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### Course Overview

Wave equations describe transient phenomena commonly encountered in all areas of engineering. This course covers: (i) elastic waves, such as response of offshore structures to wind or wave loading, earthquakes; (ii) acoustic waves such as water hammer in pipelines, micro-pressure waves in railway tunnels; (iii) electromagnetic waves, such as signals in transmission lines, transient states in DC cables. These phenomena in real world engineering applications are simulated using several numerical methods. Students develop their own simulation codes using Matlab or any other programming language, and run a series of simulations for the problem of their choice.

### Course Details

Study Type Level Undergraduate 5 First Sub Session 15 credits (7.5 ECTS credits) Old Aberdeen No Professor Dubravka Pokrajac

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### What courses & programmes must have been taken before this course?

• Engineering (EG) (Studied)
• Either Any Undergraduate Programme (Studied) or Master Of Science In Advanced Mechanical Engineering

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### Course Description

Course Aims

To introduce MEng students to a selection of advanced engineering analysis techniques, applicable over a range of engineering disciplines.

Main Learning Outcomes

By the end of the course students should:

A) have knowledge and understanding of:
• simple harmonic oscillations of single mechanical or electrical  systems; coupled oscillations of two systems; coupled oscillations of many systems
• longitudinal and transverse travelling waves
• waves in elastic media and electromagnetic waves
• boundary conditions and reflection of waves
• characteristic form of linear hyperbolic equation and its meaning in context of wave propagation in a physical system
• fundamental principles underlying numerical methods suitable for solving linear hyperbolic equations

B) have gained intellectual skills so that they are able to:
• confidently analyse engineering problems related to waves, based on a sound understanding of fundamental principles
• analyse elastic waves in solids
• analyse transient states in transmission lines
• analyse transient pipe flow
• formulate approximate form of linear hyperbolic equations according to various numerical methods

C) have gained practical skills so that they are able to:
• use MATLAB to solve numerically wave equations describing various difficult engineering problems (water hammer, propagation of acoustic waves, elastic waves, electro-magnetic waves)

D) have gained or improved transferable skills so that they are able to:
• use MATLAB as a general tool to aid analysis and understanding of engineering problems
• carry out critical analysis and assessment of the results of numerical solution of wave equations

### Further Information & Notes

This is a level 5 course available only to candidates following an Engineering degree programme or with the permission of the Head of School of Engineering.

Main Learning Outcomes

By the end of the course students should:

A) have knowledge and understanding of:
• simple harmonic oscillations of single mechanical or electrical  systems; coupled oscillations of two systems; coupled oscillations of many systems
• longitudinal and transverse travelling waves
• waves in elastic media and electromagnetic waves
• boundary conditions and reflection of waves
• characteristic form of linear hyperbolic equation and its meaning in context of wave propagation in a physical system
• fundamental principles underlying numerical methods suitable for solving linear hyperbolic equations

B) have gained intellectual skills so that they are able to:
• confidently analyse engineering problems related to waves, based on a sound understanding of fundamental principles
• analyse elastic waves in solids
• analyse transient states in transmission lines
• analyse transient pipe flow
• formulate approximate form of linear hyperbolic equations according to various numerical methods

C) have gained practical skills so that they are able to:
• use MATLAB to solve numerically wave equations describing various difficult engineering problems (water hammer, propagation of acoustic waves, elastic waves, electro-magnetic waves)

D) have gained or improved transferable skills so that they are able to:
• use MATLAB as a general tool to aid analysis and understanding of engineering problems
• carry out critical analysis and assessment of the results of numerical solution of wave equations

Course Content

The course covers a range of numerical methods suitable for solving wave equations. The theoretical part of the course deals with the derivations of the wave equations using the principles of solid mechanics, fluid mechanics and electromagnetic theory. In order to explain the transition from discrete to continuous systems the derivations start from free oscillations of first single and then coupled simple systems. This is followed by extrapolation to a large number of coupled systems whose interaction results in travelling waves.

The applied part of the course focuses on the following engineering problems: elastic waves in a rigid body, transient pipe flow and transient states in transmission lines.  Fundamental laws studied in the first part are generalised and expressed in conservative form. Characteristic form of hyperbolic equations is developed and related to the propagation of disturbances in physical systems. The final part of the course covers several numerical methods suitable for solving hyperbolic equations. The methods are used to build simulation codes which can be used for solving a broad range of engineering problems.

Students carry out practical exercises using MATLAB for coding numerical solutions of wave equations.

### Contact Teaching Time

Information on contact teaching time is available from the course guide.

### Summative Assessments

1st Attempt

1 three hour written examination (80%); continuous assessment (20%).
The continuous assessment is based on the report on computing exercises.

Resit

1 three hour written examination (80%), and the continuous assessment mark from the 1st attempt (20%)

### Formative Assessment

Based on the work carried out during computing class, the students write a report and submit it after week 19, in order for formative assessment and feedback to be provided.

### Feedback

a) Students can receive feedback on their progress with the Course on request at the weekly computing class or tutorial/feedback sessions.

b) Generic exam feedback will be emailed to the whole class at their University email address.

c) Students requesting individual feedback on their exam performance should make an appointment within 3 weeks of the publication of the exam results.

d) Reports will be returned with marks and comments after week 19.

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