Experimental self-generation of axisymmetric internal wave super-harmonics

In a cylindrical framework ($\mathbf{e_{r}}$, $\mathbf{e_{\theta}}$, $\mathbf{e_{z}}$), $\mathbf{e_{z}}$ being the ascendent vertical, small amplitude inertia gravity waves in an inviscid fluid with a constant background stratification satisfy the following equations under the Boussinesq approximation

	$\displaystyle\rho_{0}\left(\dfrac{\partial\mathbf{v}}{\partial t}+\left(\mathbf{v}\cdot\mathbf{\nabla}\right)\mathbf{v}\right)$	$\displaystyle=$	$\displaystyle-\mathbf{\nabla}p-(\rho-\bar{\rho})g\mathbf{e_{z}},$		(1)
	$\displaystyle\dfrac{\partial\rho}{\partial t}+\left(\mathbf{v}\cdot\mathbf{\nabla}\right)\rho$	$\displaystyle=$	$\displaystyle 0,$		(2)
	$\displaystyle\mathbf{\nabla}\cdot\mathbf{v}$	$\displaystyle=$	$\displaystyle 0,$		(3)

where $\mathbf{v}=(v_{r},\leavevmode\nobreak\ v_{\theta},\leavevmode\nobreak\ v_{z})$, $p$, $\rho$, and $\bar{\rho}$ are the velocity, pressure, density, and background density fields, respectively. Setting $\rho_{0}$ a reference density, the buoyancy field $b$ and frequency $N$ can be introduced as

$$b=-\frac{g}{\rho_{0}}\rho^{\prime}\mathrm{\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\ and\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\ }N^{2}=-\frac{g}{\rho_{0}}\dfrac{\partial\bar{\rho}}{\partial z},$$

(4)

where $\rho^{\prime}=\rho-\bar{\rho}$ is the perturbation to the density field.

In an axisymmetric geometry, the velocity and the buoyancy fields are $\theta$-independent and the vertical and radial velocities are therefore given by derivatives of a stream function $\psi$ as

$$v_{r}=-\frac{1}{r}\dfrac{\partial(r\psi)}{\partial z}\mathrm{\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\ and\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\ }v_{z}=\frac{1}{r}\dfrac{\partial(r\psi)}{\partial r}.$$

(5)

Using this formulation, the system of equations (1), (2), and (3) can be written as coupled equations in $\psi$, $v_{\theta}$, and $b$,

	$\displaystyle\partial_{t}\Delta_{h}\psi+\mathcal{J}^{\odot}\left(r\psi,\frac{\Delta_{h}\psi}{r}\right)$	$\displaystyle=$	$\displaystyle-\frac{2}{r}v_{\theta}\partial_{z}v_{\theta}+\partial_{r}b,$		(6)
	$\displaystyle\partial_{t}v_{\theta}+\frac{1}{r^{2}}\mathcal{J}^{\odot}(r\psi,rv_{\theta})$	$\displaystyle=$	$\displaystyle 0,$		(7)
	$\displaystyle\partial_{t}b+\frac{1}{r}\mathcal{J}^{\odot}(r\psi,b)$	$\displaystyle=$	$\displaystyle-N^{2}\frac{1}{r}\partial_{r}(r\psi),$		(8)

where the truncated Laplacian $\Delta_{h}\psi$ is defined by

$$\Delta_{h}\psi=\dfrac{\partial^{2}\psi}{\partial z^{2}}+\dfrac{\partial}{\partial r}\left(\frac{1}{r}\dfrac{\partial(r\psi)}{\partial r}\right)=\dfrac{\partial^{2}\psi}{\partial z^{2}}+\dfrac{\partial^{2}\psi}{\partial r^{2}}+\frac{1}{r}\dfrac{\partial\psi}{\partial r}-\frac{\psi}{r^{2}},$$

(9)

and the cylindrical Jacobian $\mathcal{J}^{\odot}$ of two functions $f$ and $g$ is given by

$$\mathcal{J}^{\odot}(f,g)=\dfrac{\partial f}{\partial r}\dfrac{\partial g}{\partial z}-\dfrac{\partial g}{\partial r}\dfrac{\partial f}{\partial z}.$$

(10)

The system of equations (6), (7), and (8) can be linearised by setting all Jacobians and the $v_{\theta}\partial_{z}v_{\theta}$ terms equal to zero, leading to the linear problem

	$\displaystyle\partial_{t}\Delta_{h}\psi$	$\displaystyle=$	$\displaystyle\partial_{r}b,$		(11)
	$\displaystyle\partial_{t}v_{\theta}$	$\displaystyle=$	$\displaystyle 0,$		(12)
	$\displaystyle\partial_{t}b$	$\displaystyle=$	$\displaystyle-N^{2}\frac{1}{r}\partial_{r}(r\psi).$		(13)

Equation (12) is self-consistent, and yields an orthoradial velocity that does not evolve in time. On the other hand, cross-derivatives of equations (11) and (13) give a linear equation in $\psi$

$$\mathcal{L}[\psi]=0,$$

(14)

where $\mathcal{L}$ is an operator defined as

$$\mathcal{L}[\psi]=\partial^{2}_{t}\Delta_{h}\psi+N^{2}\partial_{r}\left(\frac{1}{r}\partial_{r}(r\psi)\right).$$

(15)

As discussed in Maurer et al. maurer2017 and in Boury et al. (boury2018, ), the linear solution of equations (12) and (14) that fulfills the boundary conditions imposed by an axisymmetric forcing at the surface and zero velocities normal to the other sides, with no orthoradial velocity, is

	$\displaystyle\psi(r,z,t)$	$\displaystyle=$	$\displaystyle\psi_{0}J_{1}(lr)\cos(\omega t-mz),$		(16)
	$\displaystyle v_{\theta}(r,z,t)$	$\displaystyle=$	$\displaystyle 0,$		(17)
	$\displaystyle b(r,z,t)$	$\displaystyle=$	$\displaystyle-\frac{N^{2}l}{\omega}\psi_{0}J_{0}(lr)\sin(\omega t-mz),$		(18)

with $\omega$ the frequency of the wave field, $l$ and $m$ the radial and the vertical wave numbers, and $\psi_{0}$ a constant amplitude. The radial structure of the wave field is given by Bessel functions of zeroth and first orders, namely $J_{0}$ and $J_{1}$. With the proposed solution (16) for $\psi$, the linear operator (15) gives the dispersion relation for gravity waves

$$(l^{2}+m^{2})\omega^{2}=N^{2}l^{2}.$$

(19)

Back to equations (6), (7), and (8), the same cross-derivation that led to the linear system can be used to derive the fully non-linear equation

$$\mathcal{L}[\psi]=\mathcal{N}[\psi,v_{\theta},b],$$

(20)

where $\mathcal{L}$ is still defined by equation (15) and $\mathcal{N}$ is the non-linear operator

$$\mathcal{N}[\psi,v_{\theta},b]=-\partial_{t}\mathcal{J}^{\odot}\left(r\psi,\frac{\Delta_{h}\psi}{r}\right)-\partial_{r}\left(\frac{1}{r}\mathcal{J}^{\odot}(r\psi,b)\right)-\frac{1}{r}\partial_{t}\partial_{z}v_{\theta}^{2}.$$

(21)

As discussed in the appendix of boury2019 , when the velocity amplitudes become too large, these non-linear terms cannot be neglected. If we consider the linear solution, we note that $v_{\theta}=0$ so $\mathcal{N}[\psi,v_{\theta},b]=\mathcal{N}[\psi,b]$. The non-linear right-hand side of equation (20) therefore becomes only dependent on two Jacobians that can be evaluated. Using that nist2010

$$\dfrac{\partial(J_{0}(lr))}{\partial r}=-lJ_{1}(lr),\mathrm{\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\ }\dfrac{\partial(rJ_{1}(lr))}{\partial r}=lrJ_{0}(lr),\mathrm{\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\ and\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\ }\dfrac{\partial(r^{-1}J_{1}(lr))}{\partial r}=-lr^{-1}J_{2}(lr),$$

(22)

and $\psi$ and $b$ as derived in equations (16) and (18), the non-linear terms involved in $\mathcal{N}[\psi,b]$ become

	$\displaystyle\partial_{t}\mathcal{J}^{\odot}\left(r\psi,\frac{\Delta_{h}\psi}{r}\right)$	$\displaystyle=$	$\displaystyle C\frac{\left[J_{1}(lr)\right]^{2}}{r}\cos(2\omega t-2mz),$		(23)
	$\displaystyle\partial_{r}\left(\frac{1}{r}\mathcal{J}^{\odot}(r\psi,b)\right)$	$\displaystyle=$	$\displaystyle C\left[\cos^{2}(\omega t-mz)J_{0}(lr)\partial_{r}J_{0}(lr)+\sin^{2}(\omega t-mz)J_{1}(lr)\partial_{r}J_{1}(lr)\right],$		(24)

with

$$C=2\omega(l^{2}+m^{2})m\psi_{0}^{2}.$$

(25)

As a result, the non-linear right-hand side of equation (20) is

$$\mathcal{N}[\psi,b]=CJ_{1}(lr)\left[J_{2}(lr)\sin^{2}(\omega t-mz)-J_{1}(lr)\cos^{2}(\omega t-mz)\right]\neq 0$$

(26)

We can go a step further if, in the general case, inspired by the work of Thorpe thorpe1966 , we write $\psi$ as a sum over possible solutions of various frequencies $\omega_{j}$ as

$$\psi(r,z,t)=\sum_{j}\chi_{j}(r,z)\cos(\omega_{j}t),$$

(27)

very much like a discrete Fourier Transform, with $\chi_{j}$, $j\in\mathbb{Z}$, a spatial function. The linear part of equation (20) is therefore

$$\mathcal{L}[\psi]=\sum_{j}\left[(N^{2}-\omega_{j}^{2})\dfrac{\partial}{\partial_{r}}\left(\frac{1}{r}\dfrac{\partial(r\chi_{j})}{\partial_{r}}\right)-\dfrac{\partial^{2}\chi_{j}}{\partial z^{2}}\right]\cos(\omega_{j}t).$$

(28)

As the non-linear terms are only second order products of the stream function, we assume the following development

$$\mathcal{N}[\psi]=\sum_{g,h}C_{g,h}(r,z)\cos((\omega_{g}+\omega_{h})t),$$

(29)

where $\omega_{g}$ and $\omega_{h}$ are frequencies, and $C_{g,h}$ is a spatial fonction. Projection of equation (21) over frequencies leads to

$$\forall j\in\mathbb{Z},\leavevmode\nobreak\ (N^{2}-\omega_{j}^{2})\dfrac{\partial}{\partial_{r}}\left(\frac{1}{r}\dfrac{\partial(r\chi_{j})}{\partial_{r}}\right)-\dfrac{\partial^{2}\chi_{j}}{\partial z^{2}}=C_{j},$$

(30)

where $C_{j}$ is defined as the function $C_{g,h}$ where $g$ and $h$ verify $\omega_{j}=\omega_{g}+\omega_{h}$. The $C_{j}$ functions act as forcing terms produced by non-linear wave-wave interactions. The impact of this forcing term can be further explored theoretically by using Green functions (see appendix) and a Taylor expansion of the stream function, or with Direct Numerical Simulations, which is beyond the scope of this study.

Cylindrical cavities have the ability to produce enhanced modal wave fields, likely to trigger instabilities at high frequencies boury2018 . In such a confined configuration, radial and vertical modes can be excited if they satisfy the boundary condition, namely a zero orthogonal velocity at the boundaries. Hence, a mode can be described by the stream function

$$\psi(r,z,t)=\psi^{0}J_{1}(lr)\sin(mz)\cos(\omega t).$$

(31)

As we will see next, the radial confinement sets the possible values of $l$, and so does the vertical confinement for the values of $m$.

We consider a set of nine experiments indexed from $0$ to $9$, with forcing frequency from $\omega/N=0.305$ to $0.449$, using a mode $1$ configuration at the generator boury2018 . For each frequency, a $10$ minute forcing leads to non-linear wave-wave interactions, where higher frequency waves are created. This phenomenon is illustrated in figure 2, showing the frequency spectrum from experiment number $9$, forced at $\omega/N=0.449$. The spectrum contains not only the forcing frequency at $\omega/N$, but also a mean flow at $\omega=0$ and several harmonics at $2\omega/N$, $3\omega/N$, and so on. Here, the first harmonic would be propagating as $\omega/N<0.5$ and therefore $2\omega/N<1$. This behaviour is observed for all frequencies, regardless of any resonant cavity aspect identified in Boury et al. boury2018 . In the performed experiments, due to the range of frequencies choosen, the first harmonic at $2\omega/N$ is always propagating and, except for experiments $1$ and $2$, the second harmonic at $3\omega/N$ is evanescent.

Figure 3 gives snapshot examples of such an experiment, for a forcing at $\omega/N=0.449$, corresponding to the spectrum presented in figure 2. The first row shows the wave field filtered at the forcing frequency $\omega$ with, from left to right, $v_{z}$ and $v_{r}$ in the vertical plane, and $v_{r}$ and $v_{\theta}$ in the horizontal plane. The excited wave field corresponds to a mode $1$ and presents all the features of an axisymmetric wave field boury2018 : right-left symmetry of $v_{z}$ and right-left antisymmetry of $v_{r}$ in the vertical plane, invariance by rotation of center $(0,0)$ for $v_{r}$ in the horizontal plane, and a negligible orthoradial velocity $v_{\theta}$ when observed in the horizontal plane ($v_{r}$ dominates $v_{\theta}$ by a factor $\sim 10$ in magnitude).

The first harmonic created at $2\omega/N$ is filtered and shown in the second row of figure 3. Though no spatial wave lengths can be directly infered from the wave field, we notice that the created harmonic field shares the same axisymmetric properties as the excitation wave field. The right-left symmetry/antisymmetry of $v_{z}$ and $v_{r}$ in the vertical plane is also consistent with an axisymmetric description of the wave field. More specifically, a zero of radial velocity is observed at the center of the experimental domain whereas filtered wave fields in Triadic Resonant Instability (TRI) maurerPhD have shown, in some cases, non-zero velocity, breaking the axisymmetry of the system. The orthoradial velocity $v_{\theta}$ is, as compared to the excitation wave field, a noisy signal and does not show any sign of symmetry breaking.

In order to describe quantitatively the wave field at a given frequency and, more particularly, the wave field created in the harmonic at twice the forcing frequency, we used a Proper Orthogonal Decomposition (POD) method. As we know the relevant stream functions (modes) in the cavity, the POD process simply consists of projecting the experimental wave field over an orthogonal basis built accordingly.

The series of $\left\{r\mapsto J_{0}(l_{p}r)\leavevmode\nobreak\ |\leavevmode\nobreak\ p\in\mathbb{N}^{*}\right\}$ and $\left\{z\mapsto\sin(m_{q}z)\leavevmode\nobreak\ |\leavevmode\nobreak\ q\in\mathbb{N}^{*}\right\}$ forms an orthogonal basis in an axisymmetric geometry, which is appropriate for this study. A direct consequence is that the stream function can be written in terms of normalised modes $(p,q)$, denoted $\psi_{p,q}$, in the axisymmetric domain $(r,\theta,z)\in\mathcal{C}=[0;\leavevmode\nobreak\ R]\times[0;\leavevmode\nobreak\ 2\pi]\times[0;\leavevmode\nobreak\ -L]$, defined as

$$\psi_{p,q}(r,z)=\psi_{p,q}^{0}J_{1}(l_{p}r)\sin(m_{q}z),\mathrm{\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\ with\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\ }\psi_{p,q}^{0}=\frac{1}{J_{1}^{\prime}(l_{p}R)}\sqrt{\frac{2}{L\pi}},$$

(41)

satsifying the condition

$$\int_{\mathcal{C}}\left[\psi_{p,q}(r,z)\right]^{2}r\text{d}r\,\text{d}\theta\,\text{d}z=1,$$

(42)

from which we can derive the vertical and radial velocities associated to a mode $(p,q)$, using equations (5), as follows

	$\displaystyle v_{z}^{p,q}(r,z)$	$\displaystyle=$	$\displaystyle\frac{1}{r}\dfrac{\partial(r\psi_{p,q})}{\partial r}=-l_{p}\psi_{p,q}^{0}J_{0}(l_{p}r)\sin(m_{q}z),$		(43)
	$\displaystyle v_{r}^{p,q}(r,z)$	$\displaystyle=$	$\displaystyle-\frac{1}{r}\dfrac{\partial(r\psi_{p,q})}{\partial z}=m_{q}\psi_{p,q}^{0}J_{1}(l_{p}r)\cos(m_{q}z).$		(44)

Hence, the kinetic energy contained in one mode $(p,q)$ is given by

$$K_{p,q}=\left[\int_{\mathcal{C}}v_{z}(r,z)\cdot v_{z}^{p,q}(r,z)r\text{d}r\,\text{d}\theta\,\text{d}z\right]^{2}+\left[\int_{\mathcal{C}}v_{r}(r,z)\cdot v_{r}^{p,q}(r,z)r\text{d}r\text{d}\,\theta\,\text{d}z\right]^{2},$$

(45)

which, accounting for the discretisation of our domain, can be written as follows

$$K_{p,q}=\left[\sum_{r,z}2\pi rv_{z}(r,z)\cdot v_{z}^{p,q}(r,z)\right]^{2}+\left[\sum_{r,z}2\pi rv_{r}(r,z)\cdot v_{r}^{p,q}(r,z)\right]^{2},$$

(46)

with sums over all spatial grid points. As a result, if we denote $K_{0}$ the kinetic energy of the total wave field, the fraction of energy in a mode $(p,q)$ is given by the scalar product (46) normalised by $K_{0}$. The higher the quantity, the more dominant the mode is in the observed field. Note that the prefactors for $v_{z}^{p,q}$ and $v_{r}^{p,q}$ can differ if the normalisation process is directly applied to the stream function or to the velocities, though it will not affect the conclusions.

The left panel of figure 4 shows an example of such a POD decomposition using the excitation wave field previously presented in the top part of figure 3, and the right panel of figure 4 shows the decomposition of the first harmonic wave field, respectively at $\omega/N=0.449$ and $2\omega/N=0.898$. Figure 4 (top) depicts the colormap of the energetic distribution onto different modes and figure 4 (bottom) plots the kinetic energy contained in every tested mode as a function of the quantity $p.q$, defined as $p.q=p+0.1q$, used to identify the different modes with a single number or, in other words, to transform the $2$D plot of the top row of figure 4 into the more easily quantifiable $1$D plot of the bottom of figure 4. As expected, figure 4 (left) shows that nearly all the energy of the wave field produced by the generator lies in a radial mode $1$ wave, more exactly a $(1,7)$ mode, that can be clearly identified in figure 3 with $7$ zeros along the vertical direction and one along the horizontal direction. In addition, while the filtered wave field at $2\omega/N$ in figure 3 does not explicitly display a mode, we see from figure 4 (right) that the energy is mostly split into mode $(1,2)$ ($47\%$), mode $(3,5)$ ($32\%$), and mode $(5,8)$ ($8\%$), the other contributions being negligible. The first harmonic generated in experiment $9$ can therefore be described as a sum of modes $(1,2)$, $(3,5)$, and $(5,8)$, with given prefactors to account for the distribution of energy between the three modes.

The POD discrimination process is applied to experiments $1$ to $9$ and its results are summarised in table 5. In some cases, the first harmonic generated by the non-linear wave-wave interaction can be clearly identified by eye as a single mode $(p,q)$, as shown for experiments $5$ (with $2\omega/N=0.755$, see figure 5) and $6$ (with $2\omega/N=0.791$, see figure 6), with the generation of a mode $(1,3)$ and a mode $(2,6)$, respectively. In such cases, this clearly observed mode is the one identified as the most energetic in the POD decomposition. Figure 7 illustrates this point with experiment $5$ and $6$, showing a single mode containing about $80$ to $90\%$ of the total kinetic energy of the first harmonic. The remaining fraction of energy, from $10$ to $20\%$, is sparsely distributed into lower modes that contain less than $5\%$ of the total energy each and do not contribute significantly to the general form and behaviour of the wave field. In other cases, as described in the previous section for experiment $9$, the energy is more evenly distributed between several modes.

The structure of the generated harmonic modes does not always match the one of the excitation mode. For example, experiments $3$ and $6$ lead to generation of a dominant mode with a radial structure similar to the excitation wave field ($l_{1}=19\mathrm{\leavevmode\nobreak\ m^{-1}}$), whereas experiments $2$, $5$, and $8$, generate a dominant higher order radial structure ($l_{2}=35\mathrm{\leavevmode\nobreak\ m^{-1}}$). For a given radial structure, we note that the vertical wave number increases with the frequency $2\omega/N$, consistently with the dispersion relation (19). Interestingly, the mode frequencies are not ordered by lower $p$ or lower $q$ but seem randomly distributed: for example, by increasing $\omega$, the harmonics can be dominated by a radial mode $1$, then a radial mode $2$, and then a radial mode $1$ again.

Table 6 reproduces the theoretical results of table 3, highlighting the frequencies corresponding to the observed modes of experiments $1$ to $9$, which can be compared to the values in table 5. This comparison shows that, when a dominant mode $(p,q)$ is observed, the frequency of the harmonic (third column in table 5) is close to the frequency of the same mode $(p,q)$ stated by the dispersion relation (table 6). For instance, in experiment $6$, $78\%$ of the energy of the super-harmonic is in a mode $(1,3)$, which is the mode observed in figure 5, and the frequency predicted for such a mode is $\omega_{1,3}/N=0.773$ according to table 6, close to the harmonic frequency observed in the experiment $2\omega/N=0.791$ (table 5). Similarly, in experiment $5$, $82\%$ of the energy of the super-harmonic is in a mode $(2,6)$, which is the mode observed in figure 6, and the frequency predicted for such a mode is $\omega_{2,6}/N=0.745$, also close to the harmonic frequency observed in the experiment $2\omega/N=0.755$.

To generalize this observation, we summarised our results in figure 8. The blue circles show, in the phase space $(x=\omega/N,y=p.q)$, all the theoretical cavity modes that can be created according to table 3 (for $p=1$ through $5$ and $q=1$ through $9$). In parallel, each first harmonic frequency for experiment $1$ to $9$ is represented by a vertical dashed line at constant $\omega/N$. At each of these frequencies, modes extracted from the POD projection of the filtered harmonic wave fields are represented by red dots in the phase space, where the color intensity shows the energetic contribution of each dominant mode. To guide the eye, black rectangles link together an experimentally observed mode with its theoretical counterpart whose frequency is within $1$ to $7\%$ of the experimental frequency in all cases. This margin is of the order of the error on the buoyancy frequency $N$ measured with the probe (about $4\%$). On a vertical dashed line, when a single red dot is plotted (see for example experiment 5), it means that the harmonic wave field looks like a single mode which, as shown by the linking rectangle, corresponds to the mode selected by the non-linear interaction. On the contrary, when several dots appear for a given harmonic frequency (see, for example, experiment 1), resonant conditions stated by the dispersion relation are not exactly fulfilled and the harmonic wave field is a combination of two or more modes, the system being unable to select a solitary one.