What is a Waterfall Plot (Cascade Diagram)?

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A waterfall plot, also called a cascade diagram, is a three-dimensional graph that shows how a vibration spectrum evolves over time or against another variable — most often machine speed. It is built by stacking a series of individual FFT spectra one behind another, forming a 3D surface that resembles a cascading sheet of water. That single picture lets an analyst watch each vibration component grow, shrink, appear or vanish as the machine’s operating conditions change, which is something a single static spectrum can never reveal.

1. Definition: The Three Axes of a Waterfall Plot

The power of the cascade diagram lies in adding a third dimension to the familiar two-axis spectrum. A conventional FFT plots amplitude against frequency for one instant; the waterfall plot adds time or speed as a third axis so a whole sequence of spectra can be read at a glance.

• X-axis — Frequency: the spectral content, in Hz or, when order tracking is used, in orders of running speed.
• Y-axis — Amplitude: the magnitude of each spectral component, in velocity, acceleration or displacement.
• Z-axis — Time or RPM: the variable along which the spectra are stacked. Speed (RPM) is by far the most common and most diagnostically useful.

A close relative is the cascade plot, and the terms are frequently treated as synonyms; some analysts reserve “waterfall” for a time-based stack and “cascade” for a speed-based one, but the underlying display is identical.

2. The Primary Application: Run-up and Coast-down Testing

The most important use of a waterfall plot is to analyse vibration captured during a machine’s startup (run-up) or shutdown (coast-down). During these transient events the speed sweeps through the entire operating range, and the waterfall plot draws a complete map of the machine’s dynamic response across that range. Rather than guessing how the rotor behaves at intermediate speeds, the analyst sees every speed represented on one surface.

This makes the plot indispensable for several tasks:

• Identifying critical speeds and resonances: a resonance shows up as a ridge that stays at a fixed frequency regardless of speed. As the running-speed orders (1×, 2×, …) sweep across that fixed frequency, their amplitude climbs sharply, marking the critical speed at the intersection.
• Separating forced vibration from resonance: the plot cleanly distinguishes speed-dependent peaks — forced vibrations such as unbalance that follow the order lines — from fixed-frequency peaks (resonances) that form a straight ridge across the speed axis.
• Observing changes in rotor stability: it reveals the speed at which sub-synchronous instabilities such as oil whirl and whip appear and disappear, which is central to any rotor-dynamics investigation.

3. How to Interpret a Waterfall Plot

Reading a cascade diagram comes down to recognising two families of ridges and how they interact.

Order lines (diagonal ridges)

These ridges are tied directly to the machine’s running speed and so appear as diagonal lines that climb in frequency as the speed rises.

• The most prominent diagonal is normally the 1st order (1×), the response to rotor unbalance and the running-speed component.
• Further diagonals appear at the 2nd order (2×) — frequently linked to misalignment — and at higher harmonics, each a fixed multiple of speed.

Resonances (horizontal ridges)

These ridges sit at a constant frequency, independent of speed, so they run horizontally across the plot. They mark the rotor-bearing system’s natural frequencies.

• Where an order line (such as the 1× unbalance response) crosses a resonance ridge, the amplitude rises steeply, forming a large peak at one specific speed.
• That speed is a critical speed of the system, and the amount of amplification at the crossing reveals how much damping the system carries.

4. Data Acquisition: Order Tracking and the Tachometer

To produce a crisp waterfall plot, the data is usually acquired with order tracking. This requires a tachometer pulse so that each spectrum is synchronised to shaft angle and the spectral lines do not “smear” across bins as the speed changes between samples. Without that phase reference, transient spectra blur and the order lines lose definition. While a waterfall can be drawn against a fixed frequency axis, an order-based waterfall — with orders rather than Hz on the X-axis — keeps the order lines perfectly vertical and is often easier to read on variable-speed machines.

In the field, the same instrument that captures the spectra usually supplies the speed reference. A portable two-channel analyser such as the Balancet-1A, fitted with its optical laser tachometer triggering off a strip of reflective tape, records synchronised spectra and 1× amplitude-and-phase through a run-up or coast-down — the raw material from which a cascade diagram is assembled. Because the measurement is taken in the machine’s own bearings at operating speed, the resulting plot reflects the rotor’s true installed behaviour.

5. Related Run-up / Coast-down Plots

The very same transient data set feeds several complementary displays, and experienced analysts move between them freely:

• Bode plot: amplitude and phase of a single order plotted against speed on Cartesian axes — ideal for reading the exact RPM of a peak.
• Nyquist plot: the real-versus-imaginary trace of one order’s vector, which forms a loop at each critical speed.
• Campbell diagram: a related frequency-versus-speed map that overlays order lines on natural-frequency lines to predict interferences.

Where the Bode and Nyquist plots focus on one order at a time, the waterfall plot keeps the entire spectrum in view at every speed. That breadth is exactly why it remains an indispensable tool for in-depth rotordynamic analysis, giving a complete picture of a machine’s behaviour across its whole operating range.

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