiac and cerebral function. These additional sources of variability may have important implications for the pathogenesis of cardiovascular disease and response to treatment. The study by Segers et al1 in the current issue of Hypertension represents an important step forward in our quest to quantify the effects of regional arterial function on the pathogenesis of cardiovascular disease in the community. The authors are reporting pulsatile hemodynamics from the first round of evaluations in a planned longitudinal, observational study of >2000 initially healthy men and women between 35 and 55 years of age recruited from 2 adjacent communities in the suburbs of Brussels, Belgium. The Asklepios investigators assessed pulsatile hemodynamics using a noninvasive approach first developed by Kelly …

Triangulating the Peaks of Arterial Pressure

Increased arterial stiffness and excessive pressure pulsatility have emerged as important risk factors for a number of common disorders of aging, including cardiovascular disease, stroke, cognitive impairment, and renal disease. The triple-threat combination of robust associations between arterial stiffness and the foregoing disorders, a marked increase in arterial stiffness with advancing age and the graying of our society, has led to intensive efforts to identify mechanisms that contribute to arterial stiffening and widening pulse pressure to define interventions to prevent or reverse stiffness and potentially reduce the substantial burden of related disease. Wave reflections complicate the task of evaluating arterial hemodynamics and play an unclear role in the foregoing diseases. When the heart ejects, ventricular outflow interacts with characteristic impedance of the proximal aorta to produce the forward pressure wave, which travels down the normally compliant aorta at a finite pulse wave velocity (PWV). When the forward wave encounters impedance mismatch, such as a branch point or a change in diameter or wall properties, a partial reflection occurs. Innumerable reflections arising from locations distributed throughout the arterial tree summate into a remarkably discrete reflected pressure wave with amplitude ≈40% of the incident wave. The summated reflected wave returns to the central aorta in midsystole, creating an inflection point and secondary late systolic pressure rise that often augments central aortic systolic and pulse pressure.1 Variable timing of this retrograde-traveling reflected wave creates regional inequalities in systolic and pulse pressure and, therefore, complicates interpretation of single point pressure measurements, such as standard brachial blood pressure, which may differ from central aortic pressure. Augmentation index (AI), which expresses late systolic pressure augmentation as a percentage of pulse pressure, is frequently used to assess wave reflection (Figure). AI, which depends on both timing and amplitude of the reflected wave, has also been widely …

Heritability and a Genome-Wide Linkage Scan for Arterial Stiffness, Wave Reflection, and Mean Arterial Pressure

Arterial stiffness and mean arterial pressure variably contribute to systolic hypertension and increased cardiovascular risk. However, few prior community-based studies have evaluated the genetics of arterial stiffness and separate mean and pulsatile components of blood pressure. Using arterial tonometry, we evaluated heritability and linkage of forward and reflected wave amplitude, mean arterial pressure, and carotid-femoral pulse wave velocity (CFPWV) in 1480 participants representing 817 pedigrees in the Framingham Study offspring cohort. In 204 families with tonometry data, a genome-wide scan was performed with microsatellite markers that covered the genome at 10-cM intervals. Heritability estimates were moderate for reflected wave amplitude (h2=0.48), forward wave amplitude (h2=0.21), CFPWV (h2=0.40), and mean arterial pressure (h2=0.33). Variance components linkage analysis identified 2 regions of linkage for reflected wave amplitude: chromosome 4 at 181 cM (logarithm of odds [LOD]=4.93, permuted P=0.002) and chromosome 8 at 33 cM (LOD=3.27, permuted P=0.058). There was 1 region of linkage for forward wave amplitude on chromosome 7 at 174 cM (LOD=2.88, permuted P=0.017). There were several regions of suggestive linkage for CFPWV: chromosome 2 at 94 cM (LOD=2.46), chromosome 7 at 29 cM (LOD=2.50), chromosome 13 at 108 cm (LOD=2.10), and chromosome 15 at 108 cM (LOD=2.48). There was 1 region of suggestive linkage for mean arterial pressure on chromosome 1 at 192 cM (LOD=2.18). Arterial stiffness measures and mean and pulsatile components of blood pressure are heritable and appear to have genetic determinants that may be linked to separate genetic loci in humans.

Changes in arterial stiffness and wave reflection with advancing age in healthy men and women: the Framingham Heart Study.

With advancing age, arterial stiffness and wave reflections increase and elevate systolic and pulse pressures. An elevated central pulse pressure is generally ascribed to increased wave reflection and portends an unfavorable prognosis. Using arterial tonometry, we evaluated central (carotid-femoral) and peripheral (carotid-brachial) pulse wave velocity, amplitudes of forward and reflected pressure waves, and augmentation index in 188 men and 333 women in the Framingham Heart Study offspring cohort who were free of clinical cardiovascular disease, hypertension, diabetes, smoking within the past 12 months, dyslipidemia, and obesity. In multivariable linear regression models, advancing age was the predominant correlate of higher carotid-femoral pulse wave velocity; other correlates were higher mean arterial pressure, heart rate, and triglycerides and walk test before tonometry (model R2=0.512, P<0.001). A similar model was obtained for carotid-brachial pulse wave velocity (model R2=0.227, P<0.001), although the increase with advancing age was smaller. Owing to different relations of age to central and peripheral stiffness measures, carotid-femoral pulse wave velocity was lower than carotid-brachial pulse wave velocity before age 50 years but exceeded it thereafter, leading to reversal of the normal central-to-peripheral arterial stiffness gradient. In this healthy cohort with a minimal burden of cardiovascular disease risk factors, an age-related increase in aortic stiffness, as compared with peripheral arterial stiffness, was associated with increasing forward wave amplitude and pulse pressure and reversal of the arterial stiffness gradient. This phenomenon may facilitate forward transmission of potentially deleterious pressure pulsations into the periphery.

Extinction theorem for object scattering in a stratified medium

A simple relation for the rate at which energy is extinguished from the incident wave of a far field point source by an obstacle of arbitrary size and shape in a stratified medium is derived from wave theory. This relation generalizes the classical extinction theorem, or optical theorem, that was originally derived for plane wave scattering in free space and greatly facilitates extinction calculations by eliminating the need to integrate energy flux about the obstacle. The total extinction is shown to be a linear sum of the extinction of each wave guide mode. Each modal extinction involves a sum over all incident modes that are scattered into the extinguished mode and is expressed in terms of the object’s plane wave scatter function in the forward azimuth and equivalent plane wave amplitudes of the modes. The only assumptions are that multiple scattering between the object and wave guide boundaries is negligible, and the object lies within a constant sound speed layer. Modal extinction cross sections of an object for the extinction of the individual modes of a wave guide are then defined. Calculations for a shallow water wave guide show that, after correcting for absorption loss in the medium, the modal cross section of an object for mode 1 in a typical ocean wave guide is very nearly equal to its free space cross section. This new extinction theorem may be applied to estimate the cross section of an object submerged in a wave guide from a measurement of its forward scattered field.

Extinction theorem for object scattering in a stratified medium

A simple relation for the rate at which energy is extinguished from the incident wave of a far-field point source by an obstacle of arbitrary size and shape in a stratified medium is derived from wave theory. This relation generalizes the classical extinction theorem, or optical theorem, that was originally derived for plane wave scattering in free space and greatly facilitates extinction calculations by eliminating the need to integrate energy flux about the obstacle. The total extinction is shown to be a linear sum of the extinction of each waveguide mode. Each modal extinction involves a sum over all incident modes that are scattered into the extinguished mode and is expressed in terms of the object’s plane wave scatter function in the forward azimuth and equivalent plane wave amplitudes of the modes. The only assumptions are that multiple scattering between the object and waveguide boundaries is negligible, and the object lies within a constant sound-speed layer. Calculations for a shallow-water waveguide show that the extinction cross section is highly dependent on measurement geometry, and medium stratification, as well as the scattering properties of the object and may be significantly modified by the presence of absorption in the medium.

Separation of arterial pressure waves into their forward and backward running components.

A new separation technique has been developed to determine the forward and backward running arterial pressure wave components. It takes into account friction as well as nonlinear effects due to convective acceleration and to the pressure dependence of the arterial compliance. The new method is a combination of two methods treating friction and nonlinearities separately. The method requires the measurements of pressure and flow at one location as well as the knowledge of the area-pressure relationship. The validity of the method was tested by a simulation experiment in which the forward and backward waves were known a priori. It was shown that the new method is significantly more accurate in the predictions of the forward and backward waves when compared to the classical method assuming linearity and no dissipation. The new wave separation method was also applied to simulated aortic waves for (a) a healthy subject and (b) a subject with decreased compliance. Comparison with the classical linear method showed that neglecting nonlinearities leads to an overestimation of the forward and backward pressure wave amplitudes. The errors, however, were in the order of 5 to 10 percent. We concluded that, for most clinical purposes, the improvement using the nonlinear method is of the same magnitude as experimental errors, and thus the linear method would suffice.

Forward and backward waves in the arterial system: nonlinear separation using Riemann invariants

A nonlinear method has been developed to separate arterial pressure and flow waves into their forward and backward running components. It takes into account nonlinearities arising from the area-pressure relationship as well as convective acceleration. The method is based on the treatment of the Riemann invariants and requires measurements of pressure, flow and diameter at one arterial location. The method has been successfully tested by means of a simulation experiment in which the forward and backward waves were known a priori. It was shown that the new method is significantly more accurate in the predictions of the forward and backward waves when compared to the classical linear method. The new wave separation method was then applied to simulated aortic waves for a) a healthy subject and b) a subject with decreased compliance. Comparison with the classical linear method showed that neglecting nonlinearities leads to an overestimation of the forward and backward pressure wave amplitudes of the order of 5 to 10%.

Numerical solution of coupled mode equations for rough surface scattering

Theories of underwater acoustic signal propagation based on normal modes become quite complicated when the modes are coupled via inhomogeneities in the medium or its boundaries. The usual derivation of coupled mode equations from the Helmholtz equation, e.g., along the lines of Pierce [J. Acoust. Soc. Am. 37, 21 (1965)] leads to a coupled second‐order system of differential equations which is not amenable to numerical solution as a two‐point boundary value problem. However, C. A. Boyles [J. Acoust. Soc. Am. 73, 800 (1983)] has recently shown that, in the case of rough surface scattering, this system can be transformed into a coupled first‐order system for both forward and backward wave amplitudes. In this paper we show that this system can be further transformed into a system which can be efficiently integrated numerically using the state‐of‐the‐art software packages EISPACK and LINPACK for matrix exponentiation and inversion. Thus we are able to obtain the numerical solution of the full elliptic reduced wave equation for any realization of a frozen, one‐dimensionally [z = h(r)] rough surface. Extension to include Doppler effects due to a time‐varying surface is straightforward.

Scattering and depolarization of electromagnetic waves in irregular stratified spheroidal structures of finite conductivity – full wave analysis

Scattering and depolarization of electromagnetic waves in irregular stratified, spheroidal structures (bodies of revolution) are investigated. Using complete expansions for the electromagnetic fields, Maxwell's equations are converted into sets of first order coupled differential equations for the forward and backward wave amplitudes. To obtain these generalized telegraphists' equations, exact boundary conditions are imposed at each of the interfaces between the irregular layers of the structure and Green's theorems are used to avoid term-by-term differentiation of the complete expansions. The electromagnetic parameters of the medium are assumed to vary as functions of the radial distance and the latitude. Excitations by arbitrary distributions of electric and magnetic sources are considered. Thus, this work can be applied to propagation in irregular ionospheric and tropospheric ducts. The solutions are shown to satisfy duality and reciprocity relationships in electromagnetic theory.

Diffraction of Electromagnetic Waves by Cylindrical Structures Characterized by Variable Curvature and Surface Impedance

The electromagnetic fields around a cylindrical boundary of variable curvature and surface impedance are expressed in terms of a complete set of local cylindrical modes, and Maxwell's equations are converted into coupled differential equations for the forward and backward wave amplitudes. The local cylindrical modes are shown to merge with the Fourier type expansion of the fields over plane surfaces. As an illustrative example, the launching of a surface wave over an infinite wedge with a rounded corner is considered in detail.