1.2 What is a wave?

Before proceeding to the theory of wave intensity, it is necessary to clarify what is meant by a 'wave'.

Representation by Fourier decomposition (one of many possibilities)

Historically, the most common way to represent cardiovascular waveforms is the Fourier decomposition, which treats the measured waveforms as the superposition of sinusoidal wavetrains at the fundamental frequency and all of its harmonics. Despite of the many successes of this approach, we should not make the mistake of thinking that this is the only way to decompose arterial waveforms into independent components. In fact, there are an infinite number of ways that a waveform can be decomposed into independent components. The choice of representations, therefore, becomes a question of utility.

For example, the figure below shows a human aortic pressure waveform decomposed in two different ways. The Fourier decomposition into sinusoidal wavetrains of different frequencies is shown in the left panel and a decomposition into successive wavefronts is shown in the left panel. The measured waveform is shown at the top of each panel and the decomposition below.

Representation by Fourier components The top trace is the pressure waveform measured in the human ascending aorta. The 16 traces below show the Fourier components for the fundamental and first 15 harmonics (the mean pressure is not shown). The summation of these 16 traces give the original signal to greater than 95% accuracy (if all of the Fourier components are included, the original signal is recovered exactly).

Representation by successive wavefronts Again the top trace is the measured pressure waveform. The 16 traces below are the successive wavefronts derived from this waveform. They represent the change in pressure over a time period equalling 1/16th of the cardiac period. They sum to give a good representation of the original waveform. If wavefronts were defined using the difference measured at each sampling time of the original data, the summation would be exact.

Since Fourier analysis is carried out in the frequency domain, it can be very difficult to relate features of the Fourier representation to specific times in the cardiac cycle. This is illustrated by an example of detecting the start of systole in the pressure waveform shown in the figure [Find the start of systole].

There are, however, other waves such as tsunamis and shock waves (the sonic boom) that are best described as solitary waves. For these waves it is more convenient to consider them as a sequence of small 'wavelets' or 'wavefronts' that combine to produce the observed wave. [The term 'wavelet' has been used for decades in gas dynamics to describe these incremental waves, but the term has been appropriated recently by numerical analysts to describe a new transformation of signals and images into their component parts. Because of this, it has been necessary to abandon the word in favour of the less descriptive 'wavefronts'].

Representation by successive wavefronts

Wavefronts are the elemental waves in wave intensity analysis. They can converge on each other as they propagate, causing waves to steepen and eventually form shock waves, or they can diverge causing the the wave to become less steep as it propagates. This process is seen on the beach where the fronts of approaching waves crest and eventually break as the wavefronts at the foot of the wave are overtaken by the faster wavefronts near the crest of the wave.

In the digital era, it is convenient and accurate to describe these wavefronts as the change in properties during a sampling period Δt; e.g. dP = P(t+Δt) - P(t). Differences such as this are commonly used in gas dynamics instead of the more familiar differential because they can cope with discontinuities such as shock waves when the differential is ill-defined. The difference, unlike the differential, depends upon the sampling period and this must be remembered if differences are used.

The plot on the right of the figure above shows the same pressure waveform decomposed into 16 successive wavefronts. This representation of the original waveform is rather crude but serves to illustrate the principle. Higher resolution can be obtained simply by using more wavefronts occurring at smaller intervals during the cardiac period. An exact representation of a digitised waveform can be obtained simply by using one wavefront per sampling period.

This difference in the interpretation of what is meant by a wave is fundamental to the understanding of wave intensity analysis. Both approaches give unique, complete representations of the measured waveform and the choice of representation is determined solely by convenience; wavefronts can be represented by Fourier components and sinusoidal waves can be represented by successive wavefronts. In wave intensity analysis, the successive wavefront representation is used implicitly.

Terminology

To avoid confusion, I will use the following definitions throughout this introduction.

wave will be used in a completely general way,
waveform will be used to describe the measured pressure and velocity waves in the arteries,
wavetrain will be used to describe sinusoidal waves and
wavefront will be used to describe the successive incremental waves.

Finally, it is important to observe that the fact that a waveform can be decomposed into any particular form does not imply that these basic forms are in any way intrinsic to the initial waveform. Any waveform can be decomposed without any loss of information in an infinite number of different ways. Any complete, orthogonal set of functions (the number of these functions is infinite) can be used as the basis for a unique decomposition. Similarly, any orthogonal wavelet transformation (the number of these is also infinite) can also be used to provide a unique decomposition of any waveform. No particular decomposition is inherently better than any other; their value depends solely on their utility.

What is a wave? (continued)