7.5 Boundaries, Transmission, and Reflection

A boundary for a wave is a change in medium. When a wave encounters a boundary, some energy is transmitted across the boundary and some is reflected. Consider a pulse wave travelling along a string:

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At a fixed end, the string exerts an upward force on the fixed end which pulls back down on the string according to Newton III. This generates an upside-down wave pulse travelling in the opposite direction (what we actually see while this is happening is a superposition of both pulses).
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At a free end, the same thing happens except that the reflected pulse is the same way up. Because of the superposition, the free end reaches twice the peak amplitude of the pulse.
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When we have a mix between these two cases, for example a change in linear density of the string, we will see partial transmission and reflection.

7.5.1 Reflection and Transmission Coefficients

To figure out how much gets reflected and transmitted, we must make some assumptions about what happens to the wave at the boundary. At the boundary $x_{b}$ we have:

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$y(x_{b},t)$ must be continuous (no gaps in the string).
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$\frac{\partial y(x_{b},t)}{\partial x}$ must be continuous (no sharp kinks).

Therefore, we must have the following two conditions at the boundary (they do not have to hold anywhere else!):

	$\displaystyle y_{i}(x_{b},t)+y_{r}(x_{b},t)=y_{t}(x_{b},t)$		(7.75)
	$\displaystyle\frac{\partial y_{i}(x_{b},t)}{\partial t}+\frac{\partial y_{r}(x% _{b},t)}{\partial t}=\frac{\partial y_{t}(x_{b},t)}{\partial t},$		(7.76)

where subscript $i$ , $r$ , and $t$ represent the incident, reflected, and transmitted waves respectively. Solving for $y_{r}$ and $y_{t}$ using the wave equation, we get

	$\displaystyle y_{r}(x_{b},t)$	$\displaystyle=\frac{v_{2}-v_{1}}{v_{2}+v_{1}}y_{i}(x_{b},t)=ry_{i}(x_{b},t)$		(7.77)
	$\displaystyle y_{t}(x_{b},t)$	$\displaystyle=\frac{2v_{2}}{v_{1}+v_{2}}y_{i}(x_{b},t)=\tau y_{i}(x_{b},t),$		(7.78)

where we have defined the reflection coefficient $r$ and the transmission coefficient $\tau$ as

r=\frac{v_{2}-v_{1}}{v_{1}+v_{2}},\quad\tau=\frac{2v_{2}}{v_{1}+v_{2}},

(7.79)

where $v_{1}$ is the wave velocity in the left medium and $v_{2}$ is the velocity on the right. Note that $-1\leq r\leq 1$ , and $0\leq\tau\leq 2$ . These coefficients can also be defined in terms of the wave impedance $Z=\sqrt{\mu T}$ :

r=\frac{Z_{1}-Z_{2}}{Z_{1}+Z_{2}},\quad\tau=\frac{2Z_{1}}{Z_{1}+Z_{2}}.

(7.80)

Because $r$ and $\tau$ are defined as the ratios of the amplitudes of the reflected and transmitted waves to the incident wave, we have a constraint $\lvert r\rvert+\lvert\tau\rvert=1$ . Note that this has nothing to do with energy conservation, it holds even when energy is not conserved!

7.5.2 Changes in Wave Properties at the Boundary

Consider a boundary where the density increases. The wave speed is slower on the other side of the boundary, $v_{2}<v_{1}$ , so $r<0$ and $\tau<1$ . The pulse also becomes narrower because the wave speed is slower. What about where the density decreases? The new wave speed is faster, $v_{2}>v_{1}$ , so $r>0$ and $\tau>1$ . The reflected pulse has the same width but with smaller amplitude and the transmitted pulse is broader because its speed is higher. The maths for these scenarios is general to all shapes of waves. For periodic waves, the left-hand side of the boundary consists of the superposition of the incident and reflected waves. For the transmitted wave, since the frequency is fixed by the source the wavelength must be the quantity to change. If energy is conserved, it is conserved at the boundary, so we also have the constraint

\lvert P_{\text{inst}_{i}}(x_{b},t)\rvert=\lvert P_{\text{inst}_{r}}(x_{b},t)% \rvert+\lvert P_{\text{inst}_{t}}(x_{b},t)\rvert.

(7.81)

Note that this is different to the equations above where incident and reflected were on the same side, here reflected and transmitted are on the same side.