We present a spectral method for solving the two‐dimensional equations of dynamic elasticity, based on a Chebychev expansion in the vertical direction and a Fourier expansion for the horizontal direction. The technique can handle the free‐surface boundary condition more rigorously than the ordinary Fourier method. The algorithm is tested against problems with known analytic solutions, including Lamb’s problem of wave propagation in a uniform elastic half‐space, reflection from a solid‐solid interface, and surface wave propagation in a haft‐space containing a low‐velocity layer. Agreement between the solutions is very good. A fourth example of wave propagation in a laterally heterogeneous structure is also presented. Results indicate that the method is very accurate and only about a factor of two slower than the Fourier method.

BackgroundPaediatric chronic fatigue syndrome or myalgic encephalomyelitis (CFS/ME) is a relatively common and disabling condition. The National Institute for Health and Clinical Excellence (NICE) recommends Cognitive Behavioural Therapy (CBT) as a treatment option for paediatric CFS/ME because there is good evidence that it is effective. Despite this, most young people in the UK are unable to access local specialist CBT for CFS/ME. A randomised controlled trial (RCT) showed FITNET was effective in the Netherlands but we do not know if it is effective in the National Health Service (NHS) or if it is cost-effective. This trial will investigate whether FITNET-NHS is clinically effective and cost-effective in the NHS.MethodsSeven hundred and thirty-four paediatric patients (aged 11–17 years) with CFS/ ME will be randomised (1:1) to receive either FITNET-NHS (online CBT) or Activity Management (delivered via video call). The internal pilot study will use integrated qualitative methods to examine the feasibility of recruitment and the acceptability of treatment. The full trial will assess whether FITNET-NHS is clinically effective and cost-effective. The primary outcome is disability at 6 months, measured using the SF-36-PFS (Physical Function Scale) questionnaire. Cost-effectiveness is measured via cost-utility analysis from an NHS perspective. Secondary subgroup analysis will investigate the effectiveness of FITNET-NHS in those with co-morbid mood disorders.DiscussionIf FITNET-NHS is found to be feasible and acceptable (internal pilot) and effective and cost-effective (full trial), its provision by the NHS has the potential to deliver substantial health gains for the large number of young people suffering from CFS/ME but unable to access treatment because there is no local specialist service. This trial will provide further evidence evaluating the delivery of online CBT to young people with chronic conditions.Trial registrationISRCTN registry, registration number: ISRCTN18020851. Registered on 4 August 2016.Electronic supplementary materialThe online version of this article (10.1186/s13063-018-2500-3) contains supplementary material, which is available to authorized users.

A pseudo-spectral method for a solution of the equations of dynamic elasticity in cylindrical coordinates is based on the Chebychev expansion in the radial direction and the Fourier expansion in the angular direction and is suitable for simulating wave propagation in the vicinity of cylindrical objects. The numerical grid consists of a series of concentric rings, each one with a separate Chebychev-Fourier mesh. One numerical grid is defined for the cylindrical cavity and another grid for the medium around the cavity. Combining these two numerical grids allows reduction of the number of grid points in the angular direction in the interior grid and thus increases the time step. This makes the use of polar coordinates much more economic.The numerical algorithm is applicable to any arbitrary heterogeneous medium.

We present a new method for synthesizing and analyzing laser systems based on the use of the y? diagram. The diagram is commonly used to represent two rays, the marginal and the chief rays, as they propagate through an optical system. Since a Gaussian beam can be represented by two rays, it is possible to use the y? diagram to represent these rays. This results in a representation of the beam as a single ray line on the y? diagram with simple graphic interpretations for the beam parameters. An equivalent representation of the Gaussian beam on the uū diagram is also presented and discussed. Complex design problems may be reduced to simple graphic problems, which often lead directly to algebraic solutions. Examples of y? diagrams are given for beam transfer through simple optical systems, including gradient optics. Diagram transformations are discussed and design examples are given of a three-element afocal system and a three-element collimator.