The study of arterial wave propagation is essential to understand the physiopathology of the cardiovascular system, as waves carry clinically relevant information. Impedance analysis was used for such type of studies, where results were presented in the frequency domain, but it was difficult to relate specific events to time points within the cardiac cycle. Therefore, a mathematical tool called wave intensity analysis was developed, initially using measurements of pressure and velocity (PU approach). However, the need to acquire such measurements in a non-invasive, direct and simultaneous fashion led to the development of the DU approach, a type of wave intensity analysis carried out using vessel diameter and flow velocity waveforms, thus giving up the pressure measurement. It is the only available technique, at present, able to extract wave intensity information without relying on distally recorded pressure measurements and on non-simultaneous recordings. Due to its non-invasive nature for collecting the required measurements, this technique has a potential use in clinical and research settings to investigate physiological changes under rapid perturbations, such as the ones introduced by exercise. In this thesis, the DU approach is performed by only using an ultrasound device and to extract information about cardio-arterial interaction in the human common carotid artery. In the first experimental chapter of this thesis, a reproducibility study of common carotid DU-derived wave intensity parameters was conducted on a healthy young cohort, both at rest and during exercise (semi-recumbent cycling). Carotid diameter and blood flow velocity features, as well as wave intensity parameters such as forward compression, backward compression and forward expansion wave peaks and energies, were overall fairly reproducible. In particular, diameter variables exhibited higher reproducibility and lower dispersion than corresponding velocity variables, whereas wave intensity energy variables exhibited higher reproducibility and lower dispersion than corresponding peaks. Local wave speed, calculated via lnDU-loop, a technique based only on local measurements of diameter and velocity and often associated with the DU approach, was also reproducible. It is possible to conclude that the DU-derived wave intensity analysis is reliable both at rest and during exercise. In a subsequent study, DU-derived wave intensity analysis was performed on a young trained cohort to investigate the contribution of cardiac and peripheral vascular alterations to common carotid wave intensity parameters, under rapid physiological perturbations, such as semi-recumbent cycling at incremental workrates, and subsequent recovery. Judging by the increase in local wave speed, the common carotid artery stiffened substantially as workrate increased whilst peak and energy of the forward and backward compression waves also increased, due to enhanced ventricular contractility, which was associated with larger reflections from the cerebral microcirculation and other vascular beds in the head. However, the reflection indices remained unchanged during exercise, highlighting that the increased magnitude of reflections is mainly due to the enhanced contractility, rather than changes in vascular resistance, at least at the carotid artery in young healthy individuals. The forward expansion wave increased during exercise, as the left ventricle actively decelerated blood flow in late systole, potentially improving filling time during diastole. In the early recovery, the magnitude of all waves returned to baseline value. Finally, the X wave, attributed to the reflection of the backward compression wave, had a tendency to increase during exercise and to return to baseline value in early recovery. A further development of wave intensity analysis came with the reservoir-wave approach, able to separate, from the pressure and velocity waveforms, the component solely due to the reservoir volume, for the correct evaluation of backward- and forward-travelling waves. A number of issues, however, still remains, involving specifically the lack of consensus over the fitting technique and over the value of the asymptotic pressure value (P ∞),used for the determination of the reservoir waveform. Therefore, to give a contribution to the debate involving the more correct model for the pressure and velocity reservoir-wave approach, a study aimed to investigate various common carotid hemodynamic and wave intensity parameters, using different fitting techniques and values of P ∞ currently available in literature, was performed and described in the last chapter of this thesis. The study demonstrated that different fitting method and values of P ∞ could bring significant variations in values and trends of hemodynamic and wave intensity parameters. However, despite the changes in the shape of the reservoir pressure waveform, its peak and integral with respect to time tended to remain constant. This is an important feature, because both reservoir peak pressure and its integral have been used in clinical settings for the calculation of diagnostic indicators. The reservoir and excess velocity peaks, instead, changed more significantly. This outcome, together with the concomitant substantial change in excess pressure peak and integral, may greatly affect wave intensity parameters. Wave intensity parameters were, in fact, significantly more sensitive to fitting techniques and values of P ∞ than pressure parameters. Finally, the wave speed did not substantially change, leading to the conclusion that the calculation of local vessel distensibility and/or compliance, when calculated from the excess components of the waveforms, seemed insensitive to fitting techniques and values of P ∞.

To understand the function of the cardiovascular system, the propagation of waves in arteries has to be investigated, since they carry information which can be used for the prevention and diagnosis of cardiovascular diseases. The main goal of this thesis is to improve the understanding of wave propagation in central and peripheral arteries studying the local hemodynamics of the ascending aorta, the carotid artery and the femoral artery by analysing human, animal and in vitro data. Also, another aim is to introduce a technique for non-invasive determination of the local arterial distensibility, the wave speed, and wave intensities. Arterial hemodynamics is here studied using wave intensity analysis, a time domain technique based on pressure and velocity measurements that is derived from the 1D theory of wave propagation in elastic tubes. Also, variations of this technique were used, such as (i) the non-invasive wave intensity analysis that relies on diameter and velocity measurements and (ii) the reservoir-wave approach in which pressure is considered the sum of a pressure due to the elastic properties of the arteries and a pressure due to the travelling wave. To identify the correct analysis to describe the wave propagation in the ascending aorta using pressure and velocity measurements, the hemodynamics of the canine ascending aorta was studied invasively using the traditional wave intensity (or waveonly) analysis and the reservoir-wave approach in both control condition and during total aorta occlusions in order to provide clear reflection sites. The models produced a remarkably similar wave intensity curves, although the intensity magnitudes were different. The reservoir-wave model always yielded lower values for all hemodynamic parameters studied. Both models led to the conclusion that distal occlusions have little or no effect on hemodynamics in the ascending aorta. Since the ascending aorta is not an accessible vessel its examination in clinical routine is challenging. More superficial arteries, such as carotid, radial, brachial and femoral arteries, might be easier to examine, in particular using ultrasound equipment that is normally available in the clinic. These considerations led to the second study of this thesis that is the introduction of a new technique for the non-invasive determination of arterial distensibility, local wave speed and wave intensities to study arterial hemodynamics in humans. The technique relies only on diameter and velocity measurements that can be obtained using ultrasound. In particular, the technique was used for the first time to study the hemodynamic of the carotid and femoral arteries in a large population of healthy humans to investigate the changes with age and gender. The carotid artery was more affected by the aging process than the femoral artery, even in healthy subjects. Local wave speed, distensibility and hemodynamic wave intensity parameters (except the reflection index) had strong correlations with age at the carotid artery. The mechanical properties and hemodynamic parameters of the femoral artery were not significantly age-dependent, but local wave speed, distensibility and forward wave intensity were significantly gender-dependent. The findings of the first and second studies contributed to the design of the third study. The carotid artery is an elastic artery relatively close to the heart and thus the hemodynamics of this vessel is related to left ventricular function. For this reason, the carotid hemodynamics of the same healthy population was investigated for the first time using the reservoir-wave approach. Pressure and velocity measurements were separated into their reservoir and excess components and the effects of age and gender on these parameters were studied. It was found that in the carotid artery reservoir and excess components are strongly affected by the ageing process. From the above studies some questions about the hemodynamics of central arteries remained unsolved. For this reason it was decided to carry out in vitro experiments in a mock circulatory system to investigate the effects of variation of compliance and stroke volume on the reservoir and excess pressure components of the ascending aorta. This allows for the study of different physiological and pathological conditions, such as age, hypertension, atherosclerosis and ventricular dysfunction in relation to vascular compliance and stroke volume. The reservoir and excess components of the measured pressure wave were both significantly related to aortic compliance and stroke volume, but the reservoir pressure had a stronger relationship with aortic compliance compared with the excess pressure and its magnitude increased more significantly when the aorta became stiffer. Wave speeds, calculated using measured and excess pressures, followed the same pattern, but the one calculated using excess pressure was smaller than the other. Wave speed was strongly related to aortic compliance, but not to the change of stroke volume. In conclusion, the use of the wave-only and the reservoir-wave models led to different values of wave speed and intensities that can be explained considering the anatomy of the arterial system. Notably, elastic and muscular arteries are differently affected by age and gender. The hemodynamics of the carotid artery are strongly related to age also in healthy subjects. Pressure and flow velocity in the carotid artery can be separated into their reservoir and excess components. The new non-invasive technique based on diameter and velocity measurements could be relevant in clinical practice as a screening tool.

The scientific literature on airflow in the respiratory system is usually associated with rigid ducts. Many studies have been conducted in the frequency domain to assess respiratory system mechanics. Time-domain analyses appear more independent from the hypotheses of periodicity, required by frequency analysis, providing data that are simpler to interpret since features can be easily associated to time. However, the complexity of the bronchial tree makes 3-D simulations too expensive computationally, limiting the analysis to few generations. 1-D modelling in space-time variables has been extensively applied to simulate blood pressure and flow waveforms in arteries, providing a good compromise between accuracy and computational cost. This work represents the first attempt to apply this formulation to study pulse waveforms in the human bronchial tree. Experiments have been carried out, in this work, to validate the model capabilities in modelling pressure and velocity waveforms when air pulses propagate in flexible tubes with different mechanical and geometrical properties. The experiments have shown that the arrival of reflected air waves occurs in correspondence of the theoretical timing once the wave speed is known. Reflected backward compression waves have generated an increase of pressure (P) and decrease of velocity (U) while expansion backward waves have produced a decrease of P and increase of U according to the linear analysis of wave reflections. The experiments have demonstrated also the capabilities of Wave intensity analysis (WIA), an analytical technique used to study wave propagation in cardiovascular system, in separating forward and backward components of pressure and velocity also for the air case. After validating the 1-D modelling in space and time variables, several models for human airways have been considered starting from simplified versions (bifurcation trachea- main bronchi, series of tubes) to more complex systems up to seven generations of bifurcations according to both symmetrical and asymmetrical models. Calculated pressures waveforms in trachea are shown to change accordingly to both peripheral resistance and compliance variations, suggesting a possible non-invasive assessment of peripheral conditions. A favourable comparison with typical pressure and flow waveforms from impulse oscillometry system, which has recently been introduced as a clinical diagnostic technique, is also shown. The results suggested that a deeper investigation of the mechanisms underlying air wave propagation in lungs could be a useful tool to better understand the differences between normal and pathologic conditions and how pathologies may affect the pattern of pressure and velocity waveforms.

The wall properties of the arteries play an important role in cardiovascular function. Stiffness of large artery is predictive of cardiovascular events. To understand the function of the cardiovascular system, special attention should be paid to the understanding of pulse wave propagation, because pulse waves carry information of the cardiovascular function, and provide information which can be useful for the prevention and diagnosis of diseases. This thesis presents a series of in vitro experimental studies of wave propagation, wave reflection and determination of mechanical properties of flexible vessels. In this thesis, several studies have been included: 1) applied and compared foot-to-foot, PU-loop and lnDU-loop methods for determination of wave speed in flexible tubes and calf aortas; 2) investigated the variation of local wave speed determined by PU-loop with proximity to the reflection site; 3) investigated using wave intensity analysis (WIA) as the analytical technique to determine the reflection coefficient; 4) developed a new technique which based on one-point simultaneous measurements of diameter and velocity to determine the mechanical properties of flexible tubes and calf aortas. In the first study, it is found wave speeds determined by PU-loop and lnDU-loop methods are very similar, and smaller than those determined by foot-to-foot method. The timing of arrival time of reflected wave based on diameter and velocity technique highly agreed with the corresponding timing based on pressure and velocity technique. The shapes of forward and backward non-invasive wave intensities based on diameter and velocity are very similar with the corresponding shapes based on pressure and velocity. Although the density term is not part of the equation, the lnDU-loop method for determining local wave speed is sensitive to the fluid density. In the second study, it is found wave speed measured by PU-loop is varied with proximity to the reflection site. The closer the measurement site to the reflection site, the greater the effect upon measured wave speed; a positive reflection caused an increase in measured wave speed; a negative reflection caused a decrease in measured wave speed. Correction iteration process was also considered to correct the affected measured wave speed. In the third study, it is found, reflection coefficient determined by pressure, square roots of wave intensity and wave energy are very close, but they are different from reflection coefficient determined by wave intensity and wave energy. Due to wave dissipation, the closer the measurement site to the reflection site, the greater is the value of the local reflection coefficient. The local reflection coefficient near the reflection site determined by wave intensity and wave energy are very close to the theoretical value of reflection coefficient. In the last study I found that distensibility determined by the new technique which utilising lnDU-loop is in agreement with that determined from the pressure and area which obtained from tensile test in flexible tubes; distensibility determined by the new technique is similar to those determined in the static and dynamic distensibility tests in calf aortas; Young’s modulus determined by the new technique are in agreement with that those determined by tensile tests in both flexible tubes and calf aortas. In conclusion, wave speed determined by PU-loop and lnDU-loop methods are very similar, the new technique lnDU-loop provides an integrated noninvasive system for studying wave propagation; wave speed determined by PU-loop is affected by the reflection, the closer the measurement site to the reflection site, the greater the change in measured wave speed; WIA could be used to determine local reflection coefficient when the measurement site is close to the reflection site; the new technique using measurements of diameter and velocity at one point for determination of mechanical properties of arterial wall could potentially be non-invasive and hence may have advantage in the clinical setting.

Os projetos de redes hidráulicas dimensionam a capacidade de vazão e pressão para atender a uma demanda e prever a resistência necessária para garantir a estabilidade da instalação. Os esforços solicitantes não deverão ultrapassar as resistências estruturais, mantendo-se um coeficiente de segurança que garanta a estabilidade, mesmo nos casos de falhas operacionais. Deve-se proceder à simulação hidráulica tanto para o regime permanente quanto para o regime transitório. Na fase operacional haverão situações transitórias ocasionadas por manobras intencionais ou falhas não previstas que provocarão mudanças do regime permanente para o regime transitório, cuja variação de carga é diretamente proporcional à velocidade de propagação da onda ou celeridade e, dependendo da intensidade do transitório hidráulico, haverá a necessidade de instalar dispositivos de proteção de redes. A concepção dos dispositivos de proteção baseia-se na dissipação do fenômeno transitório por meios de descargas, amortecimento em câmaras de ar comprimido e por reservatórios intermediários que absorvem e suprem os picos de pressão e vazão no início da perturbação, de tal forma que a onda de pressão seja controlada. A celeridade, por sua vez, é função das características do meio fluido, do material da tubulação e da geometria. Ao provocar uma mudança na compressibilidade do meio fluido, tem-se redução no coeficiente de compressibilidade volumétrico da mistura fluida que vai ocasionar uma redução na velocidade de propagação da onda ou celeridade, amortecendo o impacto da variação de carga para níveis que possam ser absorvidos pela instalação. Esta dissertação aborda um dispositivo de redução da velocidade de propagação ou celeridade que pode ser usado como proteção de redes hidráulicas ou atenuador de celeridade, que junto com outros dispositivos e associados em série, distribuídos ao longo da tubulação, podem absorver os excessos de pressão e vazão gerados pelos transitórios e manter os esforços solicitantes inferiores à resistência oferecida pela instalação. / The projects of hydraulic networks define the capacity of outflow and pressure to meet a demand and to achieve the necessary resistance to guarantee the installation stability. The applied loading must not exceed structural resistance, there remaining a safety coefficient which guarantees the stability even in case of operational failure. The hydraulic simulation must be performed both for permanent and transitory regime. In the operational phase, there will be transitory situations caused by intentional maneuvers or unpredicted failures which will cause changes from permanent to transitory regime, with changes in load directly proportional to the propagation speed of the wave or celerity and, depending on the intensity of the hydraulic transient, it will be necessary to provide devices for the network protection. The conception of protection devices is based on the transient phenomena dissipation by means of discharges, shock absorption in air compressed chambers and though intermediate reservoirs, which absorb and supply the outflow and pressure peaks in the beginning of the disturbance. This dissertation approaches a device so that the pressure wave is controlled. The celerity, in its turn, is a function of the fluid characteristics, the network material and geometry. Provoking a change in the mean fluid compressibility there is a reduction in the volumetric compressibility coefficient and also in the specific mass of the mixture, which will cause a reduction in the speed of propagation of the wave or celerity, cushioning the impact of the pressure head variation to a level that can be absorbed by the installation. Therefore, the reduction in the speed of propagation or celerity canbe used is as a protection device for hydraulic nets, or celerity attenuation, which together with other devices, associated in series and distributed along the duct may absorb the excess of pressure and outflow generated by the transitory and keep the applied loading inferior to the resistance supplied by the installation.

Submitted by ANA KARLA PEREIRA RODRIGUES (anakarla_@hotmail.com) on 2017-08-29T15:12:58Z No. of bitstreams: 1 arquivototal.pdf: 1939955 bytes, checksum: f6ebf1c4db19321eabce5c175a32c2aa (MD5) / Approved for entry into archive by Viviane Lima da Cunha (viviane@biblioteca.ufpb.br) on 2017-08-29T15:43:41Z (GMT) No. of bitstreams: 1 arquivototal.pdf: 1939955 bytes, checksum: f6ebf1c4db19321eabce5c175a32c2aa (MD5) / Made available in DSpace on 2017-08-29T15:43:41Z (GMT). No. of bitstreams: 1 arquivototal.pdf: 1939955 bytes, checksum: f6ebf1c4db19321eabce5c175a32c2aa (MD5) Previous issue date: 2015-07-30 / Coordenação de Aperfeiçoamento de Pessoal de Nível Superior - CAPES / This work intends to show the hyperbolic functions, analyzing their similarities and contrasts with the circular trigonometric functions. To this, we start showing a short review about the circular trigonometry and hyperbole, describing their main elements and properties. Then, we made a study about hyperbolic functions, with the de nitions of sine, cosine and the other hyperbolic functions and their main properties. We nished this work with some applications of these functions on everyday. / Este trabalho tem como objetivo apresentar as funções hiperbólicas, analisando suas semelhanças e diferenças com as funções trigonométricas circulares. Para tanto, iniciamos apresentando uma breve revisão sobre a trigonometria circular e a hipérbole, descrevendo seus principais elementos e propriedades. Posteriormente, realizamos um estudo sobre as funções hiperbólicas, apresentando as de nições do seno, cosseno e das demais funções hiperbólicas e suas principais propriedades. Concluí- mos com algumas aplicações destas funções no cotidiano.

Os projetos de redes hidráulicas dimensionam a capacidade de vazão e pressão para atender a uma demanda e prever a resistência necessária para garantir a estabilidade da instalação. Os esforços solicitantes não deverão ultrapassar as resistências estruturais, mantendo-se um coeficiente de segurança que garanta a estabilidade, mesmo nos casos de falhas operacionais. Deve-se proceder à simulação hidráulica tanto para o regime permanente quanto para o regime transitório. Na fase operacional haverão situações transitórias ocasionadas por manobras intencionais ou falhas não previstas que provocarão mudanças do regime permanente para o regime transitório, cuja variação de carga é diretamente proporcional à velocidade de propagação da onda ou celeridade e, dependendo da intensidade do transitório hidráulico, haverá a necessidade de instalar dispositivos de proteção de redes. A concepção dos dispositivos de proteção baseia-se na dissipação do fenômeno transitório por meios de descargas, amortecimento em câmaras de ar comprimido e por reservatórios intermediários que absorvem e suprem os picos de pressão e vazão no início da perturbação, de tal forma que a onda de pressão seja controlada. A celeridade, por sua vez, é função das características do meio fluido, do material da tubulação e da geometria. Ao provocar uma mudança na compressibilidade do meio fluido, tem-se redução no coeficiente de compressibilidade volumétrico da mistura fluida que vai ocasionar uma redução na velocidade de propagação da onda ou celeridade, amortecendo o impacto da variação de carga para níveis que possam ser absorvidos pela instalação. Esta dissertação aborda um dispositivo de redução da velocidade de propagação ou celeridade que pode ser usado como proteção de redes hidráulicas ou atenuador de celeridade, que junto com outros dispositivos e associados em série, distribuídos ao longo da tubulação, podem absorver os excessos de pressão e vazão gerados pelos transitórios e manter os esforços solicitantes inferiores à resistência oferecida pela instalação. / The projects of hydraulic networks define the capacity of outflow and pressure to meet a demand and to achieve the necessary resistance to guarantee the installation stability. The applied loading must not exceed structural resistance, there remaining a safety coefficient which guarantees the stability even in case of operational failure. The hydraulic simulation must be performed both for permanent and transitory regime. In the operational phase, there will be transitory situations caused by intentional maneuvers or unpredicted failures which will cause changes from permanent to transitory regime, with changes in load directly proportional to the propagation speed of the wave or celerity and, depending on the intensity of the hydraulic transient, it will be necessary to provide devices for the network protection. The conception of protection devices is based on the transient phenomena dissipation by means of discharges, shock absorption in air compressed chambers and though intermediate reservoirs, which absorb and supply the outflow and pressure peaks in the beginning of the disturbance. This dissertation approaches a device so that the pressure wave is controlled. The celerity, in its turn, is a function of the fluid characteristics, the network material and geometry. Provoking a change in the mean fluid compressibility there is a reduction in the volumetric compressibility coefficient and also in the specific mass of the mixture, which will cause a reduction in the speed of propagation of the wave or celerity, cushioning the impact of the pressure head variation to a level that can be absorbed by the installation. Therefore, the reduction in the speed of propagation or celerity canbe used is as a protection device for hydraulic nets, or celerity attenuation, which together with other devices, associated in series and distributed along the duct may absorb the excess of pressure and outflow generated by the transitory and keep the applied loading inferior to the resistance supplied by the installation.

Le dépistage et le diagnostic de l’insuffisance placentaire (IP), qu’il s’agisse du retard de croissance intra-utérin (RCIU) ou de la prééclampsie (PE), sont des enjeux majeurs de santé publique. En pratique clinique, les propriétés mécaniques du placenta ne sont pas explorées, pourtant des modifications de son architecture tissulaire pourraient engendrer des variations d’élasticité. Parmi les méthodes d’élastographie ultrasonore (US), l’élastographie transitoire paraît adaptée pour une telle application. Cette technique consiste à calculer la vitesse de l’onde de cisaillement (Cs) générée par une vibration externe se propageant dans le milieu considéré. Les valeurs d’élasticité obtenues à partir des méthodes US actuelles ne sont calculées qu’à une fréquence unique. Une modification structurelle du tissu pouvant correspondre à une loi de puissance particulière de la dispersion fréquentielle de Cs, nous avons évalué l’intérêt d’une approche multifréquentielle pour distinguer l’élasticité des placentas normaux et celle de placentas présentant des signes d’IP au troisième trimestre de la grossesse. Nous avons développé un dispositif préliminaire en onde plane (pour l’exploration ex vivo) permettant de valider le principe de la méthode proposée, puis un dispositif d’élastographie transitoire 2D (exploration ex vivo et in vivo). Les données sont ajustées au moyen d’un modèle rhéologique fractionnaire dans lequel le comportement en fréquence est modélisé par une loi de puissance (exposant n du modèle). Nous avons montré que les placentas RCIU présentent des valeurs de Cs et n inférieures à celles des placentas normaux ou des PE. Cette diminution de n pourrait s’expliquer par les lésions anatomopathologiques du RCIU et la diminution de Cs est cohérente avec l’étude d’un modèle murin de RCIU par ligature utérine. Enfin, l’analyse de la dispersion fréquentielle est faisable chez la femme enceinte. La valeur ajoutée de la méthode développée devrait désormais être testée lors d’une large étude clinique. / Screening and diagnosis of placental insufficiency (PI), whether intrauterine growth restriction (IUGR) or preeclampsia (PE) are major public health issues. In clinical practice, the mechanical properties of the placenta are not explored; however changes in its tissue architecture could cause variations in elasticity. Among the ultrasound (US) elastography methods, transient elastography seemed suitable for such an application. This technique consists in calculating the shear wave speed (Cs) generated by an external vibration propagating in the medium under consideration. Elasticity values obtained from current US methods are calculated at a single frequency. As a structural modification of the tissue may correspond to a particular power law of frequency dispersion of Cs, we evaluated the interest of a multifrequency approach to distinguish the elasticity of healthy placentas and that of placentas with PI signs in the third trimester of pregnancy. We have developed a preliminary plane wave device (for ex vivo exploration) to validate the principle of the proposed method, and then a 2D transient elastography device (ex vivo and in vivo exploration). The data is adjusted using a fractional rheological model where the frequency behavior is modeled by a power law (exponent n of the model). We have shown that IUGR placentas have Cs and n values lower than those of healthy placentas or PE. This decrease in the n value could be explained by histopathological lesions of IUGR. As for the decrease of Cs in cases of IUGR, this result is consistent with the study of an IUGR rat model by uterine ligation. Finally, the frequency dispersion analysis is feasible in pregnant women. The added value of this method should now be tested in a large clinical study.

This thesis discusses properties arising when finite differences are implemented forsolving the two dimensional wave equation on media with various properties. Both homogeneous and heterogeneous surfaces are considered. The time derivative of the wave equation is discretised using a weighted central difference scheme, dependenton a variable parameter gamma. Stability and convergence properties are studied forsome different values of gamma. The report furthermore features an introduction to solving large sparse linear systems of equations, using so-called multigrid methods.The linear systems emerge from the finite difference discretisation scheme. Aconclusion is drawn stating that values of gamma in the unconditionally stable region provides the best computational efficiency. This holds true as the multigrid based numerical solver exhibits optimal or near optimal scaling properties.

No description available.