Traveling Waves (Attenuation, Dispersion, Wave Velocity)

We'll now add the next widget to the wave transmission story.  Real fluids and vessel walls possess viscosity, that property that results in energy loss due to deformation of the material.  In the above, fluid viscosity has been added to the model shown in the previous article ( no wall viscosity ).  This results in an exponential decay of both the pressure and flow waves with distance traveled.  At a given frequency, it's the same attenuation for both pressure and flow. After all, we've already seen that the ratio of $$P/Q$$ has to remain constant – the characteristic impedance ($$Z_0$$).

We could go to the same process as in the last article and some of the images we would obtain are below for a single frequency sinusoid.  DON'T FORGET – THIS IS A FREEZE-FRAME FROM THE REAL-TIME VIDEO SHOWING VALUES ALL ALONG THE LENGTH OF THE TUBE. For negative pressure gradient:

Cross-sectional area:

Phase Velocity

We agreed previously that an input of a single sinusoidal frequency would result in outputs of the same frequency only.  The waves shown above do not look quite like sinusoids anymore.  However what is meant by the single frequency statement is that pressure and flow ( indeed ALL of the dependent variables ) have sinusoidal output at each location.  Here's another video where specific locations of measurement are shown using colored vertical lines.

Here's a still frame of the video for reference:

In the next figure, the color-coded, time domain ("strip chart") recordings of pressure and flow are shown. A moments thought and this is all perfectly obvious. At each location, all the pressure and flow waves show up as perfect sinusoids ( they happen to have a frequency of 2 Hz).  There is a time delay for the wave to travel from one location to the next so we see that the peak of the traveling wave shows up successively later at downstream locations.  It turns out that for this simple situation ( single sinusoid and no wave reflection ),  we could figure out the speed of the wave with 2 pressure transducers and a "stopwatch".  Really all you do is find out how long it takes for the peak to go from location 1 to location 2 and divide the distance by that time.  It would turn out that we could get confused if we weren't careful about this.  As a thought experiment, imagine if the 2 pressure transducers were exactly 1 wavelength apart so that the peak seemed to pass both transducers at the same moment.  The problem in this scenario is that we don't know which peak is which and we have to make sure we have determined the time for the same peak to pass the measurement locations. Yup, this is another example of an aliasing phenomenon.

P-Q phase relationships, attenuation, phase velocity,  dispersion

The wave velocity of an individual frequency in a uniform medium is referred to as the phase velocity.  We would find the velocity a little more difficult to determine if the wave includes multiple frequencies; you can't necessarily look for the passage of a peak, trough, or any other apparent feature of the composite wave which changes shape as it progresses through the system.  In the presence of reflected waves, determining phase velocity becomes even more difficult as we will soon see.

When there are multiple frequencies in the measured waveform ( or even when there are not ), a commonly used method to determine apparent phase velocity starts with a Fourier transform of pressure recordings made at 2 or more sites.  The recordings below are from locations 1 and 3 in the video above.  The Fourier transforms of the 2 pressure recordings give us the modulus ($$M_1,M_2$$) and phase ($$\theta_1,\theta_2$$) for each/every sinusoid in the recording.

If we know the distance between the measurement sites we are readily able to determine the apparent wave velocity from the the following:

$$\Large b = \frac{\Delta \theta}{\Delta z}$$

$$\Large v = \frac{\omega}{b}$$

$$\omega$$ has units of radians per second and $$b$$ is in radians per centimeter (for example); hence the result of the calculation is in centimeters per second.

Progressively increasing phase velocity (due to increasing modulus of elasticity of the tube material) is depicted in the next 3 figures pertaining to a length of compliant tubing and in the absence of wave reflection. The inputs to the 3 tubes is the same 2 Hz wave. This is to help visualize the fact that a slower traveling wave has a greater change in wave angle per unit of distance ($$b$$) in the above.  Changing the stiffness of the wall apparently affects attenuation also.

Wave Attenuation

It's also fairly obvious that the magnitude ( modulus ) of the pressure waves diminished from one location to the next.  The magnitude diminishes exponentially with distance but we could still figure out the value of it from the above data.  Here is the formula we're trying to fit:

$$\Large |P_2| = |P_1|\,e^{-a\,\Delta z}$$

$$\Large a = - \ln{\left[ \frac{|P_2|}{|P_1|}\right]} / \Delta z$$

$$|P_1|$$ is the magnitude of an upstream pressure wave at a specific frequency, $$|P_2|$$ is the magnitude of corresponding wave in the downstream recording.   $$a$$ is called the attenuation coefficient and is a positive real number (always).  It has physical units of 1/length, e.g. "per cm".  You see that it multiplies a distance $$\Delta z$$ ( the distance between the 2 pressure transducers for example ) and that multiplication must yield a pure number  since we're going to raise $$e$$ to that number.  The flow waves attenuate the same way so we should get the same value for the attenuation coefficient whether we are measuring pressure or flow.  Of course accurate flow meters are more difficult to come by then pressure transducers.

It looks like the peak of the pressure wave passes each measurement location at the same moment as the flow wave. That's what we saw in the last article when there was no friction - no viscosity in the fluid. However this is not quite true anymore.  The addition of viscosity to the fluid causes the pressure and flow waves to be out of phase, at least slightly.  We can see that best by plotting pressure against flow in a hysteresis loop.

Viscosity increased to 0.1; p and q clearly out of phase, q leads p.

p-q Hysteresis loop for above at 0.2 Hz.

Dispersion

Here's something that's already part of your repertoire (since we all use ultrasound for imaging).  High-frequency waves are attenuated more than low-frequency ones.  In the circulation, that means that high frequency components of the pressure and flow waves will be attenuated more and change the shape of the wave with distance traveled.  This aspect of wave transmission is known as dispersion and is one of several reasons for the changing shape of the pressure and flow pulses ("pulsed wave transformation").  Dispersion comes about as a result of the viscosity in the fluid which results in frequency dependence of both attenuation and phase velocity.  Hence both size and phase relationships between the frequency components of the waves are altered with transmission.

This figure shows a freeze frame view of a composite pressure wave ( 2 frequencies, actually of equal magnitude at the upstream / left side of the figure) traveling in a compliant tube ( no wave reflection ).  While both frequencies are attenuated, the higher frequency is at attenuated to a greater extent with propagation. Consequently the shape of the wave changes with transmission.

The next figure shows time varying pressure "recordings" made at the upstream location and also at the location designated above by the magenta vertical line.  Both frequencies are attenuated by transmission, but the higher frequency oscillation is attenuated more. Consequently the shape of the recorded pressure wave changes.  The greater attenuation of high frequency wave components is important in selecting an ultrasound transducer, making underwater photographs, and other every day life experiences.

$$a$$ and $$b$$ shown above make up the real and imaginary parts of the propagation coefficient, $$\gamma$$.  The transformation of a single sinusoidal wave ( pressure or flow ) is described as simply as this:

$$\Large A = A_0 e^{-a\,z}e^{-j\,b\,z} = A_0 e^{-\gamma \,z}$$

$$\Large \gamma = a+j\,b$$

An item that occasionally comes up at conferences is shown below. The figure below is a classic (from Hemodynamics. WR Milnor 1989)  depicting "pulse wave transformation" - the change in form of the pressure recorded at various locations in the arterial system.  I've included some annotations to indicate the "dicrotic notch" which is also referred to in other texts and on the web as the "incisura" or "aortic incisura".   For the femoral artery you will also see a secondary dip in pressure and subsequent wave referred to as a dicrotic notch and dicrotic wave respectively.

As near as I can tell from the literature, terms like dicrotic notch and dicrotic wave are used generally; the notch marks the end of systole (if measured at the aortic valve) and the wave is the thing that follows.  I want to be definitive here however, that the high-frequency twitch in the pressure recording of the ascending aorta can have nothing to do physically with the low frequency dip and wave in the distal recordings. The wave frequencies involved for these 2 separate aspects of the pressure recordings are entirely different and cannot be physically related except in a highly nonlinear system.  I prefer to call the glitch due to aortic valve closure the aortic incisura.  This high-frequency wave component is attenuated beyond detection by the time the wave reaches the periphery.  I reserve the terms dicrotic notch and wave for the low frequency disturbances recorded from the periphery.  Veterinarians who pay attention will be able to actually palpate the peripheral dicrotic wave in normal horses and pressure recordings often reveal a second dicrotic notch and wave as well.  These features of the pressure pulse are thought to be the result of wave reflection which will be discussed soon.