Mathematical Modelling of a flute

Abstract

The transverse flute can be studied on many aspects. In this thesis the sound waves that can be generated inside the flute are studied on their shape and frequencies. To achieve this, two mathematical models of the sound waves inside a flute are developed. To develop these models, solutions to the second dimensional wave equation are calculated analytically and numerically. The exact model is found within a rectangular parallelepipid region with homogeneous boundary conditions. By the method of separation of variables the second-dimensional wave equation is solved and a function of the air pressure inside the flute is calculated. This model can be used to visualise the waves inside the flute. It allows approximate true frequencies of measured musical notes in the flute. The second model, the numerical model is found in the same domain as the exact model, but with non-homogeneous boundary conditions.These are more realistic boundary conditions and therefore this model might be more accurate than the exact model. By the method of finite difference the numerical second-dimensional wave equation is solved. A system of two numerical equations is found. One of the equations calculates the pressure of the air inside the flute by inserting the previous pressure and velocity of the wave inside the flute. The other one calculates the velocity of the sound wave inside the flute by inserting the previous pressure of the sound wave inside the flute. The visualisation of this model however is not yet an accurate model to visualise waves inside the transverse flute. Therefore a conclusion is made that the exact model best describes the sound waves that can be generated inside the transverse flute.