Engineering Acoustics/Boundary Conditions and Forced Vibrations

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.

The sequence of wave reflection for incident, reflected and combined waves are illustrated below. Please note that the wave is traveling to the left (negative x direction) at the beginning. The reflected wave is ,of course, traveling to the right (positive x direction).

t=0

t=t1

t=t2

t=t3

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

Again for free boundary, the sequence of wave reflection for incident, reflected and combined waves are illustrated below:

t=0

t=t1

t=t2

t=t3

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.

Here is a website with nice movies of wave reflection at different BC's: Wave Reflection

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

Forced Vibrations
1.forced vibrations of infinite string suppose there is a string very long, at x=0 there is a force exerted on it.

F(t)=Fcos(wt)=Real{Fexp(jwt)}

use the boundary condition at x=0,

neglect the reflected wave

it is easy to get the wave form

where w is the angular velocity, k is the wave number.

according to the impedance definition

it represents the characteristic impedance of the string. obviously, it is purely resistive, which is like the resistance in the mechanical system.

The dissipated power

Note: along the string, all the variables propagate at same speed.

link title a useful link to show the time-space property of the wave.

Some interesting animations of the wave at different boundary conditions.

1.hard boundary (which is like a fixed end)

2.soft boundary (which is like a free end)

3.from low density to high density string

4.from high density to low density string

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