Transmission and reflection of waves at boundaries

[albega]

Homework Statement

An infinite string is made of three sections, a single intermediate section of length a and density p1 between two semi-infinite sections of density p2.

A wave of frequency f is incident on the intermediate section. If a is an integer multiple of the wavelength in that section, show that the incident wave is not reflected.

Determine the amplitudes of the forward and backward traveling waves in the intermediate section in terms of the incident amplitude

Homework Equations

I can work out the amplitude and relfection coefficients as
r=k1-k2/k1+k2, t=2k1/k1+k2
in terms of wavenumbers (k1 is the incident wavenumber).

The Attempt at a Solution

In terms of showing the lack of reflection:
k1=k2 gives r=0. However that would imply no change in wavespeed across the boundary, so as the densities aren't equal, the tensions would have to be different, but I think they are the same. I think I'm misunderstanding something...

 

[dauto]

I assume you mean r=(k₁-k₂)/(k₁+k₂), t=2k₁/(k₁+k₂).

Note that there will be two reflected waves interfering with each other. What does the amplitude of the resultant of that interference depends on?

 

[albega]

I assume you mean r=(k₁-k₂)/(k₁+k₂), t=2k₁/(k₁+k₂).

Note that there will be two reflected waves interfering with each other. What does the amplitude of the resultant of that interference depends on?

Why stop at 2 reflected waves?

I thought it would just depend on the wavenumbers but that wouldn't help.

 

[dauto]

Why stop at 2 reflected waves?

I thought it would just depend on the wavenumbers but that wouldn't help.

You stop at two reflected waves because there are only two reflecting boundaries. And yes, you need one more piece of information (knowledge) which is whether or not the phase of the reflected wave is shifted from the phase of the original wave. There are two possible shifts. 1st possibility: No shift. Second possibility: 180° shift.

 

[albega]

You stop at two reflected waves because there are only two reflecting boundaries. And yes, you need one more piece of information (knowledge) which is whether or not the phase of the reflected wave is shifted from the phase of the original wave. There are two possible shifts. 1st possibility: No shift. Second possibility: 180° shift.

But if there's an incident wave, it reflects at the first boundary, and then the transmitted wave reflects at the second boundary. This reflected wave is transmitted through the first boundary, but also reflects back towards the second. This then is transmitted and reflected, and then moves back towards the first boundary again, which can be transmitted and reflected again. Surely there would be an infinite number of reflected waves.

Doesn't that just depend on the sign of r and t though, so thus on the wavenumbers. I still can't see how the intermediate section length enters the calculation.

 

[albega]

Any help please :)?