Engineering Acoustics/Boundary Conditions and Wave Properties

Boundary Conditions

The functions representing the solutions to the wave equation previously discussed,

i.e. $y(x,t)=f(\xi )+g(\eta )\,$ with $\xi =ct-x\,$ and $\eta =ct+x\,$

are dependent upon the boundary and initial conditions. If it is assumed that the wave is propagating through a string, the initial conditions are related to the specific disturbance in the string at t=0. These specific disturbances are determined by location and type of contact and can be anything from simple oscillations to violent impulses. The effects of boundary conditions are less subtle.

The most simple boundary conditions are the Fixed Support and Free End. In practice, the Free End boundary condition is rarely encountered since it is assumed there are no transverse forces holding the string (e.g. the string is simply floating).

- For a Fixed Support:

The overall displacement of the waves travelling in the string, at the support, must be zero. Denoting x=0 at the support, This requires:

$y(0,t)=f(ct-0)+g(ct+0)=0\,$

Therefore, the total transverse displacement at x=0 is zero.

- For a Free Support:

Unlike the Fixed Support boundary condition, the transverse displacement at the support does not need to be zero, but must require the sum of transverse forces to cancel. If it is assumed that the angle of displacement is small,

$\sin(\theta )\approx \theta =\left({\frac {\partial y}{\partial x}}\right)\,$

and so,

$\sum F_{y}=T\sin(\theta )\approx T\left({\frac {\partial y}{\partial x}}\right)=0\,$

But of course, the tension in the string, or T, will not be zero and this requires the slope at x=0 to be zero:

i.e. $\left({\frac {\partial y}{\partial x}}\right)=0\,$

- Other Boundary Conditions:

There are many other types of boundary conditions that do not fall into our simplified categories. As one would expect though, it isn't difficult to relate the characteristics of numerous "complex" systems to the basic boundary conditions. Typical or realistic boundary conditions include mass-loaded, resistance-loaded, damping-loaded, and impedance-loaded strings. For further information, see Kinsler, Fundamentals of Acoustics, pp 54–58.

Wave Properties

To begin with, a few definitions of useful variables will be discussed. These include; the wave number, phase speed, and wavelength characteristics of wave travelling through a string.

The speed that a wave propagates through a string is given in terms of the phase speed, typically in m/s, given by:

$c={\sqrt {T/\rho _{L}}}\,$ where $\rho _{L}\,$ is the density per unit length of the string.

The wave number is used to reduce the transverse displacement equation to a simpler form and for simple harmonic motion, is multiplied by the lateral position. It is given by:

$k=\left({\frac {\omega }{c}}\right)\,$ where $\omega =2\pi f\,$

Lastly, the wavelength is defined as:

$\lambda =\left({\frac {2\pi }{k}}\right)=\left({\frac {c}{f}}\right)\,$

and is defined as the distance between two points, usually peaks, of a periodic waveform.

These "wave properties" are of practical importance when calculating the solution of the wave equation for a number of different cases. As will be seen later, the wave number is used extensively to describe wave phenomenon graphically and quantitatively.

For further information: Wave Properties

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Edited by: Mychal Spencer