2 Mathematical model

The kinematic wavepacket sound-source model is based on Lighthill’s acoustic analogy, where the fluctuating sound pressure, $p$ , is given by the inhomogeneous wave equation

(2.1) $$\begin{eqnarray}\unicode[STIX]{x1D6FB}^{2}p-\frac{1}{c_{0}^{2}}\frac{\unicode[STIX]{x2202}^{2}p}{\unicode[STIX]{x2202}t^{2}}=S(\boldsymbol{y},t),\end{eqnarray}$$

where $\boldsymbol{y}$ are the source coordinates, $c_{0}$ is the ambient speed of sound, $t$ is time and $S(\boldsymbol{y},t)$ is the source term expressed as the double divergence of Lighthill’s stress tensor along the jet axis. The Helmholtz equation can be obtained via a Fourier transform of (2.1) to arrive at

(2.2) $$\begin{eqnarray}\unicode[STIX]{x1D6FB}^{2}p+k^{2}p=S(\boldsymbol{y},\unicode[STIX]{x1D714}),\end{eqnarray}$$

where $k=\unicode[STIX]{x1D714}/c_{0}$ .

Using a free-field Green’s function $G(\boldsymbol{x},\boldsymbol{y},\unicode[STIX]{x1D714})$ , the integral solution to the Helmholtz equation (2.2) is,

(2.3) $$\begin{eqnarray}p(\boldsymbol{x},\unicode[STIX]{x1D714})=\int _{V}S(\boldsymbol{y},\unicode[STIX]{x1D714})G(\boldsymbol{x},\boldsymbol{y},\unicode[STIX]{x1D714})\,\text{d}\boldsymbol{y},\end{eqnarray}$$

where the integration is carried out over the region $V$ where $S\neq 0$ and $\boldsymbol{x}$ are the observer coordinates.

As proposed by Tam (Reference Tam1990) and Lele (Reference Lele2005), the full three-dimensional source term for BBSAN can be represented as a one-dimensional multiplicative combination of the shock-cell $u_{s}$ and turbulent $u_{t}$ velocity fluctuations

(2.4) $$\begin{eqnarray}S(\boldsymbol{y},t)\simeq {\hat{S}}(y,t)=u_{s}(y)u_{t}(y,t),\end{eqnarray}$$

where ${\hat{S}}(y,t)$ is a line-source model (azimuthal and radial dependences are not considered). The coordinate vector $\boldsymbol{y}$ is now replaced by the axial position coordinate $y$ . This modelling of acoustic sources along a line thus only accounts for axisymmetric wavepacket fluctuations.

One approach to model the disturbances due to the quasi-periodic train of shock cells is to regard the jet mixing layer as a waveguide (Prandtl Reference Prandtl1904; Pack Reference Pack1950). By approximating the mixing layer of the jet as a vortex sheet, the disturbances due to the shock cells can be decomposed into a set of spatially periodic functions. Each of these periodic functions can be thought of as a waveguide mode of the jet. In one dimension, the velocity fluctuations related to the shock-cell waveguide modes along the axial direction, $u_{s}(y)$ , is represented as (Prandtl Reference Prandtl1904; Pack Reference Pack1950)

(2.5) $$\begin{eqnarray}u_{s}(y)=U_{s}\mathop{\sum }_{n}c_{s_{n}}\frac{1}{2}\{\text{e}^{\text{i}k_{s_{n}}y}+\text{e}^{-\text{i}k_{s_{n}}y}\}.\end{eqnarray}$$

The shock-cell waveguide modes are described by the wavenumbers $k_{s_{n}}$ and the amplitude terms $c_{s_{n}}$ where we adopt the expression from Prandtl & Pack’s vortex sheet model,

(2.6) $$\begin{eqnarray}\displaystyle & \displaystyle k_{s_{n}}=\frac{2\unicode[STIX]{x1D70E}_{n}}{D_{j}(M_{j}^{2}-M_{d}^{2})^{1/2}},\quad n=1,2,3\ldots , & \displaystyle\end{eqnarray}$$

(2.7) $$\begin{eqnarray}\displaystyle & \displaystyle c_{s_{n}}=\frac{2\unicode[STIX]{x1D6E5}p}{\unicode[STIX]{x1D70E}_{n}p_{\infty }},\quad n=1,2,3\ldots , & \displaystyle\end{eqnarray}$$

where $\unicode[STIX]{x1D70E}_{n}$ is the $n$ th zero crossing of the zeroth-order Bessel function, $D_{j}$ and $M_{j}$ are respectively, the ideally expanded diameter and Mach number of the jet. $M_{d}$ is the design Mach number which is equal to unity for a convergent nozzle. The amplitude term $c_{s_{n}}$ is the ratio between the pressure imbalance $\unicode[STIX]{x1D6E5}p$ at the throat of the nozzle and the ambient pressure $p_{\infty }$ . The amplitude decay of the shock modes over axial distance as seen in experimental measurements, however, is not calculated or accounted for. The overall scaling amplitude of the shock cells is represented by $U_{s}$ . The complete solution for the vortex sheet shock-cell model can be found in Pack (Reference Pack1950).

To model $u_{t}$ , Lele (Reference Lele2005) used a wavepacket whose amplitude is modulated in both space and time. The wavepacket, at a given axial position $y$ , is defined by its envelope length scale $L$ , hydrodynamic wavenumber $k_{h}$ and frequency $\unicode[STIX]{x1D714}$ ; the two latter quantities being related by the convection velocity $k_{h}=\unicode[STIX]{x1D714}/U_{c}$ . The explicit single-point form of $u_{t}$ with amplitude $U_{t}$ is,

(2.8) $$\begin{eqnarray}u_{t}(y,t)=U_{t}\text{e}^{-(y/L)^{2}}\text{e}^{\text{i}(k_{h}y-\unicode[STIX]{x1D714}t)}.\end{eqnarray}$$

To model the jitter of the wavepackets, Lele introduced a temporal modulation term involving stochastic realisations. The work of Cavalieri & Agarwal (Reference Cavalieri and Agarwal2014), however, established that coherence decay can provide an alternative statistical representation of jitter. Therefore, rather than including a temporal dependence, we use two-point statistics to model the wavepacket’s stochastic behaviour. After taking the Fourier transform of the proposed source model ${\hat{S}}(y,\unicode[STIX]{x1D714})$ , the source term at a single point $y$ is now given by

(2.9) $$\begin{eqnarray}{\hat{S}}(y,\unicode[STIX]{x1D714})=A(\unicode[STIX]{x1D714})\text{e}^{-(y/L(\unicode[STIX]{x1D714}))^{2}}\{\text{e}^{\text{i}k_{h}(\unicode[STIX]{x1D714})y}\}\mathop{\sum }_{n}c_{s_{n}}\{\text{e}^{\text{i}k_{s_{n}}y}+\text{e}^{-\text{i}k_{s_{n}}y}\},\end{eqnarray}$$

where an implicit factor of $\exp (-\text{i}\unicode[STIX]{x1D714}t)$ is assumed and the $U_{s}$ and $U_{t}$ amplitude terms in (2.5) and (2.8) have been absorbed into the amplitude term $A(\unicode[STIX]{x1D714})$ .

While equation (2.9) allows direct computation of the fluctuating pressure field, the Fourier transform that would provide the source term $S(\boldsymbol{y},\unicode[STIX]{x1D714})$ cannot be evaluated as it involves an integrand that is not square integrable (Landahl & Mollo-Christensen Reference Landahl and Mollo-Christensen1992; Cavalieri & Agarwal Reference Cavalieri and Agarwal2014). Fluctuations in a turbulent jet comprise a stationary random process, and are best described through statistical metrics such as variance, autocorrelation and cross-correlation. In the frequency domain, a particularly rich statistical metric is the cross-spectral density, which is the Fourier transform of the cross-correlation, but which can also be defined as the expected value ${\mathcal{E}}(\hat{u} \hat{u} ^{\ast })$ , where $\hat{u}$ is the Fourier transform taken for a given realisation, and ${\mathcal{E}}$ is the expected-value operator, which is the asymptotic limit of an ensemble average. Using power-spectral densities (PSDs) and cross-spectral densities (CSDs), the Fourier transforms of the autocorrelation and cross-correlation functions, respectively, we express the far-field sound pressure level (SPL) as

(2.10) $$\begin{eqnarray}\langle p(\boldsymbol{x},\unicode[STIX]{x1D714})p^{\ast }(\boldsymbol{x},\unicode[STIX]{x1D714})\rangle \approx \int _{V}\int _{V}\langle S(y_{1},\unicode[STIX]{x1D714})S^{\ast }(y_{2},\unicode[STIX]{x1D714})\rangle G(\boldsymbol{x},y_{1},\unicode[STIX]{x1D714})G^{\ast }(\boldsymbol{x},y_{2},\unicode[STIX]{x1D714})\,\text{d}y_{1}\,\text{d}y_{2},\end{eqnarray}$$

where both PSDs and CSDs are obtained by multiplying by the complex conjugate between position $y_{1}$ and $y_{2}$ and the hats have been dropped for convenience. The free-field Green’s function is

(2.11) $$\begin{eqnarray}G(\boldsymbol{x},\boldsymbol{y},\unicode[STIX]{x1D714})=\frac{1}{4\unicode[STIX]{x03C0}}\frac{\text{e}^{\text{i}k|\boldsymbol{x}-\boldsymbol{y}|}}{|\boldsymbol{x}-\boldsymbol{y}|}.\end{eqnarray}$$

To obtain the two-point source term with unit coherence at position $y_{1}$ , we multiply equation (2.9) by its complex conjugate at position $y_{2}$

(2.12) $$\begin{eqnarray}\displaystyle S(y_{1},\unicode[STIX]{x1D714})S^{\ast }(y_{2},\unicode[STIX]{x1D714}) & = & \displaystyle A^{2}(\unicode[STIX]{x1D714})\text{e}^{-(y_{1}/L(\unicode[STIX]{x1D714}))^{2}}\text{e}^{-(y_{2}/L(\unicode[STIX]{x1D714}))^{2}}\{\text{e}^{\text{i}k_{h}(\unicode[STIX]{x1D714})(y_{1}-y_{2})}\}\nonumber\\ \displaystyle & & \displaystyle \times \,\mathop{\sum }_{n}c_{s_{n}}\{\text{e}^{\text{i}k_{s_{n}}y_{1}}+\text{e}^{-\text{i}k_{s_{n}}y_{1}}\}\mathop{\sum }_{m}c_{s_{m}}\{\text{e}^{\text{i}k_{s_{m}}y_{2}}+\text{e}^{-\text{i}k_{s_{m}}y_{2}}\}.\end{eqnarray}$$

We now define the coherence function between points $y_{1}$ and $y_{2}$ as

(2.13) $$\begin{eqnarray}\unicode[STIX]{x1D6FE}^{2}(y_{1},y_{2},\unicode[STIX]{x1D714})=\frac{|\langle S(y_{1},\unicode[STIX]{x1D714})S^{\ast }(y_{2},\unicode[STIX]{x1D714})\rangle |^{2}}{\langle |S(y_{1},\unicode[STIX]{x1D714})|^{2}\rangle \langle |S(y_{2},\unicode[STIX]{x1D714})|^{2}\rangle }\end{eqnarray}$$

modelled as a Gaussian function following Cavalieri & Agarwal (Reference Cavalieri and Agarwal2014),

(2.14) $$\begin{eqnarray}\unicode[STIX]{x1D6FE}^{2}(y_{1},y_{2},\unicode[STIX]{x1D714})=\exp \left(-2\frac{(y_{1}-y_{2})^{2}}{L_{c}^{2}(\unicode[STIX]{x1D714})}\right).\end{eqnarray}$$

The coherence decay between points $y_{1}$ and $y_{2}$ is now defined by the characteristic coherence length scale $L_{c}$ . Introducing this effect by multiplying equations (2.12) and (2.14), we arrive at the CSD of the two-point source model for broadband shock-associated noise

(2.15) $$\begin{eqnarray}\displaystyle S(y_{1},\unicode[STIX]{x1D714})S^{\ast }(y_{2},\unicode[STIX]{x1D714}) & = & \displaystyle A^{2}(\unicode[STIX]{x1D714})\text{e}^{-(y_{1}/L(\unicode[STIX]{x1D714}))^{2}}\text{e}^{-(y_{2}/L(\unicode[STIX]{x1D714}))^{2}}\{\text{e}^{\text{i}k_{h}(\unicode[STIX]{x1D714})(y_{1}-y_{2})}\}\text{e}^{(-(y_{1}-y_{2})^{2}/L_{c}(\unicode[STIX]{x1D714})^{2})}\nonumber\\ \displaystyle & & \displaystyle \times \,\mathop{\sum }_{n}c_{s_{n}}\{\text{e}^{\text{i}k_{s_{n}}y_{1}}+\text{e}^{-\text{i}k_{s_{n}}y_{1}}\}\mathop{\sum }_{m}c_{s_{m}}\{\text{e}^{\text{i}k_{s_{m}}y_{2}}+\text{e}^{-\text{i}k_{s_{m}}y_{2}}\}.\end{eqnarray}$$

For a given pair of points $(y_{1},y_{2})$ , the source term is described by two wavepacket envelope terms $\exp (-y_{1}/L(\unicode[STIX]{x1D714}))^{2}$ and $\exp (-y_{2}/L(\unicode[STIX]{x1D714}))^{2}$ at the two points respectively. The term $\exp [\text{i}k_{h}(\unicode[STIX]{x1D714})(y_{1}-y_{2})]$ describes the phase difference between $y_{1}$ and $y_{2}$ while the coherence decay is modelled by $\exp [-(y_{1}-y_{2})^{2}/L_{c}^{2}(\unicode[STIX]{x1D714})]$ . Finally this is multiplied by the shock-cell modes at points $y_{1}$ and $y_{2}$ by the expression $\sum [\exp (\text{i}k_{s}y_{1})+\exp (-\text{i}k_{s}y_{1})]\sum [\exp (\text{i}k_{s}y_{2})+\exp (-\text{i}k_{s}y_{2})]$ . The frequency dependence notation of $k_{h}$ , $L$ and $L_{c}$ , while implied, is now hereafter omitted for convenience.

Similar to Cavalieri & Agarwal (Reference Cavalieri and Agarwal2014), the model is now governed by two characteristic length scales. The first length scale, $L$ , characterises the wavepacket amplitude envelope. The second, $L_{c}$ , is the coherence length scale which characterises the decay of coherence between two points along the axial direction. It should be noted that as $L_{c}\rightarrow \infty$ , the two-point model (2.15) reduces to the unit-coherence model (2.12).

The far-field sound pressure for both models can now be found by inserting equations (2.15) and (2.12) into equation (2.10). Using the usual Fraunhofer far-field approximation where $|\boldsymbol{x}-\boldsymbol{y}|\approx |\boldsymbol{x}|-\hat{\boldsymbol{x}}\boldsymbol{\cdot }\boldsymbol{y}$ (Crighton Reference Crighton1975; Howe Reference Howe2003), we arrive at the expression

(2.16) $$\begin{eqnarray}\langle p(\boldsymbol{x},\unicode[STIX]{x1D714})p^{\ast }(\boldsymbol{x},\unicode[STIX]{x1D714})\rangle \approx \frac{A^{2}(\unicode[STIX]{x1D714})}{4\unicode[STIX]{x03C0}x^{2}}\int \int \langle S(y_{1},\unicode[STIX]{x1D714})S^{\ast }(y_{2},\unicode[STIX]{x1D714})\rangle \text{e}^{\text{i}k\cos \unicode[STIX]{x1D703}(y_{1}-y_{2})}\,\text{d}y_{1}\,\text{d}y_{2},\end{eqnarray}$$

where $\unicode[STIX]{x1D703}$ is taken as the angle from the downstream jet axis. Due to the line-source approximation for this model, the double volume integral reduces to a double integral in the streamwise direction.

3 Acoustic efficiency and directivity

3.1 Parameters of the source model

Our objective here is to study the impact of coherence decay on BBSAN in a model problem. To do so, we must first specify values of the hydrodynamic terms $(k_{h},A(\unicode[STIX]{x1D714}),L)$ in (2.16). The coherence length term $L_{c}$ must also be specified. The chosen modelling parameters are listed in table 1 and given in appendix A; the justification of these values is discussed below.

The first parameter, $k_{h}$ is the hydrodynamic wavenumber of the wavepacket defined as $k_{h}=\unicode[STIX]{x1D714}/u_{c}$ . We consider the average convection velocity of the wavepackets to be $u_{c}\approx 0.6U_{j}$ , consistent with the literature (Harper-Bourne & Fisher Reference Harper-Bourne and Fisher1973; Lau Reference Lau1980; Troutt & Mclaughlin Reference Troutt and Mclaughlin1982; Kerhervé, Fitzpatrick & Jordan Reference Kerhervé, Fitzpatrick and Jordan2006; Morris & Zaman Reference Morris and Zaman2010).

The second term $A(\unicode[STIX]{x1D714})$ represents the wavepacket amplitude. While there is no theoretical form for shock-containing flows, it has been previously measured in experimental campaigns (Bridges & Wernet Reference Bridges and Wernet2008; Savarese et al. Reference Savarese, Jordan, Girard, Royer, Fourment, Collin, Gervais and Porta2013). More recently, Antonialli et al. (Reference Antonialli, Cavalieri, Schmidt, Colonius, Jordan, Towne, Brés and Agarwal2018) were able to determine the frequency dependence of wavepacket amplitudes in a subsonic Mach 0.9 jet by comparing large-eddy simulation data of Brès et al. (Reference Brès, Ham, Nichols and Lele2017) to the fluctuation fields predicted from the parabolised stability equations model of Sasaki et al. (Reference Sasaki, Piantanida, Cavalieri and Jordan2017b ). We therefore model the wavepacket amplitude term using an energy-spectrum function as proposed by Antonialli et al. (Reference Antonialli, Cavalieri, Schmidt, Colonius, Jordan, Towne, Brés and Agarwal2018)

(3.1) $$\begin{eqnarray}A(\unicode[STIX]{x1D714})=C_{1}\text{e}^{-C_{2}\unicode[STIX]{x1D714}},\end{eqnarray}$$

where terms $C_{1}$ and $C_{2}$ are fitting parameters with values $3.4\times 10^{-7}$ and $0.58$ respectively. The value of $C_{2}$ has been normalised based on the Strouhal number of $St=fD/U_{j}$ . A similar exponential decay spectrum was also used by Suzuki (Reference Suzuki2016). The numerical value of $A(\unicode[STIX]{x1D714})$ is obtained by fitting the model (3.1) to the measured velocity spectra (Savarese et al. Reference Savarese, Jordan, Girard, Royer, Fourment, Collin, Gervais and Porta2013) obtained along the shear layer at an axial position $y/D\approx 3$ for a jet operating at $NPR=2.5$ . The amplitude term is normalised to yield a source strength of unity at the peak frequency.

As we do not have at our disposal the wavepacket length scales $L$ and $L_{c}$ for supersonic jets, we adopt values from previous work on subsonic flows. The objective of this study is not to develop a predictive capability but rather to determine the impact of coherence decay in a model problem. Hence, in the same spirit as Cavalieri & Agarwal (Reference Cavalieri and Agarwal2014) who used average values independent of frequency, we evaluate the source term in (2.16) for $L=1.0D$ and $k_{h}L=5$ . From the two-point measurements of Jaunet et al. (Reference Jaunet, Jordan and Cavalieri2017) for a $M=0.4$ jet, it is evident that coherence lengths have a frequency and axial position dependence. However, without prior measurements of this dependence in shock-containing flows, we adopt for this study an average value of $L_{c}=1.0D$ similar to Cavalieri & Agarwal (Reference Cavalieri and Agarwal2014) and Baqui et al. (Reference Baqui, Agarwal, Cavalieri and Sinayoko2015). It should be noted, however, that Suzuki (Reference Suzuki2016) did extract the coherence length of the near-field pressure in an underexpanded supersonic jet. An approximately constant coherence length scale was found over a range of frequencies, further suggesting that an average value is suitable for the purposes of this study. The sensitivity of the modelling choices have been investigated with some results presented in appendix A.

3.2 Far-field acoustic predictions

The far-field sound is obtained by numerical evaluation of (2.16). Figure 1 shows the variation of far-field SPL as a function of emission angle and frequency for an underexpanded jet operating at $NPR=3.4$ . The modelled jet issues from a convergent nozzle, which corresponds to an off-design parameter of $\unicode[STIX]{x1D6FD}=1.04$ . In comparing the case between unit coherence and coherence decay, all tuning parameters as specified in § 3.1 with the exception of $L_{c}$ are kept constant. The far-field sound pressure contours were obtained using the first ten shock-cell modes ( $n=10$ ). The use of the number of modes is justified in § 5. It is clear from figure 1 that coherence decay has a significant effect on the BBSAN spectrum. Consistent with experimental observations, both plots comprise a peak frequency that increases with decreasing observation angle; though the effect is more evident in the unit-coherence case.

Figure 1. Contours of sound pressure level (arbitrary dB) as a function of frequency ( $St$ ) and directivity ( $\unicode[STIX]{x1D703}$ ) for (a) unit coherence and (b) with coherence decay. The jet issues from a converging nozzle ( $M_{d}=1.0$ ) at a nozzle pressure ratio of $NPR=3.4$ which corresponds to a fully expanded jet Mach number of $M_{j}=1.45$ and an off-design parameter of $\unicode[STIX]{x1D6FD}=1.04$ . The dashed line indicates the peak frequency as predicted by Harper-Bourne & Fisher (Reference Harper-Bourne and Fisher1973).

The first stage of the analysis considers cases involving a single shock-cell mode ( $n=1$ ); the centreline pressure fluctuations in a moderately underexpanded jet are reasonably well represented by a single mode (Tam, Jackson & Seiner Reference Tam, Jackson and Seiner1985; Ray & Lele Reference Ray and Lele2007). Figure 2 shows the directivity for far-field SPL at $St=0.3$ and $St=0.6$ for the same conditions as figure 1 but with only the fundamental shock-cell mode ( $n=1$ ) included. The Strouhal number is defined as $St=fD/U_{j}$ . Models with unit coherence and coherence decay are plotted on the same figure for comparison. At  $St=0.3$ , both models predict the highest amplitude of radiation in the direction slightly upstream of perpendicular, consistent with previous findings. At $St=0.6$ , the BBSAN peaks shift downstream but with a smaller sound amplitude. With all other terms equal, the introduction of coherence decay broadens the radiation lobe, increasing the SPL in the downstream direction (low $\unicode[STIX]{x1D703}$ values). Contrary to the subsonic jet case (Cavalieri et al. Reference Cavalieri, Jordan, Agarwal and Gervais2011; Baqui et al. Reference Baqui, Agarwal, Cavalieri and Sinayoko2015, amongst others), however, the introduction of coherence decay reduces the peak amplitude by approximately 2–5 dB. The reason for this behaviour will be further explored in § 4. It is also evident, from the peak frequency trends in figures 1 and 2, that the current wavepacket model agrees with the predictions made by localised phased-array models such as Harper-Bourne & Fisher (Reference Harper-Bourne and Fisher1973).

Figure 2. Sound pressure level at a distance of $100D$ for a wavepacket frequency of (a) $St=0.3$ and (b) $St=0.6$ as a function of observation angle $\unicode[STIX]{x1D703}$ for a wavepacket with $k_{h}L=5$ .

Figure 3 shows a comparison of the noise spectra between the two models and experimental data for an underexpanded jet operating at $NPR=3.4$ (Norum & Seiner Reference Norum and Seiner1982). The modelled peak amplitudes are adjusted to match experimental data in order to facilitate comparison of the spectral shape. As can be seen, while there is reasonable agreement between the two models and the measured data in terms of the peak frequency, the overall agreement between the two models and the data is moderate. Both the unit-coherence and coherence-decay models capture the BBSAN peak frequency dependence and the narrowing of the spectral peak with increasing angle. With the inclusion of coherence decay, however, the BBSAN spectral peak width broadens leading to a more favourable agreement for frequencies greater than the peak. Below $\unicode[STIX]{x1D703}=75^{\circ }$ , both models fail to capture the correct peak frequency. A possible explanation for this disagreement in peak frequency at downstream angles could be due to the choice of the $u_{c}$ and $L$ modelling variables, as discussed in appendix A, or the dominance of jet mixing noise close to the jet axis.

Figure 3. Power spectrum at a distance of $100D$ through a range of observation angles between $\unicode[STIX]{x1D703}=60^{\circ }$ and $\unicode[STIX]{x1D703}=150^{\circ }$ measured from the downstream jet axis for a wavepacket with $k_{h}L=5$ . Each measurement angle is offset by $\unicode[STIX]{x0394}dB=25$ . NASA experimental data from Norum & Seiner (Reference Norum and Seiner1982).

It is evident from figure 3 that the jitter of wavepackets, modelled by coherence decay, broadens the acoustic spectrum. The model suggests, however, that coherence decay does not have a major impact on the sound amplitude. Unlike in subsonic flows, it provides little contribution to the peak SPL. This is also consistent with the results presented by Sinha et al. (Reference Sinha, Rodríguez, Brès and Colonius2014) for isothermal fully expanded supersonic jets, where it was found that the far-field noise spectrum at downstream observation angles is well captured even without considering the jitter of wavepackets. The reason for this is because in supersonic ideally expanded flows, wavepackets propagate downstream with supersonic phase velocities. As a result, noise in the form of Mach wave radiation is generated efficiently (Tam Reference Tam1995) in the downstream direction. On the other hand, in shock-containing flows, the presence of shock cells generates an additional component which travels upstream. The effect of jittering, on both upstream- and downstream-travelling components, is discussed in more detail in § 4.

Using a single shock-cell waveguide mode, both wavepacket models show reasonable agreement with the peak of the measured spectrum. However, they suffer from the same issue discussed by Ray & Lele (Reference Ray and Lele2007) where the high-frequency sound at upstream angles is ‘missing’. In their study, for a $M_{j}=1.22$ isothermal jet, the frequency range of interest was restricted to $St<1$ due to the assumed breakdown of linear theory at high frequencies. More recently, however, it has been shown by Sasaki et al. (Reference Sasaki, Cavalieri, Jordan, Schmidt, Colonius and Brès2017a ) that linear theory still yields good agreement for frequencies up to $St=4$ . Therefore, we argue here that the drop off in high frequency is not due to the breakdown of linear theory but rather neglecting to include higher-order shock-cell modes. This is further supported by Suzuki’s wavepacket model where a similar underestimation of high-frequency SPL was observed when using an empirical representation of the shock cells. The effect of including higher-order modes is discussed in § 5.

4 Interpretation of sound-radiation characteristics

4.1 Fourier transform into wavenumber space

In order to explore how coherence affects the source structure and sound-radiation characteristics, the CSD of the models with and without coherence decay are transformed to wavenumber space. This transformation is achieved with the double Fourier transform,

(4.1) $$\begin{eqnarray}{\mathcal{F}}(k_{y_{1}},k_{y_{2}})=\frac{1}{(\sqrt{2\unicode[STIX]{x03C0}})^{2}}\int _{-\infty }^{\infty }\int _{-\infty }^{\infty }F(y_{1},y_{2})\text{e}^{\text{i}k_{y_{1}}y_{1}}\text{e}^{\text{i}k_{y_{2}}y_{2}}\,\text{d}y_{1}\,\text{d}y_{2},\end{eqnarray}$$

where $F(y_{1},y_{2})$ is the two-point expression of the CSD. If we take coherence as unity for the entire domain by inserting (2.12) into (4.1), the Fourier transform for the perfectly coherent model for a single shock-cell mode is

(4.2) $$\begin{eqnarray}\displaystyle {\mathcal{F}}(k_{y_{1}},k_{y_{2}}) & = & \displaystyle \frac{1}{(\sqrt{2\unicode[STIX]{x03C0}})^{2}}\int _{-\infty }^{\infty }\int _{-\infty }^{\infty }A_{n=1}^{2}(\unicode[STIX]{x1D714})\text{e}^{-(y_{1}/L)^{2}}\text{e}^{-(y_{2}/L)^{2}}\{\text{e}^{\text{i}k_{s}y_{1}}+\text{e}^{-\text{i}k_{s}y_{1}}\!\}\nonumber\\ \displaystyle & & \displaystyle \times \,\{\text{e}^{\text{i}k_{s}y_{2}}+\text{e}^{-\text{i}k_{s}y_{2}}\}\{\text{e}^{\text{i}k_{h}(y_{1}-y_{2})}\}\text{e}^{\text{i}k_{y_{1}}y_{1}}\text{e}^{\text{i}k_{y_{2}}y_{2}}\,\text{d}y_{1}\,\text{d}y_{2}.\end{eqnarray}$$

Evaluating the above integral gives

(4.3) $$\begin{eqnarray}\displaystyle {\mathcal{F}}(k_{y_{1}},k_{y_{2}}) & = & \displaystyle \frac{A_{n=1}^{2}(\unicode[STIX]{x1D714})}{8}(\text{e}^{-(1/4)(k_{h}-k_{s}+k_{y_{1}})^{2}L^{2}}+\text{e}^{-(1/4)(k_{h}+k_{s}+k_{y_{1}})^{2}L^{2}})\nonumber\\ \displaystyle & & \displaystyle \times (\text{e}^{-(1/4)(k_{h}+k_{s}-k_{y_{2}}\!)^{2}L^{2}}+\text{e}^{-(1/4)(-k_{h}+k_{s}+k_{y_{2}})^{2}L^{2}})L^{2},\end{eqnarray}$$

which is the wavenumber spectrum of the perfectly coherent source CSD. Likewise, by inserting (2.15) into (4.1) and evaluating the resulting integral, the Fourier-transformed CSD of the coherence decay model can also be obtained (the solution is presented in appendix B). Equation (4.3) is obtained from (B 1) by taking the limit $L_{c}\rightarrow \infty$ .

Both source CSD models in physical space are shown in figure 4(a,b). In this model problem, the jet nozzle is not present and the sources are simply centred at the origin and extended in both positive and negative directions along the jet axis. The peaks in the contour map are similar to the freckled appearance seen in the two-point pressure correlations obtained by Suzuki (Reference Suzuki2016). As noted by Suzuki, the spacing between the peaks on the diagonal approximately correspond to the shock-cell spacings. The off-diagonal peaks correspond to the wavepacket interacting with adjacent shock cells. The introduction of coherence decay concentrates the sources in space; along the axis $y_{1}=y_{2}$ . This behaviour is also seen in the non-shock-containing case, as found by Cavalieri & Agarwal (Reference Cavalieri and Agarwal2014).

Figure 4. The real part of the CSD of the unit coherence (a) and with statistical decay (b) models for $L_{c}=1.0D$ . The diagonal line represents $y_{1}=y_{2}$ . The corresponding Fourier-transformed CSD in wavenumber space with the unit-coherence (c) and statistical decay (d) models. Diagonal line corresponds to $k_{y1}=-k_{y2}$ . The square represents the acoustic matching criterion $|k|/k_{h}=0.6M_{j}$ . The amplitudes of both models have been normalised to highlight the effect of coherence decay. The wavepacket frequency is $St=0.4$ for a jet operating at $\unicode[STIX]{x1D6FD}=1.04$ .

A comparison of the models’ Fourier-transformed CSDs as given by (4.3) and (B 1) is shown in figure 4(c,d) respectively. The contour scale is logarithmic and both axes are normalised by the hydrodynamic wavenumber of the wavepacket $k_{h}$ . The source term in both cases produces four distinct lobes. The existence and ramifications of these are discussed in detail in § 4.2. The introduction of coherence decay stretches the lobes parallel to the $k_{y1}=-k_{y2}$ axis. This is consistent with what has been observed in subsonic jets (Cavalieri & Agarwal Reference Cavalieri and Agarwal2014; Jaunet et al. Reference Jaunet, Jordan and Cavalieri2017).

As noted by Crighton (Reference Crighton1975), only certain spectral components of the source term corresponding to supersonic phase speeds can contribute to far-field noise. In order to isolate only the radiating wavenumbers of the source term, the following conditions need to be met

(4.4a ) $$\begin{eqnarray}\displaystyle & \displaystyle \frac{|k_{y1}|}{k_{h}}\leqslant M_{c} & \displaystyle\end{eqnarray}$$

(4.4b ) $$\begin{eqnarray}\displaystyle & \displaystyle \frac{|k_{y2}|}{k_{h}}\leqslant M_{c}, & \displaystyle\end{eqnarray}$$

where $M_{c}=\unicode[STIX]{x1D714}/(k_{h}c_{0})$ is the convective Mach number. These conditions in wavenumber space are represented by the squares in figure 4(c,d). Source energy that lies outside the square does not contribute to the far-field sound. Unlike the subsonic jet case, where the unit-coherence source lies completely outside the radiation square, the supersonic shock-containing case already has a source lobe satisfying the radiation criterion. This is similar to what is observed in ideally expanded supersonic jets (Cavalieri & Agarwal Reference Cavalieri and Agarwal2014; Sinha et al. Reference Sinha, Rodríguez, Brès and Colonius2014). The other three lobes are silent as they lie outside the radiation square.

When coherence decay is introduced we see that the stretching of the radiation lobe actually removes a small portion of the source energy from the radiating region. Unlike the subsonic case, where jittering of the wavepacket causes the source energy to be stretched into the radiation square, coherence decay removes energy in the BBSAN case. This explains why the introduction of coherence decay decreases the peak SPL compared to the unit-coherence case as seen in figure 2.

The Fourier transform contour plots can also be used to explain the directivity behaviour observed in figure 2. For a given value of $\unicode[STIX]{x1D703}$ , the Fourier-transformed wavenumbers $k_{y_{1}}$ and $k_{y_{2}}$ are given by

(4.5) $$\begin{eqnarray}\displaystyle \left(\frac{k_{y_{1}}}{k_{h}},\frac{k_{y_{2}}}{k_{h}}\right)=(-M_{c}\cos \unicode[STIX]{x1D703},M_{c}\cos \unicode[STIX]{x1D703}). & & \displaystyle\end{eqnarray}$$

Therefore, for $\unicode[STIX]{x1D703}$ values corresponding to the perpendicular direction, the relevant part of the Fourier transform is close to the origin. Moving away from the origin along this axis will correspond to angles upstream and downstream of the jet axis respectively. Hence, the stretching of the source lobe along the $k_{y_{1}}=-k_{y_{2}}$ axis also broadens the directivity of the jet as depicted in figure 2. This broadening is due to the source energy being stretched in both directions from the origin within the radiation square. To summarise, coherence decay is not a sound amplifier as found in the subsonic case but rather broadens the directivity of BBSAN. This broadening will be seen to be even more important when we consider higher-order shock-cell modes in § 5.

4.2 Nonlinear interaction terms

By transforming the CSD from physical space to wavenumber space, it has been demonstrated that not all wavelengths of the line-source model are responsible for sound generation. The source lobes seen in wavenumber space are due to the nonlinear interactions present in this BBSAN model. As previously mentioned, only source components corresponding to supersonic phase speeds relative to the ambient speed of sound can be effective BBSAN noise generators. In order to test whether the acoustically matched source component is supersonic, we aim to identify the phase speeds of the four source lobes in wavenumber space.

Recall that the kinematic model comprises the multiplicative combination of the wavepacket and the shock-cell structure as described in (2.4). Using the unit-coherence case for simplicity, after expanding the shock-cell components, the source term in (2.12) can be written as

(4.6) $$\begin{eqnarray}\displaystyle S(y_{1},\unicode[STIX]{x1D714})S^{\ast }(y_{2},\unicode[STIX]{x1D714})=\mathop{\sum }_{n}\{\text{e}^{\text{i}k_{s}(y_{1}+y_{2})}+\text{e}^{\text{i}k_{s}(y_{1}-y_{2})}+\text{e}^{\text{i}k_{s}(-y_{1}+y_{2})}+\text{e}^{\text{i}k_{s}(-y_{1}-y_{2})}\}\{\text{e}^{\text{i}k_{h}(y_{1}-y_{2})}\},\quad & & \displaystyle\end{eqnarray}$$

where we have ignored the amplitude and wavepacket envelope terms for this analysis. Expanding again we obtain four terms defined as

(4.7) $$\begin{eqnarray}S(y_{1},\unicode[STIX]{x1D714})S^{\ast }(y_{2},\unicode[STIX]{x1D714})=A_{1,2}^{+}+A_{1,2}^{-}+B_{1,2}^{+}+B_{1,2}^{-},\end{eqnarray}$$

where the terms are shown to be

(4.8a ) $$\begin{eqnarray}\displaystyle & \displaystyle A_{1,2}^{+}=\text{e}^{\text{i}(y_{1}-y_{2})(k_{h}+k_{s})}, & \displaystyle\end{eqnarray}$$

(4.8b ) $$\begin{eqnarray}\displaystyle & \displaystyle A_{1,2}^{-}=\text{e}^{\text{i}(y_{1}-y_{2})(k_{h}-k_{s})}, & \displaystyle\end{eqnarray}$$

(4.8c ) $$\begin{eqnarray}\displaystyle & \displaystyle B_{1,2}^{+}=\text{e}^{\text{i}y_{1}(k_{s}+k_{h})+\text{i}y_{2}(k_{s}-k_{h})}, & \displaystyle\end{eqnarray}$$

(4.8d ) $$\begin{eqnarray}\displaystyle & \displaystyle B_{1,2}^{-}=\text{e}^{\text{i}y_{1}(-k_{s}+k_{h})+\text{i}y_{2}(-k_{s}-k_{h})}. & \displaystyle\end{eqnarray}$$

By grouping the terms in this manner, we see that the $A_{1,2}^{+}$ and $A_{1,2}^{-}$ terms, which are respectively the sum and difference nonlinear wave interactions, have phase velocities

(4.9a ) $$\begin{eqnarray}\displaystyle & \displaystyle v^{+}=\frac{\unicode[STIX]{x1D714}}{k_{h}+k_{s}}, & \displaystyle\end{eqnarray}$$

(4.9b ) $$\begin{eqnarray}\displaystyle & \displaystyle v^{-}=\frac{\unicode[STIX]{x1D714}}{k_{h}-k_{s}}. & \displaystyle\end{eqnarray}$$

Since $\unicode[STIX]{x1D714}/(k_{h})>\unicode[STIX]{x1D714}/(k_{h}+k_{s})$ , the $A_{1,2}^{+}$ component travels slower than the ambient speed of sound and is not acoustically matched. It corresponds to the top left lobe in figure 4(c,d) along the diagonal. Conversely, the term $A_{1,2}^{-}$ is capable of either subsonic or supersonic phase speeds and is represented by the bottom right lobe in figure 4(c,d). This explains why only the $A_{1,2}^{-}$ component of the total source term is capable of generating far-field noise and is consistent with previous findings (Tam & Tanna Reference Tam and Tanna1982).

The terms $B_{1,2}^{+}$ and $B_{1,2}^{-}$ , on the other hand, involve combined effects of the sum and difference interaction and are a complex conjugate pair. These terms, however, lie off the $k_{y1}=-k_{y2}$ axis and outside the radiation square and therefore do not contribute to the far-field sound. A similar depiction of these off-diagonal lobes are observed in the Fourier-transformed CSD maps of Baqui et al. (Reference Baqui, Agarwal, Cavalieri and Sinayoko2015) and nonlinear wavenumber interactions in low Reynolds number subsonic jets as discussed by Sandham, Morfey & Hu (Reference Sandham, Morfey and Hu2006).

5 Coherence decay and higher-order shock-cell modes

We now focus on the more general model where higher-order shock-cell modes are included. It has been demonstrated that these additional shock-cell modes do make significant contributions to high frequency sound in the upstream directions (Tam et al. Reference Tam, Jackson and Seiner1985; Ray & Lele Reference Ray and Lele2007). We here test the effect of coherence decay when these higher-order shock-cell modes are included.

Tam et al. (Reference Tam, Jackson and Seiner1985) have argued that it should only be necessary to include the first four shock-cell modes and any higher modes are unnecessary to describe the shock-cell structure. Ray & Lele (Reference Ray and Lele2007), on the other hand, justified the use of only the fundamental shock-cell mode as source activity associated with higher-order modes lie outside the range of radiating wavenumbers. This is true when coherence decay is not accounted for.

The sensitivity of the acoustic spectrum to the addition of higher-order shock-cell modes was investigated for the current source model. It was found that the far-field spectrum and source structure did not change significantly beyond the first ten modes. Hence only the first ten shock-cell modes, as defined by (2.6), were used for this study.

By including higher-order modes, as shown by figure 5, an improvement in spectral shape is observed for all observation angles. Higher waveguide modes are required to describe the acoustic spectra for frequencies greater than the broadband peak, consistent with Tam et al. (Reference Tam, Jackson and Seiner1985). While there are still discrepancies for the downstream angle at $\unicode[STIX]{x1D703}=60^{\circ }$ , both models predict the peak frequency with reasonable accuracy, although the case with coherence decay slightly underpredicts the peak frequency compared to the data. The spectral peak shape predicted when coherence decay is included is more accurate than the case with perfect coherence. For the perfectly coherent wavepacket, oscillations start to occur at higher Strouhal numbers resulting in ‘narrow-banded’ secondary peaks. This deficiency of the perfectly coherent model at higher frequencies for upstream angles agrees with Ray & Lele (Reference Ray and Lele2007) where it was found that the linear model (unit coherence by construction) becomes ‘increasingly suspect at higher frequencies’.

Conversely, the two-point model with coherence decay ‘smooths’ out these narrow-banded peaks. This observation is consistent with Ray & Lele’s (Reference Ray and Lele2007) assertion that nonlinear effects can rectify these artificial peaks introduced from the instability wave interacting with the higher-order shock-cell modes. This finding is also consistent with the effect of coherence decay when using localised acoustic sources (Lele Reference Lele2005). We turn to the Fourier-transformed CSD maps to gain an insight into this behaviour.

Figure 5. Power spectrum at a distance of $100D$ through a range of observation angles between $\unicode[STIX]{x1D703}=60^{\circ }$ and $\unicode[STIX]{x1D703}=150^{\circ }$ measured from the downstream jet axis for a wavepacket with $k_{h}L=5$ for multiple shock-cell waveguide modes. Each measurement angle is offset by 25 dB. NASA experimental data from Norum & Seiner (Reference Norum and Seiner1982).

Figure 6. The corresponding Fourier-transformed CSD in wavenumber space for $St=0.4$ and $St=0.6$ . (a,c) Correspond to the unit-coherence model while (b,d) are for the two-point model with a coherence-decay length $L_{c}=1.0D$ . Diagonal line corresponds to $k_{y_{1}}=-k_{y_{2}}$ . The square represents the acoustic matching criterion $|k|/k_{h}=0.6M_{j}$ . The amplitude of both models has been normalised to highlight the effect of coherence decay. The jet is operating at $\unicode[STIX]{x1D6FD}=1.04$ .

The Fourier-transformed CSD maps are computed for two frequencies, $St=0.4$ (figure 6 a,b) and $St=0.6$ (figure 6 c,d) where we have now included the first ten shock-cell waveguide modes. Compared to the previous single-mode ( $n=1$ ) case, additional lobes are now visible throughout the wavenumber domain. The additional source-energy lobes are due to the wavepackets interacting with the higher-order Fourier shock-cell modes.

For the unit-coherence case (figure 6 a,c), the far-field sound radiation is still dominated by the wavepacket interaction with the $n=1$ waveguide mode. For $St=0.4$ , interaction with the second mode does not contribute to the far-field radiation as the source energy from this interaction lies outside the radiation square. This is what was observed by Ray & Lele (Reference Ray and Lele2007) and it was the reason modes higher than the fundamental were not considered in their study. At a higher frequency ( $St=0.6$ ), however, the Fourier lobes become more compact and the $n=2$ modes migrate into the radiation square as seen in figure 6(c). This is consistent with the behaviour seen in the far-field acoustic spectrum of figures 3 and 5. For frequencies below the broadband peak, we do not see an increase in SPL when higher-order shock modes are added.

For the model where coherence decay is taken into account (figure 6 b,d), we observe once again a stretching of the source lobes parallel to the $k_{y_{1}}=-k_{y_{2}}$ line. The stretching causes the fundamental ( $n=1$ ) mode to merge with the second mode and spreads the source energy within the radiating square. In contrast to the unit-coherence case, the $n=2$ shock-cell mode is now also responsible for far-field sound production. This means that coherence decay, which accounts for stochastic effects, can rectify the ‘missing sound’ at higher frequencies in the upstream direction seen in figure 3. It is clear, when compared to the perfectly coherent wavepacket, coherence decay ‘spreads and smooths’ the source energy in wavenumber space. This leads to a more uniform far-field spectrum for all observation angles and, indeed, a smoother directivity also results. This levelling effect in both directivity and frequency is evident when comparing the SPL contour plots in figure 1(a,b). Nevertheless, the overprediction in SPL for frequencies below the spectral peak is still present for all observation angles. The overprediction at low frequencies is most likely a result of the ad hoc constant coherence length assumption.

Figure 7. Fourier transform of CSD for (a)  $St=0.4$ and (b)  $St=0.6$ extracted along the line $k_{y_{1}}=-k_{y_{2}}$ . The vertical lines represent different observation angles; dashed line $=60^{\circ }$ , dash-dotted line $=90^{\circ }$ and dotted line $=150^{\circ }$ . Solid (green) vertical lines represent the acoustic matching criterion.

Suzuki (Reference Suzuki2016) has also incorporated the effect of coherence decay of wavepackets into his model. However, the empirical treatment of the shock cells as Gaussian humps does not allow the previously discussed effects of coherence decay to be observed. Similar to Ray & Lele (Reference Ray and Lele2007), the loss of high-frequency sound can be attributed to the lack of representation of the higher-order shock-cell modes. The predictions from both Suzuki’s and Ray & Lele’s models are closer to those of figure 3 and hence lack the same agreement to experimental data as other phased-array models such as Harper-Bourne & Fisher (Reference Harper-Bourne and Fisher1973) and Morris & Miller (Reference Morris and Miller2010).

To show the impact of coherence decay on noise directivity more clearly, the Fourier-transformed points along the $k_{y_{1}}=-k_{y_{2}}$ line are extracted and shown as dashed lines in figure 7(a,b). Vertical lines corresponding to different radiation directions as specified by (4.5) are also shown for reference. For frequencies slightly greater than the BBSAN peak ( $St=0.4$ ), coherence decay amplifies the radiated sound in the upstream direction and smooths out the artificial peaks seen in the perfectly coherent case. On the other hand, for higher frequencies ( $St=0.6$ ), the second shock-cell mode is now also contributing to the far-field sound and the effect of coherence decay is minimal. Comparing this to linear instability model results from Ray & Lele (Reference Ray and Lele2007), the impact of coherence decay on higher frequencies is now apparent.

Due to the ad hoc nature of fixing the modelling parameters to constant values, it is important to establish that changing the combinations and values does not lead to different conclusions. This is done in appendix A. We find, amongst other effects, that the suppression of the narrow-banded peaks cannot be accounted for by the tuning of $u_{c}$ , $A(\unicode[STIX]{x1D714})$ and $L$ within values realistic for supersonic underexpanded jets. We conclude that the inclusion of coherence decay, in the wavepacket framework, is thus imperative. In order to improve predictions at higher frequencies for BBSAN, not only do we need high-order shock modes but also the stochastic forcing term, which in the kinematic model is represented by coherence decay.

The importance of coherence in BBSSAN source modelling has been well recognised since Harper-Bourne & Fisher’s original model. Using the present BBSAN wavepacket model, however, we suggest that the overall two-point coherence contains much non-acoustically efficient information and that only the coherence decay of the low-order azimuthal modes (wavepackets) is critical in predicting far-field sound. This further suggests a mechanistic explanation for coherence decay in these systems: the coherence decay across sound sources of existing BBSAN models results from the jittering of wavepackets (Williams & Kempton Reference Williams and Kempton1978; Cavalieri et al. Reference Cavalieri, Jordan, Agarwal and Gervais2011).

6 Conclusions and perspectives

Motivated by the pioneering work of Tam (Reference Tam1987), the BBSAN models proposed by Ray & Lele (Reference Ray and Lele2007) and Suzuki (Reference Suzuki2016) indicate the suitability of using wavepackets for this component of jet noise. While successful in predicting many of the known features, the simplified models could not accurately capture the high-frequency sound produced at upstream angles. This is in contrast to the largely accurate predictions made by wavepacket models for supersonic ideally expanded flows (Sinha et al. Reference Sinha, Rodríguez, Brès and Colonius2014). Wavepacket modelling of subsonic jets suggest that coherence decay of the large-scale structures is critical in those flows. Previous BBSAN models (Harper-Bourne & Fisher Reference Harper-Bourne and Fisher1973; Morris & Miller Reference Morris and Miller2010) have demonstrated the importance of two-point coherence (or correlation), but only considered it in the context of bulk-turbulent statistics. By constructing a two-point wavepacket model, using the same methodology as Cavalieri & Agarwal (Reference Cavalieri and Agarwal2014), we test the impact of wavepacket coherence decay in shock-containing flows.

The semi-empirical kinematic model presented here provides physical insights into the underlying flow physics: the demonstration that wavepacket jitter is central to the sound generation process in shock-containing flows. The model allows us to test the hypothesis of the suitability in using low-order azimuthal structures with coherence decay to predict BBSAN.

By transforming the single-point model of Lele (Reference Lele2005) into a two-point framework, we show that coherence decay may be crucial in predicting higher-frequency noise in the upstream direction. Unlike in subsonic jets, the inclusion of coherence decay for BBSAN decreases the acoustic efficiency of wavepackets at peak frequency, but spreads source energy over a greater directivity range.

More significantly, however, is the finding that wavepacket jitter is vital for predicting frequencies above the peak. By capturing this jitter as coherence decay, clear improvements in prediction accuracy are made, especially in the upstream and sideline directions where BBSAN dominates. As exemplified in the results of § 5, coherence decay enables sound generation from higher-order shock-cell modes by stretching the source energy into the radiating square.

In addition to offering insight into the physical mechanisms of shock noise, these results also suggest directions for future modelling efforts. The combined effect of coherence decay and higher-order shock-cell modes need to be incorporated into a dynamic modelling approach that obtains wavepackets and shock-cell modes from linear stability calculations. We show that such calculations would need to be forced stochastically, to produce jittering solutions with coherence decay, and they would be required to include multiple shock-cell modes. The findings from this study are expected to help guide both future kinematic and dynamic wavepacket models of BBSAN.