45 Impedance

Snell’s Law applies to a wave that is transmitted through some internal boundary where the phase speed changes. If the changes are sharp, some of the wave energy may also be reflected. Before we consider the full problem of reflection and transmission, we need to define a property of the medium called its impedance.

The impedance, Z, is a measure of how much force is required to create a given cyclical velocity of particles in the medium.  For a wave in a fluid, impedance is the ratio of the amplitudes of the wave’s pressure field and its water parcel velocity field. (It is important to remember that it is not the phase speed of the wave that is involved in quantifying impedance, but instead the cyclical velocity of water parcels.) For now, we will just consider one component of the fluid vector velocity, the x-component, u.

    \[ Z=\frac{pressure \hspace{4pt} oscillation \hspace{4pt} amplitude}{velocity \hspace{4pt} oscillation \hspace{4pt} amplitude}=\frac{P}{u}. \]

We will be considering waves in a fluid, and thus we can use scaling arguments applied to the x-direction momentum equation to find the impedance. We consider the acceleration created by a pressure gradient force,

    \[ \frac{\partial u}{\partial t}=-\frac{1}{\rho}\frac{\partial p}{\partial x}. \]

The velocity scales, U and P, will be the amplitude of the wave field for these two variables.  The time scale, T, will be the period of the wave, and the length scale, L, will be the wavelength of the wave.  The scaled momentum equation becomes

    \[ \frac{U}{T}=\frac{1}{\rho}\frac{P}{L}, \]

and so the impedance, Z, the ratio of wave pressure to velocity amplitudes is

    \[ Z=\frac{P}{U}=\rho\frac{L}{T}=\rho C, \]

where C, the wave phase speed, is substituted for the ratio of wavelength to time.

Relevant examples, that we will consider in detail later, are acoustic impedance differences at the top and the bottom boundaries of the ocean.  Sound waves travel faster in water than in air, and water density is higher.  Thus water has a higher impedance than air.  Similarly, sediment has a higher impedance that water.

Key Takeaways

The fluid impedance (sometimes called the specific impedance) is the ratio between pressure (P) and fluid parcel velocity (U) oscillations. It can be calculated from density (\rho) and wave speed (C),

    \[ Z=\frac{P}{U}=\rho C, \]