Now let's see what happens when an incident wave train encounters a boundary with an impedance that is somewhere between hard and soft. Now, only part of the incident wave is reflected; the rest is either absorbed by or transmitted into the second medium with the different impedance. The orientation of the reflected wave (whether upright or inverted) depends on whether the new material has a larger or smaller impedance. For this particular animation, the reflected wave has half of the amplitude of the incident wave, and is inverted.

The resulting superposition looks kind of like a standing wave, but it doesn't stay still. It appears to lurch forward toward the right. This should make sense since only a part of the energy carried by the original wave is reflected from the boundary; the rest of the incident energy is either absorbed by the impedance termination or transmitted into the second medium. As a result there is more energy moving to the right than there is being reflected to the left. There is a net energy flow to the right.

If you watch the above animation very carefully you will notice that the locations where the wave pattern reaches a maximum value are fixed in space. Likewise, the locations where the wave pattern has a minimum value are also fixed locations. So, while there is a net transfer of energy toward the right, there is an aspect of this wave pattern that does not change with time or space.

Mathematically, we can write the sum of the incident and reflected waves as,

$$ \xi(x,t)=\xi_o \sin(\omega t - kx) + \Gamma \xi_o \sin(\omega t + kx) $$

If we apply some trigonometric identities, we can show that the amplitude of this "pseudo-standing wave" depends on position according to

$$ C(x)=\sqrt{1 + \Gamma^2 + 2\Gamma \cos(2kx)} $$

The dashed curves in the animation at left represent the envelope defined by \(C(x)\), and you can see that as the "standing" wave is set up, it remains "fixed" within the confines of this envelope. The ratio of the maximum amplitude divided by the minimum amplitude is called the Standing Wave Ratio (SWR) and depends on the relative amount, \(\Gamma\), of the incident wave that is reflected from the boundary.

$$ \hbox{SWR}={\hbox{maximum amplitude} \over \hbox{minimum ampliutde}} = {1 + \Gamma \over 1 - \Gamma} $$

The standing wave ratio may be used to measure the value of the mechanical impedance and/or the absorption coefficient of an unknown material.