Waves: Transmission/Reflection Coefficients/Energy Conservation

[maverick280857]

Hi

This isn't really homework but still...

x = 0 represents the junction between two media which can support traveling waves over two strings (specifically the junction at x = 0 is a "knot" which separates the two strings--the "media"). If a traveling wave (traveling to the right) of amplitude $A_{I}$ is incident at the junction from the medium on the left, given its frequency $\omega$ and angular wavenumber $k_{1}$ we can write the wave as

$$Y_{I} = A_{I}\sin (-k_{1}x+\omega t)$$

If the transmitted and reflected waves have amplitudes $A_{T}$ and $A_{R}$ respectively (and the angular wavenumber of the transmitted wave is $k_{2}$) then,

$$Y_{R} = A_{R}\sin (k_{1}x+\omega t)$$
$$Y_{T} = A_{T}\sin (-k_{2}x+\omega t)$$

we know that the boundary conditions are:

1. The frequencies of the incident, transmitted and reflected waves be equal. (This has already been accounted for, in the equations above).
2. The slope of the resultant wave in medium 1 = slope of resultant wave in medium 2, at the junction, i.e.
$$\frac{\partial(Y_{I}+Y_{R})}{\partial x}|_{x=0} = \frac{\partial Y_{T}}{\partial x}|_{x=0}$$

From these boundary conditions,

$$A_{R} = \frac{k_1 - k_2}{k_1+k_2}A_{I}$$ (equation 1)
$$A_{T} = \frac{2k_{1}}{k_1 + k_2}A_{I}$$ (equation 2)

Now, it can be shown using the conservation of energy at the junction,

$$P_{incident} = P_{transmitted} + P_{reflected}$$

that equations 1 and 2 are true.

Now this is where my question begins:

Suppose energy conservation does not hold at the junction (i.e. at x = 0). For now, let us assume that some energy from the incident wave is lost to the surroundings (perhaps as heat energy or due to disspitative forces). Then we will have

$$P_{incident} \neq P_{transmitted} + P_{reflected}$$

In other words,

$$P_{incident} = P_{transmitted} + P_{reflected} + P_{lost}$$

Using this "modified energy conservation" constraint, it can be shown that equations 1 and 2 will no longer hold. But equations 1 and 2 have been derived using just boundary conditions at the junction which are obvious, and nothing else. So the boundary condition method will not agree with energy conservation at the junction.

(a) What is the solution to this dilemma?

(b) If in a situation, I am given that some percentage of the incident energy is lost (and therefore, only a fraction of it is now available), what do I do? That is, can I apply equations 1 and 2 to find the amplitudes or is there a modification?

I think I am making a fundamental mistake in my reasoning. I would be very grateful if someone could explain it to me.

Thanks and cheers
Vivek

 

[maverick280857]

To the Moderator: Please shift this to the right forum.

Thanks and cheers
Vivek

 

[thepatientmental]

Hi Vivek,

This is an interesting question about the relationship between boundary conditions and energy conservation in the context of waves. Let's break down the dilemma and see if we can come to a solution.

First, let's understand what is meant by energy conservation in this context. In physics, energy conservation refers to the principle that energy cannot be created or destroyed, only transferred or converted from one form to another. In the case of waves, energy conservation means that the total energy of the incident wave must be equal to the sum of the energies of the transmitted and reflected waves at the junction. This is represented by the equation P_{incident} = P_{transmitted} + P_{reflected}.

Now, let's look at the boundary conditions. These are simply conditions that must be satisfied at the junction (x = 0) in order for the wave equations to be valid. In this case, the two boundary conditions are that the frequencies of the incident, transmitted, and reflected waves must be equal, and the slopes of the resultant waves in each medium must be equal at the junction. These boundary conditions are based on the assumption that energy conservation holds at the junction.

So, what happens if we modify the energy conservation equation, as you proposed, by including a term for energy lost to the surroundings? This would mean that P_{incident} = P_{transmitted} + P_{reflected} + P_{lost}. In this case, the boundary conditions would no longer be satisfied, and equations 1 and 2 would not hold. This is because the boundary conditions were derived based on the assumption of energy conservation, and if that assumption is no longer true, the boundary conditions no longer hold.

To answer your specific questions:

(a) What is the solution to this dilemma?

The solution to this dilemma is to either modify the boundary conditions to account for the lost energy, or to modify the energy conservation equation to match the boundary conditions. This can be done by including a term for the lost energy in the boundary conditions, or by adjusting the equation for energy conservation to only include the energies of the transmitted and reflected waves. Either approach will result in a consistent set of equations that take into account the lost energy.

(b) If in a situation, I am given that some percentage of the incident energy is lost, what do I do?

In this case, you can modify the boundary conditions or the energy conservation equation to account for the lost energy. For