When solving the wave equation we must look at two
possible cases.
1)
The boundaries are so far away that they are not
“visible”. In this case we look for traveling wave type solutions.
2)
The boundaries are near enough that what happens
there, effects our wave. In this case we must take these effects into account.
What happens at these boundaries determine what kind of waves can exist in the
whole system. The conditions valid at the boundary at all times are called the boundary
conditions. Now we shall look at them.
The boundary conditions
cannot be obtained from the wave equation itself. Remember that the equation is
valid inside and is even valid when the boundaries are so far away that they
cannot be seen (the traveling wave case). Thus they do not follow from the wave
equation but must be imposed in addition to the wave equation.
In this way the boundary
conditions are just like the initial conditions. In the simple harmonic
oscillator, and even in the free fall case we saw that the initial speed and
position of the particle could not be obtained from the equation of motion but
were needed in addition to the equation to determine the motion.
Similarly in the wave motion
we must have the boundary conditions in addition to the wave equation to be
able to see which waves can exist in the medium. The difference between the
initial conditions and boundary conditions is that all initial conditions are
given at the beginning of motion (say t=0). In an initial value type problem we
are not given any “final” conditions. This follows from the causality. Earlier
events cause later effects not the other way around. Boundary conditions are
given on both sides. If there are several degrees of freedom then some conditions
are given on one side and some on the other side.
Lets have a system which
extends from x=0 to x=L. We will arbitrarily call the boundary at x=0 the “left
boundary” and the other end at x=L the “right boundary”. The conditions we
shall look at are
1)
Fixed boundary.
2)
Free boundary.
Other kinds of boundary
conditions can be designed. For example two different media with different
properties may meet at a boundary or a medium may be terminated by a damping
device, instead of being fixed. However such boundaries will not interest us
much and we shall concentrate on the two types mentioned above.
Fixed Boundary: In
this type of boundary no motion is possible. Thus the solution is
y(0,t) = 0 for left
boundary
y(L,t) = 0 for right boundary.
Both conditions must hold
for all times.
Free Boundary: In
this type of boundary motion is possible, in fact since the boundary is free no
force is needed. In fact you cannot exert a force on a free boundary. A string
attached to a massless and frictionless loop on a rod is an example. Since the
loop does not have mass it can not exert an inertial force, since there is no
friction it can not exert a drag force. And because of Newton’s third law the
medium cannot exert a force on it either. Since the force, one side exerts on
the other, in wave motion, is proportional to the gradient ¶y /¶x
then we must have the gradient equal to zero.
Thus the solution is
(¶y /¶x)
(L,t) = 0 for left boundary
(¶y /¶x)(L,t)
= 0 for right boundary.
Both conditions must hold
for all times.
Now lets look at the effects
of these boundary conditions on the possible wave solutions.
First the left boundary:
i)
Fixed boundary: y(0,t)= 0 at all times.
Traveling wave solutions
like y= Asin(wt-kx),
or y= Asin(wt+kx) can not live here. y is zero sometimes but not at all times.
Similarly standing waves like y=Bcos(kx)sin(wt+j)can not exist. Thus the
only possible solutions are of the family,
y=Asin(kx)sin(wt+j).
In
these solutions “y” is always 0 at x= 0.
ii)
Free boundary: (¶y /¶x)
(0,t) = 0 at all times.
Again traveling wave
solutions are impossible but now standing wave solutions of the type
y=Acos(kx)sin(wt+j).
exist. Note that the derivative of the cos is the sin function and it is always 0 at x=0. And satisfies the boundary condition.
Now the right boundary:
iii)
Fixed boundary: y(L,t)= 0 at all times.
If the left boundary is
fixed then the solution is y=Asin(kx) sin(wt+j). The right boundary
condition then forces sin(kL) to be zero. Therefore
kL= np.
=> kn = np / L
Thus only waves with certain
wavenumbers kn are allowed. All other solutions fail the boundary
conditions and can not exist in this medium.
If the left boundary is
free, then y=Bcos(kx)sin(wt+j)
and cos(kL)=0.
kL= (n-˝)p.
=> kn = (n-˝)p / L
Again waves with a set of
wavenumbers are allowed. In both cases n is a positive integer. The wavenumbers
kn are called eigenvalues and the solution are known as eigenfunctions.
(In Turkish Özdeđer ve Özfonksiyon)
iv)
Free boundary: (¶y /¶x)
(L,t) = 0 at all times.
The x derivative of the wavefunction
must be zero here. Thus if the left boundary is fixed and y=Asin(kx)sin(wt+j)then
cos(kL)= 0 and therefore,
kL= (n-˝)p.
=> kn = (n-˝)p / L
And if the left boundary is
free so that y=Bcos(kx)sin(wt+j)
then sin(kL)= 0 and,
kL= np.
=> kn = np / L
Thus again only standing waves with
certain wavenumbers (eigenfunctions) can exist.
Thus the boundary conditions determine which solutions
are possible in this kind of environment.
Note that for wave equation and other partial
differential equations the boundary conditions and the wave equation are not
enough to determine the wave motion. Since an infinite number of modes with
different “n” numbers can be excited in a medium we also need which ones are
actually excited and with what amplitude and phase. For these we need initial
conditions in addition to the boundary conditions. The initial conditions are
imposed at time t=0 but throughout the whole medium. y(x,0), ¶y /¶t(x,0).
Wave equation + Boundary conditions + Initial
conditions => determine wave motion uniquely.