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Two equal stable density jumps of size $\Delta \rho /2$ at $z=\pm h/2$ lie in the global linear shear flow $U=\Delta Uz/h$. Decaying normal modes haveThe jump conditions for stratified inviscid shear flow give the determinant equationWriting $\alpha =kh/2$, $J=g\Delta \rho h/({\rho }_0\Delta U^2)$ and $\tilde{c}=2c/\Delta U$, this isThe two roots for ${\tilde{c}}^2$ are real; one is negative exactly when $2\alpha /(1+e^{-2\alpha })<J<2\alpha /(1-e^{-2\alpha })$. At large $\alpha$, this narrow band centres on $J=2\alpha$, where the isolated counterpropagating interfacial gravity waves have the same zero laboratory speed. This realizes counterpropagating wave instability.

At an isolated interface with continuous base velocity, a stable density drop $\Delta \rho /2$ and a jump $[U^{\prime }]$ of base shear, decaying disturbances $w\propto e^{-k|z-z_i|}$ obey the displayed dispersion relation. It follows by inserting their derivative jump $[w^{\prime }]=-2kw$ into the jump conditions for stratified inviscid shear flow. Both the density jump and the vorticity jump contribute to the intrinsic phase velocity $c-U_i$.

Write $Z=\tilde{z}$ and introduce an arbitrary constant streamfunction scale ${\Psi }_0$, with $\hat{\psi }={\Psi }_0\tilde{\psi }$. The displacement jump condition acquires only a constant prefactor $2{\Psi }_0/\Delta U$, which can be divided out:For the second of the jump conditions for stratified inviscid shear flow, the two derivative terms have common scale $\Delta U{\Psi }_0/h$. Dividing by it givesThe term proportional to the reference density has zero jump by the first condition. ThusNo physical reference-density force has been discarded inside either layer; its continuous interface contribution has canceled. The layer equation becomes ${\tilde{\psi }}_{ZZ}={\alpha }^2\tilde{\psi }$. This explains why only the density anomaly appears in the nondimensional jump condition.

Take $k>0$ without loss of generality, so $\alpha =kh/2>0$, and require decay as $Z\to \pm \infty$. The base velocity is $\tilde{U}=Z$ everywhere; there is no jump in its derivative. Let $C=\tilde{c}$ and $E=e^{-2\alpha }$. A decaying representation that already solves the layer equations isIt is continuous at both interfaces. At $Z=-1$ the value is $A+BE$ and the derivative jump is $-2\alpha A$; at $Z=1$ they are $B+AE$ and $-2\alpha B$. The density anomaly drops by one across each interface. The jump conditions for stratified inviscid shear flow therefore reduce toFor nonreal $C$ the displacement denominators do not vanish. Neutral limiting values are obtained by continuation, rather than by dividing by zero at an interface.

Set $q=J/(2\alpha )$. A nonzero mode requires the determinant to vanish:Expansion gives the dispersion relation for two density interfaces in uniform shear,Uniform shear throughout the exterior is important: replacing it with constant exterior velocities would introduce vorticity jumps and give a different dispersion relation.

Use dimensionless variables throughout this calculation and write $c$ for $\tilde{c}$. Take $k>0$ without loss of generality, so $\alpha =kh/2>0$; for the opposite Fourier orientation the decay rate is $|k|h/2$. The density profile printed alongside the velocity profile is $\tilde{\rho }=-1$ above, $0$ in the middle, and $+1$ below. It is absent from the TeX transcription but essential to the interface calculation.

The nondimensional jump conditions for stratified inviscid shear flow also follow directly from the stated scales. Divide the dimensional stress bracket by $\Delta U/h$ times the streamfunction scale. The density contrast supplies $J\tilde{\rho }\tilde{\psi }/(\tilde{U}-c)$. The term from the constant reference density is proportional to $\tilde{\psi }/(\tilde{U}-c)$ and drops out of its jump because the first condition makes this ratio continuous. This explains why only the density contrast remains.

For a vertically localized disturbance, take decaying exterior solutions and set $P=\tilde{\psi }(1)$, $Q=\tilde{\psi }(-1)$. The velocity is continuous at the interfaces, so the first jump condition gives continuity of $\tilde{\psi }$. The outer solutions are $P{e}^{-\alpha (\tilde{z}-1)}$ above and $Q{e}^{\alpha (\tilde{z}+1)}$ below. In the middle the endpoint derivative map for an evanescent wave layer givesThese expressions follow by fitting a linear combination of $\cosh (\alpha \tilde{z})$ and $\sinh (\alpha \tilde{z})$ to its two endpoint values.

At the upper interface the shear drops from one to zero and the density drops from zero to minus one. At the lower interface they change from zero to one and from one to zero. Substitute those jumps, including the exterior derivatives $-\alpha P$ and $+\alpha Q$. DefineMultiplication by the respective nonzero $U-c$ gives the homogeneous systemFor growing modes $\mathrm{Im}c>0$ the denominators cannot vanish. Neutral limiting values are interpreted by continuation of the matching calculation.

A nonzero disturbance requires the determinant to vanish. Put $T=1+J$, $\epsilon =e^{-2\alpha }$ and $d=a^2-b^2=2\alpha a$, with $a=2\alpha /(1-{\epsilon }^2)$ and $b=a\epsilon$. The determinant isDividing by $d$ and collecting powers proves the quartic dispersion relation for a three-layer stratified shear flow:whereHere $P$ as a polynomial coefficient is distinct from the endpoint amplitude used above. The result is the desired dispersion equation, derived from both shear and density jumps rather than a density-only matching rule.

Treat this as a quadratic in $X=c^2$. Whenever $Q_0<0$, its two roots are real with opposite signs: the discriminant $P^2-4Q_0$ is strictly positive and their product is negative. The negative root gives $c=+i\sqrt{-X}$, which grows under the convention $e^{ik(x-ct)}$. Factor the constant term:The unstable band of a three-layer stratified shear flow follows immediately:or equivalently $\alpha e^{\alpha }/\cosh \alpha <1+J<\alpha e^{\alpha }/\sinh \alpha$. The endpoints are neutral $c=0$ limits of this growing branch.

For large $\alpha$, the overlap of the two interface disturbances is exponentially small: $b\simeq 2\alpha e^{-2\alpha }$. Dropping this coupling leaves the two independent gravity-vorticity interface waves. Their relevant laboratory phase speeds areThe upper wave propagates backward relative to its current and the lower wave forward relative to its current. They share a stationary laboratory phase when $J=2\alpha -1$. Restoring their small evanescent coupling produces counterpropagating wave resonance in a three-layer shear flow, with the unstable intervalNear this resonance, the waves lock in phase and extract energy from the mean shear. At large wavenumber the resonance band is exponentially narrow; large wavenumber by itself does not make an arbitrary fixed-$J$ profile unstable.

Take a common constant buoyancy frequency $N$ and put $k=k_T>0$. Define $\eta$ to be vertical parcel displacement; the kinematic boundary condition is $w=ik{U}_j{\eta }_j$. In either uniform region the exact Taylor–Goldstein equation givesNo WKB approximation is needed. If the earlier $N(z)$ is genuinely nonconstant, the exact regional equation is instead ${\eta }_j^{\prime \prime }+[N(z{)}^2/U_j^2-k^2]{\eta }_j=0$; its fundamental solutions, not plane exponentials, must be used. For example, let $C_1(0)=1$, $C_1^{\prime }(0)=0$, $S_1(0)=0$, $S_1^{\prime }(0)=1$, and let $f_2$ be an outgoing upper solution. Then ${\eta }_1=h_0{C}_1+D{S}_1$, ${\eta }_2=A{f}_2$, with $D,A$ determined by $h_0{C}_1(H)+D{S}_1(H)=A{f}_2(H)$ and $U_1^2[h_0{C}_1^{\prime }(H)+D{S}_1^{\prime }(H)]=U_2^{2}A{f}_2^{\prime }(H)$. This gives the exact arbitrary-profile field implicitly, while a simple angle-only coefficient requires further stratification assumptions.

Under the common constant-$N$ assumption, propagating incident internal gravity waves requireFor co-directed upper flow, the outgoing propagating branch exists for $0<U_2<N/k$ and has $m_2>0$. The radiation condition chooses $m_j>0$ because the laboratory vertical group velocity is positive on the negative intrinsic frequency branch. Define $\tan {\theta }_j=m_j/k$, so $\cos {\theta }_j=k{U}_j/N$.

At the velocity jump, the material-interface jump conditions for stratified inviscid shear flow require continuity of displacement and of fluid pressure. Vertical velocity itself is generally discontinuous: $w_j=ik{U}_j\eta$. In terms of displacement the pressure is $p_j=U_j^2{\eta }_j^{\prime }$. The displacement impedance for an internal wave at a velocity jump is $Z_j=U_j^2{m}_j$. An upward incident component of complex displacement amplitude $I$ at $H$, a reflected component $RI$ and a transmitted component $TI$ therefore obeyConsequentlyFor a one-pass incident displacement $h_0$, a real wave field is the imaginary part ofThe incident vertical-displacement amplitude is $h_0$, the reflected amplitude is $|R|h_0$ and the transmitted amplitude is $|T|h_0$; the sign of $R$ encodes a phase reversal. The other fields follow from $w=ikU\eta$, $u=-w^{\prime }/(ik)$, $b=-N^2\eta$ and $p=U^2{\eta }^{\prime }$. If a perfectly reflecting terrain boundary is enforced after the return wave arrives, replace the incident coefficient $h_0$ by $h_0/[1+R{e}^{2i{m}_{1}H}]$. The one-pass amplitudes use the permitted neglect of repeated terrain reflections.

For common $N$, $Z_j=(N^2/k)\cos {\theta }_j\sin {\theta }_j$. The physical vertical-displacement transmission coefficient is thereforeThe printed coefficient has a different denominator and cannot be obtained for this standard displacement convention. For example, when $U_2=U_1$, there is no interface scattering and the vertical amplitude stays $h_0$, whereas the printed expression gives $h_0/\cos {\theta }_1$. A total displacement magnitude would instead be the vertical amplitude divided by $\cos {\theta }_2$; with unequal velocities it still does not give the printed expression. The transmission formula needs an explicit correction or a different, presently unspecified amplitude/interface model.

For $U_2>N/k$, choose $m_2=i\kappa$ with $\kappa =(k^2-N^2/U_2^2{)}^{1/2}$: the transmitted response decays as $e^{-\kappa (z-H)}$ and the same impedance equations give a complex $R,T$, with $|R|=1$. At $U_2=0$ the steady equation is singular. Oppositely directed upper flow formally supports propagating waves when $0<|U_2|<N/k$, but upward radiation then requires $m_2<0$ and signed $Z_2$; the positive-angle co-directed formula is not valid there. A real velocity jump can also have Kelvin-Helmholtz instability, which is excluded from this formal steady scattering calculation.

Let $\hat{\eta }$ be the displacement of a material interface and use $[f{]}_{-}^{+}=f_{+}-f_{-}$. On each side, the kinematic boundary condition isThere is one displaced material interface, so its displacement is continuous. Thus the first jump condition isIn particular, vertical velocity need not be continuous when the background velocity jumps.

With no surface tension, pressure continuity applies on the displaced interface, rather than at its undisplaced height. Linearizing gives $[\hat{p}+\hat{\eta }{\overline{p}}^{\prime }{]}_{-}^{+}=0$. Since ${\overline{p}}^{\prime }=-g\overline{\rho }$, this becomes $[\hat{p}-g\overline{\rho }\hat{\eta }{]}_{-}^{+}=0$. The horizontal momentum calculation in part (a) givesSubstitute this and $\hat{\eta }=\hat{w}/[ik(\overline{U}-c)]$ to obtain the second jump condition:These jump conditions for stratified inviscid shear flow use one-sided values of ${\overline{U}}^{\prime }$ and apply to jumps of mass density, vorticity, or velocity. If $\overline{U}$ is continuous, the first condition reduces to continuity of $\hat{w}$; a jump of vorticity generally still prevents continuity of ${\hat{w}}^{\prime }$. This derivation avoids multiplying singular distributional derivatives of a discontinuous velocity in the Taylor–Goldstein equation.

Take $k>0$, without loss for this symmetric real base state, and write $s=1+J$, $e=e^{-2\alpha }$, and $y={\tilde{c}}^2$. The given quartic becomes a real quadratic $y^2+Ay+B=0$. Its constant term factors asTherefore $B<0$ exactly whenA negative product means the two quadratic roots are real and of opposite sign: the discriminant is $A^2-4B>0$. The negative root gives a pair $\tilde{c}=\pm i\sqrt{-y_{-}}$. One sign has positive temporal growth rate $k{c}_i$ and is unstable. Using the definitions of the hyperbolic functions gives the required unstable band:This proves the unstable band of a three-layer stratified shear flow directly, without needing to solve all four quartic roots.

For the wave interpretation, consider either interface in isolation with decaying normal modes $\hat{w}=w_0{e}^{-k|z-z_i|}$. Put $S=\Delta U/h$, $U_i=\overline{U}(z_i)$, and $v=c-U_i$, the intrinsic phase velocity. At either interface $[\overline{\rho }]=-\Delta \rho /2$, while $[{\overline{U}}^{\prime }]=-S$ at the upper interface and $+S$ at the lower one. The jump conditions for stratified inviscid shear flow give the gravity-vorticity interface wave relationFor stable density jumps, $J\ge 0$, both intrinsic branches are real. In units $C=2c/\Delta U$, the isolated wave speeds areThe upper wave travelling against its positive background current has $C_u=1-[1+\sqrt{1+8\alpha J}]/(4\alpha )$. The lower wave travelling against its negative background current has the opposite speed, $C_l=-C_u$. Their counterpropagating wave resonance in a three-layer shear flow occurs at $C_u=C_l=0$, which givesAt large wavenumber, coupling across the separation $h$ is exponentially weak, of order $e^{-kh}=e^{-2\alpha }$. The unstable band becomeswith full width $4\alpha e^{-2\alpha }+O(\alpha e^{-6\alpha })$. It is a counterpropagating wave instability: the two waves propagate oppositely relative to their local currents but have almost equal laboratory phase velocities, allowing weak coupling to lock their phases and extract mean-flow energy. The unstable band of a three-layer stratified shear flow thus becomes narrowly concentrated around this resonance. For fixed $J$, arbitrarily large $k$ lies outside that band; large-$k$ resonance requires $J$ to grow with $\alpha$.

Let $\hat{\eta }$ be the displacement amplitude of an interface. Its kinematic boundary condition is $ik(U-c)\hat{\eta }=\hat{w}$, on either side, so the first of the jump conditions for stratified inviscid shear flow isFor zero surface tension, pressure is continuous at the displaced interface. Since the background hydrostatic pressure has derivative $-g\rho$, linearization gives $[\hat{p}-g\rho \hat{\eta }{]}_{-}^{+}=0$. Using $\hat{p}={\rho }_0[(U-c){\hat{w}}^{\prime }-U^{\prime }\hat{w}]/(ik)$ and the kinematic condition givesThis derivation also works when $U$ jumps: the displacement is shared, while the vertical velocity need not be. It avoids ambiguous products of distributions at a discontinuous velocity profile.

Write $V_0=\Delta U/2$, $z_{\pm }=\pm h/2$, $B=g\Delta \rho /(2{\rho }_0)$ and $d=B/(2k)$. The velocity is linear through all three layers, so $U(z_{\pm })=\pm V_0$ and $U^{\prime }$ has no jump. Each interface has density drop $\Delta \rho /2$. The jump conditions for stratified inviscid shear flow becomeBetween interfaces, the Taylor–Goldstein equation is ${\hat{w}}^{\prime \prime }-k^2\hat{w}=0$. A decaying solution can therefore be writtenIts derivative jumps are $-2k{b}_{\pm }$, while its values at the interfaces are $b_{\pm }+e^{-kh}{b}_{\mp }$. Hence the coefficients satisfyThe determinant must vanish. With $\tilde{c}=c/V_0$, $\alpha =kh/2$ and $J=g\Delta \rho h/({\rho }_0\Delta U^2)$, one has $d/V_0^2=J/(2\alpha )$. Expanding gives the dispersion relation for two density interfaces in uniform shear:

Use a common factor $e^{i(kx-\omega t)}$, with $k=k_0>0$ and $\omega ={\omega }_0$. For the upward propagating branches with $0<{\hat{\omega }}_2=\omega -k{U}_2<N_2$, defineThese are the inclinations of upward-sloping constant-phase lines of an internal gravity wave to the horizontal. Write the incident, reflected and transmitted complex velocity amplitudes as $A_0,A_r,A_t$. Incompressibility gives $(u_i,w_i)=A_0(\cos {\theta }_1,\sin {\theta }_1)$, $(u_r,w_r)=A_r(\cos {\theta }_1,-\sin {\theta }_1)$, and $(u_t,w_t)=A_t(\cos {\theta }_2,\sin {\theta }_2)$.

The jump conditions for stratified inviscid shear flow require a common interface displacement, not a common vertical velocity: $w=-i\hat{\omega }\eta$ on each side. Since the background mass density is continuous, pressure is continuous without a hydrostatic jump. The horizontal momentum equation gives $p/{\rho }_0=(\hat{\omega }/k)u$. HenceThese two matching equations determine internal-wave transmission across a velocity jump:The amplitude formula printed in the PDF is inconsistent with these material-interface matching conditions. In particular, identical layers have $A_t/A_0=1$ and $A_r/A_0=0$, whereas its expression labelled “reflected” equals one. The displayed results above distinguish reflection from transmission and retain the intrinsic-frequency factors required by the kinematic boundary condition and pressure balance.

For equal buoyancy frequencies and the specified opposing current, ${\theta }_1=\pi /6$, ${\theta }_2=\arcsin (3/4)$, andThe incident phase lines have slope $1/\sqrt{3}$, the reflected lines slope $-1/\sqrt{3}$, and the transmitted lines slope $3/\sqrt{7}$: they steepen above the interface. For the other specified current, ${\hat{\omega }}_2=0$. The transmitted vertical wavenumber then diverges and there is no regular propagating upper-layer wave of that frequency: this is the critical level of an internal gravity wave limit. Approaching it from ${\hat{\omega }}_2>0$ gives $A_r/A_0\to -1$ and vanishing transmitted vertical energy flux; setting the intrinsic frequency to zero directly is outside the regular matching calculation.

Across a jump of $U^{\prime }$ in a constant-density parallel flow with continuous $U$, the normal mode vertical velocity and pressure are continuous. The linearized momentum equation gives $\hat{p}=\rho [(U-c){\hat{w}}^{\prime }-U^{\prime }\hat{w}]/(ik)$, yielding the displayed conditions. Equivalently, away from $U=c$, $(U-c)[{\hat{w}}^{\prime }]=[U^{\prime }]\hat{w}$. Derivative continuity is generally incorrect when the background vorticity jumps. This is the constant-density, continuous-velocity specialization of jump conditions for stratified inviscid shear flow.