Wave impedance measures how strongly a medium “resists” the motion associated with a traveling wave. It connects the size of the wave’s
driving quantity to the resulting motion quantity. In acoustics, the relevant quantity is often the ratio between sound pressure and particle velocity;
on a string, it relates transverse force to transverse particle velocity. The idea is important because when a wave reaches a boundary between two media,
the amount of reflection depends strongly on how closely the impedances match.
For fluids and sound, a common simplified model uses
\[
Z=\rho v,
\]
where \(\rho\) is density and \(v\) is wave speed. This is the specific acoustic impedance, often measured in
rayl, which in SI base units is
\[
\text{kg}\,\text{m}^{-2}\,\text{s}^{-1}.
\]
A denser medium or a medium with faster sound speed gives larger impedance. This is why sound often reflects strongly at boundaries such as air-to-water:
the acoustic impedances differ a lot.
For an ideal stretched string, the characteristic impedance can be written as
\[
Z=\sqrt{\mu T},
\]
where \(\mu\) is linear mass density and \(T\) is the string tension. The units here are different from acoustic specific impedance:
\[
\sqrt{\frac{\text{kg}}{\text{m}}\cdot \text{N}} = \frac{\text{kg}}{\text{s}}.
\]
This difference is important: a string impedance should only be compared with another string impedance of the same kind, not directly with an acoustic impedance.
Why impedance matters for matching. Suppose a wave travels from medium 1 with impedance \(Z_1\) into medium 2 with impedance \(Z_2\).
If the impedances are close, the wave passes through the boundary with little reflection. If they are very different, reflection is strong.
A simple amplitude reflection coefficient model is
\[
r=\frac{Z_2-Z_1}{Z_2+Z_1}
\]
or, depending on sign convention,
\[
r=\frac{Z_{ref}-Z}{Z_{ref}+Z}.
\]
The exact interpretation depends on the wave variable being tracked, but the key message is the same: mismatch causes reflection.
A convenient engineering quantity is the mismatch ratio,
\[
\frac{Z}{Z_{ref}},
\]
which is dimensionless. A value close to 1 means the impedances are closely matched. Designers of speakers, instruments, acoustic layers, and transmission
systems often try to control this ratio to reduce unwanted reflection or improve power transfer.
In practice, this idea appears in many places: speaker matching in acoustics, anti-reflection layers, transmission lines in electronics, mechanical couplers,
and waveguides. The university-level generalization connects directly to characteristic impedance in electrical transmission lines and to impedance matching
in RF engineering, optics, and continuum mechanics.
This calculator computes the impedance for a chosen model, optionally compares it with a reference impedance, reports the mismatch ratio, and shows a simple
boundary animation to build intuition. The bar chart helps visualize whether the current impedance is lower, higher, or approximately matched to the reference.
