4.1. Integral solution

The integral solution to (4.1) is derived following the procedure described by Brevdo & Bridges (Reference Brevdo and Bridges1996) for spatially periodic flows and Brevdo & Bridges (Reference Brevdo and Bridges1997) for temporally oscillating flows. A complete outline of the derivation is included in Ramage (Reference Ramage2017), but for brevity, only the main points are described here, including the definitions for the Fourier and Laplace transforms, and the integral expression for $\boldsymbol {\psi }$.

First, a Fourier transform is applied to the streamwise $x$-direction

(4.5a,b)\begin{equation} \boldsymbol{w}(\alpha) = \int_{-\infty}^{\infty} \boldsymbol{w}(x)\,\mathrm{e}^{-\textrm{i} \alpha x} \,\mathrm{d}\kern0.06em x \quad \textrm{and} \quad \boldsymbol{w}(x) = \frac{1}{2 {\rm \pi}} \int_{-\infty}^{\infty} \boldsymbol{w}(\alpha)\,\mathrm{e}^{\textrm{i} \alpha x}\,\mathrm{d} \alpha, \end{equation}

where $\alpha$ again denotes the streamwise wavenumber that is now allowed to be complex.

Following the application of Floquet theory, a Laplace transform is utilised, which is where the current approach differs from that presented in Brevdo & Bridges (Reference Brevdo and Bridges1996, Reference Brevdo and Bridges1997). The frequency $\omega$ used in Brevdo & Bridges (Reference Brevdo and Bridges1997) is replaced here by the Floquet exponent $\mu$. One reason for the change in notation is for consistency with the Floquet method presented in § 3. A second reason is that for Floquet modes (3.2), the term ‘frequency’ is imprecise since each Floquet mode contains a range of harmonic frequencies. The notation used here can be reconciled with that of Brevdo & Bridges (Reference Brevdo and Bridges1997) by setting $\omega = - \textrm {i} \mu$, so that the subsequent analysis in the complex $\mu$-plane is a $90^{\circ }$ rotation of the complex $\omega$-plane.

The Laplace transform used here has the form

(4.6a,b)\begin{equation} \boldsymbol{w}(\mu) = \int_{0}^{\infty} \boldsymbol{w}(\tau)\,\mathrm{e}^{-\mu \tau}\,\mathrm{d} \tau \quad \textrm{and} \quad \boldsymbol{w}(\tau) = \frac{1}{2 {\rm \pi}} \int_{\sigma - \textrm{i} \infty}^{\sigma + \textrm{i} \infty} \boldsymbol{w}(\mu)\,\mathrm{e}^{\mu \tau}\,\mathrm{d} \mu, \end{equation}

where $\sigma$ in the integral limits of (4.6b) is selected so that the inversion contour lies to the right of all poles of the integrand in the complex $\mu$-plane. (In the subsequent discussion it will be demonstrated that this ensures causality is satisfied.)

Following the application of the inverse Fourier and Laplace transforms, the integral solution for $\boldsymbol {\psi }$ is given as

(4.7)\begin{equation} \boldsymbol{\psi}(x,\tau) = \frac{1}{4 {\rm \pi}^2} \int_F {\boldsymbol{\mathsf{Q}}}(\alpha,\tau) \int_L \frac{{\boldsymbol{\mathsf{G}}}(\alpha,\mu)}{D(\alpha,\mu)} \exp(\mu \tau + \textrm{i} \alpha x)\,\mathrm{d} \mu\,\mathrm{d} \alpha, \end{equation}

where the inversion contours for the Fourier and Laplace transforms are denoted by $F = (-\infty ,\infty )$ and $L = (\sigma - \textrm {i} \infty ,\sigma + \textrm {i} \infty )$, respectively. The periodicity of the problem manifests in the function ${\boldsymbol{\mathsf{Q}}}$, which is periodic in time $\tau$ with the same period as the basic state (2.2). Moreover, ${\boldsymbol{\mathsf{Q}}}$ and ${\boldsymbol{\mathsf{G}}}$ are non-singular (Brevdo & Bridges Reference Brevdo and Bridges1996, Reference Brevdo and Bridges1997). Thus, the integrand of (4.7) only has poles when the dispersion relation $D(\alpha ,\mu ) = 0$. When this condition is satisfied, the Floquet exponent $\mu$ represents an eigenvalue of (3.3) for a given wavenumber $\alpha$. Therefore, the solution to the stability equation (3.3) is equivalent to finding the set of complex-valued $\mu$ that satisfy the dispersion relation $D(\alpha ,\mu ) = 0$ for fixed $\alpha$.

The integral solution (4.7) can be used to describe two particular features of linear disturbances in the Stokes layer:

1. (i) The temporal evolution of the disturbance maximum.

2. (ii) The behaviour about fixed streamwise $x$-locations.

Thus, it is possible to classify the impulse response as stable, convectively unstable or absolutely unstable.

4.2. Pinch points

Consider a solution to the disturbance equation (4.1) for an impulsive forcing $\boldsymbol {g}$ that is localised in both the spatial $x$-direction and in time $\tau$, with $\boldsymbol {g}$ set to zero for $\tau <0$. The corresponding solution is referred to as the impulse response, and is classified as stable if it decays in the asymptotic limit, whereby the total flow returns to the unperturbed basic state (2.2). On the other hand, if the solution grows it is unstable. An unstable solution can be further classified as convective or absolute, depending on the behaviour about fixed spatial $x$-locations. Unstable disturbances that decay at each $x$-position and convect away from the region of interest are convectively unstable, whereas a disturbance that grows about every $x$-location is absolutely unstable. While it might be concluded that the overall growth or decay of the solution to (4.1) is matched to the least-stable Floquet mode (3.2), the classification of the impulse response as being convective or absolute requires further consideration using the Briggs (Reference Briggs1964) method.

The application of the Briggs method to the stability of steady shear flows is well documented (Gaster Reference Gaster1968; Lingwood Reference Lingwood1997a; Schmid & Henningson Reference Schmid and Henningson2001; Huerre Reference Huerre2002) and for temporally periodic flows the process remains broadly unchanged, provided the time dependence of the term ${\boldsymbol{\mathsf{Q}}}$ in (4.7) is treated appropriately. However, a significant result of the subsequent investigation is the manner in which the existence of absolute instability is constrained by antiperiodicity. Moreover, the symmetry arguments used here to demonstrate this behaviour rely on an in-depth understanding of Briggs’ method. Thus, an overview of the procedure outlined by Brevdo & Bridges (Reference Brevdo and Bridges1997), that differs from the Briggs (Reference Briggs1964) method through the inclusion of the time-periodic term ${\boldsymbol{\mathsf{Q}}}$, is presented below in order to contextualise the subsequent analysis.

The approach for characterising unstable disturbances is equivalent to that presented by Brevdo & Bridges (Reference Brevdo and Bridges1997) with the exception of the rotation in the complex frequency plane that arises as a consequence of the differing representations of the temporal component of the perturbation. Equation (4.7) is evaluated for contours $F$ and $L$ in the respective complex $\alpha$- and $\mu$-planes, which are indicated by the respective horizontal and vertical red lines in figure 2. Closure of the two contours, $F$ and $L$, is then achieved by adding a contour that connects the respective endpoints. Since the integrand is zero along this closure, the integral solution (4.7) along the straight-line open contour is equivalent to that obtained over the closed contour. Thus, by Cauchy's residue theorem, the integral is the sum of the residues of the poles contained within the closed contour.

Figure 2.

Sketches illustrating the initial set-up for Briggs’ method. (a) Complex $\alpha$-plane. (b) Complex $\mu$-plane.

In the complex $\alpha$-plane, depicted in figure 2(a), the integrand in (4.7) decays as $\alpha _i \rightarrow \infty$ for $x>0$. Thus, the horizontal $F$ contour can be closed by a semi-circular contour of infinite radius in the upper half-plane. Hence, when $x>0$, the integral along $F$ is determined by summing the residues of poles in the upper half-plane. Similarly, if $x < 0$, closure is achieved by a semi-circular contour in the lower half-plane and the integral along $F$ is given by summing the residues of poles in the lower half-plane.

For each $\alpha \in F$ there are $\mu$-roots to the dispersion relation $D(\alpha ,\mu ) = 0$ that correspond to poles of the integrand in (4.7). Taking the loci of these poles for all $\alpha \in F$ establishes $\mu (F)$ curves in the complex $\mu$-plane. The $\mu (F)$ curve located furthest to the right in the complex $\mu$-plane is illustrated by the solid purple curve in figure 2(b). The vertical $L$ contour is then chosen to lie to the right of this curve by selecting the real part as

(4.8)\begin{equation} \sigma > max \left\{\mu_r \,|\, D(\alpha,\mu) = 0,\ \forall\ \alpha \in \mathbb{R} \right\}. \end{equation}

Similar to the approach implemented for the $F$ contour, closure of the $L$ contour in the complex $\mu$-plane is achieved via the addition of semi-circular contours to the left or right. The integrand in (4.7) decays as $\mu _r \rightarrow \infty$ for time $\tau < 0$. Thus, provided there are no poles to the right of the $L$ contour, the integral must be zero. Hence, the disturbance is zero for all points in time prior to the application of the impulsive forcing $\boldsymbol {g}$. Therefore, it is necessary that $\sigma$ is given by (4.8) to satisfy causality. For time $\tau > 0$, closure is achieved by a semi-circular contour to the left of the $L$ contour. This closed contour then contains all poles of the integrand in (4.7).

For Floquet exponents $\mu \in L$, $\alpha ^{\pm }(L)$ curves in the complex $\alpha$-plane satisfy the dispersion relation $D(\alpha ,\mu ) = 0$ and describe poles of the integrand in (4.7), as depicted by the solid purple curves in figure 2(a). Initially, all poles are associated with spatial decay, while $\alpha ^{\pm }(L)$ represent those curves that exhibit the slowest decay along the positive and negative valued $x$-directions.

The Fourier integral (4.5) in (4.7) is computed first, which is given as

(4.9)\begin{equation} \boldsymbol{\psi}(x^{{\pm}},\mu) = \frac{1}{2{\rm \pi}} \oint_{C_F^{{\pm}}} {\boldsymbol{\mathsf{Q}}}\left(\alpha,\tau\right) \frac{{\boldsymbol{\mathsf{G}}}\left(\alpha,\mu\right)}{D\left(\alpha,\mu\right)} \exp(\textrm{i} \alpha x^{{\pm}}) \,\mathrm{d} \alpha, \end{equation}

for some $\mu$, where $x^+$ and $x^-$ are shorthand for $x > 0$ and $x < 0$, respectively. Similarly, $C_F^{\pm }$ denotes the closed contour for a particular $x$ value. Due to the adoption of this compact notation, two expressions are represented by (4.9). A single expression for all $x$ is given by

(4.10)\begin{equation} \boldsymbol{\psi}(x,\mu) = H(x) \boldsymbol{\psi}(x^+,\mu) + H({-}x) \boldsymbol{\psi}(x^-,\mu), \end{equation}

where $H(x)$ is the Heaviside function. Furthermore, by Cauchy's residue theorem

(4.11)\begin{equation} \boldsymbol{\psi}(x^{{\pm}},\mu) ={\pm} \textrm{i} \sum_{k = 1}^{\infty} {\boldsymbol{\mathsf{Q}}}\left(\alpha_k^{{\pm}}\left(\mu\right),\tau\right) \frac{ {\boldsymbol{\mathsf{G}}} \left( \alpha_k^{{\pm}}\left(\mu\right), \mu \right)}{\dfrac{\partial D}{\partial \alpha} \left( \alpha_k^{{\pm}}\left(\mu\right),\mu \right)} \exp(\textrm{i} \alpha_k^{{\pm}}\left(\mu\right) x^{{\pm}}), \end{equation}

where $\alpha ^+_k(\mu )$ and $\alpha ^-_k(\mu )$ are the roots of the dispersion relation $D(\alpha ,\mu ) = 0$ in the upper and lower halves of the complex $\alpha$-plane, respectively.

On solving the Fourier integral (4.9), the Laplace integration (4.6) is carried out. This is achieved by deforming the $L$ contour to the left in the complex $\mu$-plane. To illustrate this behaviour, we consider one particular $\mu$ value on $L$ with a fixed imaginary part $\mu _i$. The corresponding real part, $\mu _r$, is then reduced. The problem then simplifies to tracing the behaviour of poles in the complex $\alpha$-plane as the real part of $\mu$ is shifted horizontally to the left. Figure 3 demonstrates both the behaviour about a fixed $\mu _i$ value and the overall deformation of the contours and curves when all $\mu _i$ are considered. The leftwards shift of a fixed $\mu$ value and the corresponding movement of the $\alpha$-roots are illustrated by arrows in the respective $\mu$- and $\alpha$-planes; figure 3(a,b). The deformation of the $L$ contour in the $\mu$-plane and that of the corresponding $F$ contour in the $\alpha$-plane are made more obvious by the inclusion of the initial set-up (represented by dashed purple and red lines).

Figure 3.

Sketches illustrating the pinching of the $F$ contour in the complex $\alpha$-plane as the contour $L$ is shifted horizontally to the left in the complex $\mu$-plane. Arrows indicate the behaviour brought about by the horizontal movement of a fixed Floquet exponent $\mu$, dashed lines indicate the original contours and pole loci (as per figure 2), while solid curves demonstrate the deformations due to the leftward movement of the $L$ contour; (a,b) $\mu$ is reduced until one $\alpha$-root crosses the real $\alpha$-axis, (c,d) $\mu =\mu _0$ corresponds to a collision of poles at $\alpha _0$.

As the real part of the Floquet exponent is reduced, the resulting path will eventually cross the $\mu (F)$ curve, which describes the $\mu$-roots of the dispersion relation $D(\alpha ,\mu ) = 0$ for $\alpha \in \mathbb {R}$; dashed purple curve in figure 3(b). Thus, at least one $\alpha$-root of $D$ lies along the real $\alpha$-axis. Moving $\mu$ beyond the $\mu (F)$ curve allows one $\alpha$-root to cross over the $F$ contour and pass from one half of the complex $\alpha$-plane to the other; as demonstrated by the arrow originating in the upper right quadrant in figure 3(a). Since a pole has shifted from one half of the complex $\alpha$-plane to the other, there is a discontinuity in (4.11) for $\boldsymbol {\psi }$, with the jump being the size of the residue of this particular pole. However, an analytic continuation of $\boldsymbol {\psi }$ is achieved by deforming the $F$ contour, as shown in figure 3(a), to ensure that all $\alpha$-roots remain on their respective sides of $F$. The modification of the $F$ contour results in a corresponding deformation of the $\mu (F)$ curve in figure 3(b).

By allowing for the deformation of the $F$ contour, the horizontal shifting of the Floquet exponent $\mu$ can be extended to smaller valued $\mu _r$. However, in general, this procedure cannot be continued indefinitely. Once an $\alpha$-root has crossed into the opposite half-plane, the leftwards shift to some $\mu = \mu _0$ is accompanied by the coalescence at $\alpha _0$ of two poles originating in different halves of the complex $\alpha$-plane, as demonstrated in figure 3(c,d). The $F$ contour becomes pinched and passes through the pole at $\alpha _0$, but cannot be deformed further without introducing discontinuities in $\boldsymbol {\psi }$. Thus, a further reduction in $\mu _r$ is not possible. Note that a coalescence of $\alpha$-roots originating within the same half-plane does not pinch the $F$ contour.

The coalescence of two simple poles at $\alpha _0$ results in a second-order pole. Therefore, in addition to satisfying the dispersion relation

(4.12a)\begin{equation} D(\alpha_0,\mu_0) = 0, \end{equation}

it must also hold that

(4.12b)\begin{equation} \frac{\partial D}{\partial \alpha} (\alpha_0,\mu_0) = 0. \end{equation}

Thus, such collisions manifest as saddle points of the dispersion relation in the complex $\alpha$-plane. Moreover, by inverting this relationship it becomes evident that the formation of a saddle point at $\alpha _0$ is accompanied by the formation of a cusp at $\mu _0$ (Kupfer, Bers & Ram Reference Kupfer, Bers and Ram1987), as illustrated by the solid purple curve in figure 3(d).

In (3.3), which is equivalent to the dispersion relation $D(\alpha ,\mu ) = 0$, the wavenumber $\alpha$ appears to the fourth order, while the Floquet exponent $\mu$ is linear. Since the matrices in the eigenvalue problem (3.10) are very large, it was computationally impossible to determine $\alpha$ for a given $\mu$ using the currently available computational resources. Hence, it was not feasible to locate saddle points of the dispersion relation in the complex $\alpha$-plane. Nevertheless, utilising existing methods for computing $\mu$ for a given $\alpha$, the corresponding cusps of the dispersion relation in the complex $\mu$-plane may be located.

Using a Taylor series expansion for $\boldsymbol {\psi }$ about $\alpha _0$, it may be shown that if $\alpha _0$ is formed by the collision of two poles originating in the opposite halves of the $\alpha$-plane, then it contributes to the expressions (4.9)–(4.11) for both positive and negative valued $x$. Hence, the parameter set $(\alpha _0,\mu _0)$ contributes to the pointwise asymptotic behaviour of the disturbance. A similar argument can be used to demonstrate that if $\alpha _0$ represents the collision of two poles originating in the same half-plane, there is a cancellation effect and the collision does not contribute to the asymptotic behaviour (Huerre Reference Huerre2002). The condition that the collision occurs between poles originating in opposite half-planes is known as the collision criterion (Brevdo & Bridges Reference Brevdo and Bridges1997).

Finally, the Laplace integration in (4.7) is carried out along a contour that lies to the right of every $\mu _0$; there can be multiple such branch points, potentially a continuum, due to the continuous choice of $\mu _i$ at the start of the procedure. For a collision that satisfies the collision criterion, the leading-order behaviour is then described by

(4.13)\begin{equation} \boldsymbol{\psi}(x,\tau) \sim \frac{- \textrm{i}{\boldsymbol{\mathsf{Q}}}(\alpha_0,\tau) {\boldsymbol{\mathsf{G}}}\left(\alpha_0,\mu_0\right)}{\sqrt{ 2 {\rm \pi}\tau \dfrac{\partial D}{\partial \mu} \left( \alpha_0, \mu_0 \right) \dfrac{\partial^2 D}{\partial \alpha^2} \left( \alpha_0, \mu_0 \right) }} \exp\left(\textrm{i} \alpha_0 x + \mu_0 \tau\right), \end{equation}

where $\alpha _0$ is the location of the pinch-type saddle point in the complex $\alpha$-plane and $\mu _0$ is the corresponding cusp location. While there are notionally multiple such values, the leading-order behaviour of (4.13) is described by the pair $(\alpha _0,\mu _0)$ for which $\mu _0$ has the greatest real part. It is possible to account for lower-order behaviour by considering a superposition of similar expressions for further $(\alpha _0,\mu _0)$ pairs.

The expression for the streamfunction $\boldsymbol {\psi }$, as given in (4.13), matches the Floquet mode decomposition given in (3.2), up to a multiplication by a factor $1/\sqrt {\tau }$. Pointwise temporal growth occurs if $\mbox {Re}\{\mu _0\} > 0$, while $\mbox {Re}\{\mu _0\} < 0$ corresponds to pointwise decay. In the former instance, the disturbance is classified as absolutely unstable, while in the latter case the disturbance is either convectively unstable or stable (depending on the growth or decay of the disturbance maximum).

4.3. Symmetry constraints

In addition to being periodic with a period $2{\rm \pi}$, the temporal variation of the basic state (2.2) is antiperiodic with an antiperiod ${\rm \pi}$; i.e. $U_B(y,\tau +{\rm \pi} ) = -U_B(y,\tau )$. Thus, the distinction between the upstream and downstream $x$-directions is blurred, and the location of the roots of the dispersion relation $D(\alpha _0,\mu _0) = 0$ are constrained. This behaviour is illustrated by considering the governing perturbation equation (3.3), that is equivalent to the dispersion relation, whereby the solution is characterised by the complex-valued wavenumber $\alpha$, Floquet exponent $\mu$, and eigenfunction $\psi$:

(4.14a)\begin{equation} \left(\alpha,\mu,\psi\left(y,\tau\right)\right). \end{equation}

On taking the complex conjugate of (3.3), the governing perturbation equation remains unchanged but the solution is now characterised as

(4.14b)\begin{equation} \left(-\bar{\alpha},\bar{\mu},\bar{\psi}\left(y,\tau\right)\right), \end{equation}

where $\bar {z}$ denotes the complex conjugate of $z$. The existence of the solution (4.14a) implies the existence of the solution (4.14b), and this relationship holds irrespective of the antiperiodicity of the flow. Returning to (3.3) and applying a time shift of half a period, $\tau \rightarrow \tau + {\rm \pi}$, the sign of the basic state is reversed and the asymptotic behaviour for large time $\tau$ is unchanged. Thus, the solution is given as

(4.14c)\begin{equation} \left(-\alpha,\mu,\psi\left(y,\tau+{\rm \pi}\right)\right). \end{equation}

The existence of the solution (4.14a) implies the existence of the solution (4.14c) when antiperiodicity holds. Applying both complex conjugation and a time shift of half a period to (3.3), the solution is characterised as

(4.14d)\begin{equation} \left(\bar{\alpha},\bar{\mu},\bar{\psi}\left(y,\tau+{\rm \pi}\right)\right). \end{equation}

It follows from the related solutions (4.14) that for any given pair $(\alpha ,\mu )$ that satisfy (3.3),

(4.15)\begin{equation} D(\alpha,\mu) = 0 \Longrightarrow \left\{\begin{matrix} D(\bar{\alpha},\bar{\mu}) = 0, \\ D(-\alpha,\mu) = 0, \\ D(-\bar{\alpha},\bar{\mu}) = 0, \end{matrix}\right. \end{equation}

for any antiperiodic flow.

In the instance $\alpha \in \mathbb {R}$, such that $\bar {\alpha } = \alpha$, the set of relationships (4.15) reduce to

(4.16)\begin{equation} D(\alpha,\mu) = 0 \Longrightarrow \left\{\begin{matrix} D(\alpha,\bar{\mu}) = 0, \\ D(-\alpha,\mu) = 0, \\ D(-\alpha,\bar{\mu}) = 0. \end{matrix}\right. \end{equation}

Hence, for any $\alpha \in \mathbb {R}$, the $\mu$-roots of the dispersion relation $D(\alpha _0,\mu _0) = 0$, appear in complex conjugate pairs. In terms of Briggs’ method, this implies that the $\mu (F)$ curve, plotted in figure 2(b), is symmetric about the real $\mu$-axis.

A similar argument can be established for Floquet exponents $\mu$, once the harmonic decomposition of $\mu$ in (3.5) has been taken into account. The imaginary part of the Floquet exponent, $\mu _i$, is only given modulo one, since an integer shift is equivalent to a relabelling of the harmonics. Thus,

(4.17)\begin{equation} D(\alpha,\mu) = 0 \Longrightarrow D(\alpha,\mu+ik) = 0, \end{equation}

for all integers $k$. Moreover, in the instance that $\mu _i$ is a multiple of $1/2$, complex conjugation is equivalent to an integer shift in $\mu _i$. Therefore,

(4.18)\begin{equation} D(\alpha,\mu) = 0 \Longrightarrow D(\alpha,\bar{\mu}) = 0, \end{equation}

where, restricting our interest to the interval $\mu _i \in [ 0,1 )$ gives

(4.19)\begin{equation} \mu = \mu_r + \textrm{i} \left\{0,\tfrac{1}{2} \right\} \quad \textrm{for}\ \mu_r \in \mathbb{R}. \end{equation}

On applying the above symmetries, it follows that, for all $\mu$ of the form (4.19), the set of relationships (4.15) reduces to

(4.20)\begin{equation} D(\alpha,\mu) = 0 \Longrightarrow \left\{\begin{matrix} D(\bar{\alpha},\mu) = 0, \\ D(-\alpha,\mu) = 0, \\ D(-\bar{\alpha},\mu) = 0. \end{matrix}\right. \end{equation}

In the pinch-point analysis described above, the imaginary part of $\mu$ is fixed and the $\alpha$-roots of the dispersion relation $D(\alpha ,\mu ) = 0$ are traced as the real part of $\mu$ is shifted horizontally to the left (recall § 4.2). When the imaginary part of $\mu$ is fixed such that $\mu _i=\{0,1/2\}$, the movement of the $\alpha$-roots is constrained by the symmetries of the base flow (2.2) and hence (4.20) holds. In particular, if the wavenumber $\alpha$ is a root of the dispersion relation $D(\alpha ,\mu ) = 0$, then so are the wavenumbers $\bar {\alpha }$, $-\alpha$ and $-\bar {\alpha }$. Thus, once $\mu$ comes into contact with the $\mu (F)$ curve and an $\alpha$-root touches the real axis, the corresponding $\bar {\alpha }$-root must also touch the real axis. Similarly for $-\alpha$ and $-\bar {\alpha }$. Hence, the $F$ contour is pinched along the real $\alpha$-axis and no contour deformation is necessary. This behaviour is demonstrated in figure 4 for $\mu _i = 1/2$. Since the $F$ contour is not deformed, neither is the $\mu (F)$ curve, though the existence of the saddle point at $\alpha _0 \in \mathbb {R}$ (and $-\alpha _0$) indicates a cusp at $\mu _0$, as illustrated in figure 4(b). Moreover, there is a cusp indicated for $\mu _i = 0$, by the same argument. Note that the $\alpha ^{\pm }(L)$ curves are not symmetric, even though the movement of the $\alpha$-roots associated with $\mu _i = \{0,1/2\}$ are symmetric along both axes.

Figure 4.

Sketches illustrating the pinching of the $F$ contour along the real $\alpha$-axis, when $\mu = \mu _r + i/2$. (a) Complex $\alpha$-plane. (b) Complex $\mu$-plane. For the horizontal line extending from the $L$ contour to the $\mu (F)$ curve, there is a matching symmetric movement of the $\alpha$-roots that form collisions along the real $\alpha$-axis about $\pm \alpha _0$.

The existence of these pinch points is assured by the process outlined in § 4.2, when the $\mu (F)$ curve is well defined for $\mu _i = \{0,1/2\}$. The $L$ contour in figure 2(b) contains all possible values of $\mu _i$, including $0$ and $1/2$, while the $\mu (F)$ curve (corresponding to $\alpha \in \mathbb {R}$) must lie to the left of the $L$ contour to satisfy causality. Furthermore, as $\mu$ is shifted to the left for $\mu _i = \{0,1/2\}$, the $\mu (F)$ curve is eventually reached. Hence, by the symmetry arguments described above, a collision of complex conjugate $\alpha$-roots occurs. There is a possibility that in some antiperiodic flows the $\mu (F)$ curve is not well defined for $\mu _i = \{0,1/2\}$, i.e. there are no real $\alpha$ corresponding to these particular $\mu _i$ values. However, this was not the case for the Stokes layer studied herein and so is not discussed any further, but is highlighted as a potential factor in other antiperiodic flows.

The above analysis indicates that saddle points of the dispersion relation $D(\alpha ,\mu ) = 0$, satisfying the collision criteria, occur whenever $\alpha \in \mathbb {R}$ and $\mu _i = \{0,1/2\}$; collisions emerge between complex conjugate $\alpha$-roots. Moreover, for these particular $\mu _i$ values, the collision of two $\alpha$-roots originating within the same half-plane onto the real axis will be accompanied by the collision of their respective complex conjugates in a four-way collision. Thus, all saddle points of the dispersion relation with these characteristics must satisfy the collision criteria.

Further collisions may occur for $\mu _i\neq \{0,0.5\}$ (as $\mu$ crosses the $\mu (F)$ curve) that bring about greater pointwise growth than that found for $\mu _i = \{0,1/2\}$. In such instances, this may necessitate a deformation of the $F$ contour and establish a pinch point with the $\mbox {Re}\{\mu _0\}$ greater than that found using the symmetry arguments. However, while such behaviour may be significant for other flows, this was not found for the Stokes layer: the rightmost point of the $\mu (F)$ curve was matched to a cusp with $\alpha \in \mathbb {R}$ and $\mu _i = 0$ in every case considered.

4.4. Development of the disturbance maximum

In addition to determining the disturbance development about fixed spatial locations (via the Briggs method), the integral solution (4.7) provides insight into the leading-order behaviour of the disturbance maximum. The stability of a disturbance $\boldsymbol {\psi }$ to an impulse $\boldsymbol {g}$ can be determined by generalising the problem to consider the leading-order behaviour along an arbitrary spatio-temporal ray

(4.21)\begin{equation} x/\tau = V, \end{equation}

where $V$ is constant.

In this setting, the leading-order pointwise behaviour (deduced by the Briggs method) corresponds to $V=0$. On the other hand, the temporal growth associated with the disturbance maximum is matched to the spatio-temporal ray $V = V_{max}$. The growth rate along this particular ray is of interest as Ramage et al. (Reference Ramage, Davies, Thomas and Togneri2020) report that the temporal growth about fixed spatial locations matches that of the disturbance maximum.

Similar to the arguments used for steady flows, the Floquet mode (3.2) that contributes towards the growth (or decay) of the disturbance along the ray $V_{max}$ corresponds to $\alpha \in \mathbb {R}$ and

(4.22)\begin{equation} \frac{\partial \mu_r}{\partial \alpha} = 0. \end{equation}

This particular result is derived by solving the integral equation (4.7) along a spatio-temporal ray by the method of steepest descent (refer to Ramage (Reference Ramage2017), for details). It follows that the asymptotic temporal growth of the disturbance maximum can be determined by plotting the real part of the Floquet exponent $\mu$ as a function of the real wavenumber $\alpha$, and selecting the maximum.

For antiperiodic flows, such as (2.2), the asymptotic temporal growth of the disturbance maximum and the asymptotic pointwise growth correspond to $\alpha \in \mathbb {R}$. Hence, the growth rate along the ray $V_{max}$ and the pointwise growth rate along $V = 0$ can be equivalent, provided the collision location along the real $\alpha$-axis corresponds to a local maximum of $\mu _r$.