This week, we have focused on boundary value problems.  This comes up when you have two regions in which waves obey the usual differential equation for waves, but where the velocity of the waves in the two regions is different:

In region 1, you might have (for a displacement )
,
and then in region II, (for a displacement :
.

The example we are discussing in the lectures is the example where you have two strings of different mass per unit length, attached to each other (let’s say, at ).

The question at hand is how wave propagation is affected by this “defect” in the medium – a place where the medium changes from one type to another type in a discontinuous manner.

Knowing just the above two wave equations is not quite enough to answer this question. We also need to impose boundary conditions on the displacement functions. The two boundary conditions that we impose (in words) are:

1. The displacement of the string must be the same on either side of the point where the medium changes (the strings stay tied together)
2. There must not be a finite force on an infinitesimal mass (otherwise there would be an infinite acceleration)

In equations, these two conditions are:

The first equation is easy to understand – the second comes from the fact that the force on a segment of string is proportional to the slope of the string on either side of that segment. If the slopes are not equal, there will be a finite force on a vanishingly small mass.

Applying these conditions gives enough information to solve the wave equation in these joined regions. Note that the conditions themselves may be different for different types of waves! For example, in the text, in one of the HW problems, the second condition is different mathematically (although the physical motivation given in words above is still the same).

Consider, for example, sending in a wave pulse described by the function

In this scenario, there will be both a wave that is transmitted through the boundary where the media change, and a wave that is reflected back from the discontinuity. The reflected wave is also a pulse, but moves to the left:
.
There is also the transmitted wave, which moves to the right:
.
In region 1, the full solution is the superposition of the incident and reflected wave:

while the solution is region 2 is just the transmitted wave:
.

The question at hand can now be phrased as the following: Using the boundary conditions, can we solve for and given ? That is, for an arbitrary incident wave, can we extract both the reflected and transmitted wave? The answer is yes. We have second order equations, and two boundary conditions, which is enough to specify the output wave given our input incident wave.