## Appendix A. Time–frequency analysis

A time–frequency analysis of the signal is performed via a continuous wavelet transform (Debauchies 1990; Farge 1992). The wavelet coefficient
$C(t,s)$
at time
$t$
and pseudo-pulsation
$s$
are calculated by convolving the pressure signal
$p(t)$
with a dilated and translated version of an analysing wavelet
$\unicode[STIX]{x1D713}(t)$
:

In order to obtain fine resolution in both time and frequency we chose a bump analysing wavelet whose Fourier transform at scale
$s$
is given by,

where
$\unicode[STIX]{x1D707}$
is the peak pulsation of the unit scale wavelet spectrum and
$\unicode[STIX]{x1D70E}$
is its width. Smaller values of
$\text{sigma}$
lead to higher-frequency resolution with poorer time localisation. We chose
$\unicode[STIX]{x1D707}=5$
and
$\unicode[STIX]{x1D70E}=0.6$
which provides localisation in both time and frequency sufficient to resolve the resonance peaks.

The spectrograms shown in figure 15 reveal a rich and varied behaviour, providing additional insight not accessible from the PSD maps. Examples of the four kinds of behaviour typically observed. These include: (i) total dominance by a single resonance cycle, as, for example, at
$(M,L/D)=(0.8,3)$
, where this occurs despite the existence of neighbour resonance possibilities; (ii) slow competition between neighbour resonance cycles, which occurs for
$(M,L/D)=(0.8,4)$
; (iii) fast competition between neighbour resonance cycles, as seen for
$(M,L/D)=(0.78,4)$
; and (iv) uncorrelated co-existence of multiple resonance cycles, as for example at
$(M,L/D)=(0.6,2)$
.

## Appendix B. Derivation of resonance conditions

Consider two waves travelling, respectively, upstream and downstream between two boundaries, situated at
$x=0$
and
$x=L$
. The waves are coupled via reflections at these boundaries, these being characterised by complex-valued coefficients,
$R_{1}$
and
$R_{2}$
.

The ansatz for a dependent variable of interest,
$\hat{q}(x,\unicode[STIX]{x1D714})$
, is,

At
$x=0$
and
$x=L$
we have, respectively,

Combining these equations leads to the resonance condition,

where

In terms of magnitude and phase, equation (B 4) is,

which leads to the following resonance constraints,

a similar derivation of which can be found in Landau & Lifshitz (2013).

## Appendix C. Nozzle-plane reflection conditions

Figure 16. Schematic depiction of simplified jet–nozzle system comprised of connected, solid- and soft-walled cylindrical ducts.

Consider the simplified problem depicted in figure 16, in which an incident
$k^{-}$
wave of amplitude
$I$
, impinging on the nozzle plane, produces a reflected wave of amplitude
$R$
, and a transmitted wave of amplitude
$T$
. Consider mass and momentum conservation in a thin disk (much smaller than a wavelength, meaning that the flow can be considered incompressible) containing the nozzle exit plane, respectively,

where the subscripts
$1$
and
$2$
refer, respectively, to the upstream and downstream faces of the disk.

The incident, reflected and transmitted waves all satisfy the dispersion relation,

where non-dimensionalisation has been performed using the jet diameter and speed of sound. The radial structures of the waves are given by the eigenfunctions
$J_{0}(\unicode[STIX]{x1D6FC}r)$
where
$\unicode[STIX]{x1D6FC}$
is the non-dimensional radial wavenumber.

The
$k_{d}^{-}$
wave behaves, as shown by Towne *et al.* (2017), like a wave propagating in a soft-walled duct, and its first radial mode has therefore,
$\unicode[STIX]{x1D6FC}_{j}=2.4048$
. The transmitted wave propagates into the cylindrical, hard-walled nozzle, either as a plane wave,
$\unicode[STIX]{x1D6FC}_{n}=0$
, or as a wave with higher-order radial structure, characterised by
$\unicode[STIX]{x1D6FC}_{n}=3.8,\ldots$
.

The cut-on condition for nozzle modes of radial order
$\unicode[STIX]{x1D6FC}_{n}$
is,

For the Mach number range of interest,
$0.6\leqslant M\leqslant 0.82$
, the corresponding cut-on Strouhal number range for the first radial pipe mode,
$\unicode[STIX]{x1D6FC}_{n}=3.8$
, is well above the frequencies of interest. The incident
$k^{-}$
waves will therefore be transmitted as plane waves.

Using the Fourier-transform convention,

the pressure fields inside and outside the nozzle that we are interested in are, therefore, respectively,

where the
$\text{e}^{-\text{i}\unicode[STIX]{x1D714}t}$
has been dropped for convenience, and the momentum balance at the exit plane,
$x=0$
, reads,

In terms of the velocity fluctuation, upstream of the exit plane we have,

at the nozzle exit plane,
$x=0$
,

and downstream, considering momentum balance separately for the
$k^{+}$
and
$k^{-}$
components of the fluctuations,

giving,

The mass balance at the nozzle exit plane is

giving for the amplitude of the transmitted wave,

Combining (C 9) and (C 18) to eliminate
$T$
,

From which the reflection coefficient can be obtained,

## Appendix D. Complex resonance analysis

As described in § 4.6 we find triplets
$[k^{+},k^{-},\unicode[STIX]{x1D714}]\in \mathbb{C}$
that simultaneously solve (4.7), (4.6) and (4.3). This provides the resonance-frequency predictions shown in figure 17, for
$R_{1}R_{2}=0.002$
, where they are compared with the experimental data and predictions of the neutral-mode model. Similar trends are obtained for
$|R_{1}R_{2}|=0.004$
and
$0.008$
, the main difference being the progressive lowering of the resonance cutoff frequency discussed in § 4.6. Top and bottom plots show model results obtained, respectively, with assumptions of in-phase and out-of-phase reflection conditions.

For
$L=2$
the neutral- and complex-mode model predictions, shown, respectively, in cyan and red, are globally similar at frequencies for which the complex-mode model predicts resonance. For plate-edge positions
$L=3$
and
$4$
discrepancies are apparent, the complex analysis obtaining globally poorer agreement with the data. The discrepancies include: (i) resonance-frequency underprediction at low Mach number; (ii) resonance-frequency overprediction at high Mach number; (iii) a non-monotonic Mach number dependence of certain resonance-mode frequencies.

The discrepancies can be understood by looking at how and why the
$k^{+}$
and
$k^{-}$
branches are deformed with respect to the neutral-mode model. Figure 18 shows these in the
$St-k_{r}$
plane. The neutral-mode model branches are shown in cyan and red for the
$k^{-}$
waves, and in blue for the
$k^{+}$
waves. The deformed branches of the complex-mode model are shown by the black dots. The left plot of figure 20 shows a sparsed zoom for
$L=4$
, with successive complex-mode branches here shown, alternatingly, in red and black, for ease of visualisation.

At high frequency, where the upstream-travelling modes are trapped, propagative and duct-like in the neutral-mode model, there is little
$k^{-}$
branch deformation in the
$St-k_{r}$
plane. Note, on the other hand, that in the
$k_{r}-k_{i}$
plane (cf. figure 21), branch deformation occurs, with the resonance-admissible
$k^{-}$
waves becoming spatially evanescent, and the
$k^{+}$
less spatially unstable. This trend is due to the constraint,
$\text{e}^{\unicode[STIX]{x0394}k_{i}L}=R_{1}R_{2}$
, that requires the imaginary parts of the wavenumbers to approach one another as dictated by the reflection-coefficient product and the distance between the nozzle exit plane and the plate edge.

As frequency decreases a more marked deformation occurs in the
$St-k_{r}$
plane. This is greatest in the frequency range where phase and group velocities evolve from subsonic to sonic values. With further decrease in frequency the complex-mode
$k^{-}$
branches realign with the neutral-mode branches; again, this is only seen in the
$St-k_{r}$
plane, as is clear from figure 21. The complex-mode branches are truncated at the threshold frequency
$\unicode[STIX]{x1D714}_{r}|_{\unicode[STIX]{x1D714}_{i}=0}$
, providing a low-frequency resonance cutoff; for lower
$\unicode[STIX]{x1D714}_{r}$
solutions satisfying (4.7) and (4.6) are only found for negative
$\unicode[STIX]{x1D714}_{i}$
, indicating a damped resonance which should not lead to significant tones in the power spectral density of flow fluctuations. The
$St-k_{r}$
branch deformations become more pronounced as
$L$
is increased, and for
$L=4$
there is the appearance of what looks like a discontinuity. Figure 21 illustrates how this is due to the invalid (from an absolute stability point of view (Huerre & Monkewitz 1990))
$k^{-}/k^{-}$
saddle point discussed in Towne *et al.* (2017) (cf. figure 12(*d*) in that paper).

The consequences of the above for resonance-frequency prediction is illustrated in figure 19, which shows
$\unicode[STIX]{x0394}k_{r}(St)$
and the resonance criteria
$2n\unicode[STIX]{x03C0}/L$
and
$(2n+1)\unicode[STIX]{x03C0}/L$
, (solid and dash-dotted horizontal red lines, respectively). The right plot of figure 20 again shows a sparsed zoom for
$L=4$
. The branch deformation produced by the saddle-point results, for a given Mach number, in low-frequency resonant modes that occur at lower frequency in comparison to the neutral-mode model, and higher-frequency modes that can potentially occur at equal or higher frequencies. The Mach number dependence of a given resonance mode is sensitive to the details of the Mach number dependence of the
$\unicode[STIX]{x1D714}(k)$
landscape (cf. figure 21), particularly so in the neighbourhood of the saddle point; and to the influence of this on the resonance amplitude constraint,
$\text{e}^{\unicode[STIX]{x0394}k_{i}L}=R_{1}R_{2}$
, that requires the imaginary parts of the
$k^{+}$
and
$k^{-}$
waves to be adjusted as discussed earlier.

It is here that the fragility of the complex-mode model becomes apparent: in order to make accurate resonance-frequency predictions, we require not only an accurate knowledge of the Mach number dependence of
$\unicode[STIX]{x1D714}(k)$
, but also an accurate knowledge of the Mach number and frequency dependence of
$R_{1}$
and
$R_{2}$
and the further dependence of
$R_{2}$
on the edge position,
$L$
. The assumption that
$R_{1}R_{2}$
be independent of these makes the model fragile when the branch of resonance-admissible
$k^{-}$
eigenvalues enters the neighbourhood of the saddle point. Here, high gradients of
$\unicode[STIX]{x1D714}(k)$
lead to rapid variation in values of
$k_{r}$
associated with values of
$k_{i}$
as constrained by
$\text{e}^{\unicode[STIX]{x0394}k_{i}L}=R_{1}R_{2}$
.

The conclusion of the analysis is that the neutral-mode model, despite lacking certain aspects of the flow physics, can provide reasonable tone predictions over a limited frequency range, showing that in this range the resonance phenomenon is relatively insensitive to the missing physics. The complex model, on the other hand, despite its fragility in the absence of detailed information regarding the reflection conditions, provides an indication of where the neutral-mode model will fail, at low frequency for instance.

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