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For hindered settling with normalized particle volume fraction $s$, upward coordinate and downward isolated settling velocity magnitude $V_0$, the scalar conservation law has conservation law flux $f(s)=-V_{0}s(1-s{)}^{\alpha }$. For $\alpha >0$, its convexity changes at $s=2/(\alpha +1)$ when this lies in the physical interval. This change distinguishes spreading rarefaction waves from compressing characteristic curves; a flux with an inflection can produce both a shock wave and a rarefaction wave in one transition. At $\alpha =0$ the flux is linear and the profile simply translates.

A hydraulic bore is a propagating abrupt depth change described as a moving hydraulic jump. In the shock frame, an admissible rightward through-flow goes from supercritical flow to subcritical flow, with the appropriate characteristic curves entering the jump.

For a one-dimensional first-order system $v_t+A(v,x,t)v_x=s$, hyperbolicity means that the coefficient matrix has real eigenvalues and a complete set of eigenvectors. Distinct real eigenvalues give strict hyperbolicity and separate characteristic curves. Along a curve of speed $\lambda$, a left eigenvector $\ell A=\lambda \ell$ gives the compatibility equation $\ell dv/dt=\ell s$.

For a transport equation on a domain, prescribe boundary data only where the velocity points into the domain. On $x>0$ with speed $a>0$, a bounded weak solution with initial data $u_0$ and inflow data $f$ satisfiesThe integrals are over the quadrant and its two boundary rays. Backtracking characteristic curves gives $u_0(x-at)$ when $x\ge at$ and $f(t-x/a)$ otherwise. Corner values need not match for a bounded weak solution.

On $0<x<L$, $t>0$, an initial condition $u(x,0)=g(x)$ gives the starting profile. For the heat equation, Dirichlet boundary conditions $u(0,t)=a(t)$, $u(L,t)=b(t)$ can model ends held at prescribed temperatures. Alternatively, zero Neumann boundary conditions describe insulated ends in the constant-conductivity model. These choices can produce different evolutions from the same initial condition.

For a continuous classical solution reaching the corners, the Dirichlet boundary data must satisfy $g(0)=a(0)$ and $g(L)=b(0)$; higher smoothness can require further compatibility. In dimensionless variables, a complete worked heat equation problem isIts exact classical solution is $u(x,t)=x+e^{-{\pi }^{2}t}\sin (\pi x)$. The straight profile $x$ is the equilibrium; the extra bump decays. Substituting verifies the equation and every datum, and the energy estimate proves uniqueness among smooth solutions with these data.

For the transport equation with $c>0$ on the same interval, prescribe an inflow transport boundary condition only at $x=0$. Backtracking a characteristic curve gives $g(x-ct)$ if $x\ge ct$, and $a(t-x/c)$ otherwise. Values at $x=L$ are determined by this formula; independently prescribing them can make the problem inconsistent.^[ref]

The Inviscid Burgers equation is the scalar conservation lawFor a classical solution, the chain rule makes this equivalent to $u_t+u{u}_x=0$, a nonlinear transport equation with characteristic speed $u$. The conservative form remains meaningful for a discontinuous weak solution, whereas the product $u{u}_x$ cannot simply be evaluated pointwise at a jump. Its discontinuities obey the Rankine-Hugoniot condition $s=(u_L+u_R)/2$; this condition alone permits inadmissible expansion jumps. For bounded initial conditions on the infinite line, imposing the Kruzhkov entropy inequalities and the initial trace selects a unique entropy solution. For step data, incoming characteristic curves select $u_L>s>u_R$, while $u_L<u_R$ produces a rarefaction wave.^[ref]

If the vessel radius is proportional to $e^{-\beta z}$, its area is proportional to $e^{-2\beta z}$. Gas-volume conservation gives the displayed equation. Along concentration characteristic curves, $\dot{z}=V_s(1-2\varphi )$ and $\dot{\varphi }=2\beta V_s\varphi (1-\varphi )$. Uniform initial concentration stays uniform in the uninterrupted bubbly region and follows this logistic law. Individual bubbles move at $V_s(1-\varphi )$ instead: their rise distance there is $\eta =-(2\beta {)}^{-1}\log [{\varphi }_0+(1-{\varphi }_0)e^{-2\beta V_{s}t}]$. Boundary interactions and dense-mixture backflow can invalidate the simplified closure.

A one-dimensional particle suspension settling without particle diffusion or overturning obeys the scalar conservation law ${\varphi }_t+{\partial }_{z}j(\varphi )=0$. The upward flux is negative for downward settling. Characteristic curves, compressive shocks and rarefaction waves describe concentration transport rather than individual particle paths.

Let $f$ be smooth and strictly concave, with $f(0)=0$ and $f^{\prime }$ bijective, and let $u_0$ be smooth with compact support. For $0<t<t_{\ast }$ in the smooth concave-flux characteristic lifespan, the primitive $U_x=u$ solves the Hamilton-Jacobi equation $U_t+f(U_x)=0$ and obeyswhere $g$ is the concave Legendre dual. The tangent inequality for a concave function bounds every candidate by $U(t,x)$, and the backward characteristic curve attains equality. Its foot $x_0$ is the unique maximizer, and $u(t,x)=h((x-x_0)/t)$ for $h=(f^{\prime }{)}^{-1}$. This statement concerns the smooth solution, without asserting a post-crossing continuation.

The method of characteristics solves a first-order partial differential equation by reducing it to ordinary differential equations along characteristic curves.

If a continuous nondecreasing modulus $\mu$ satisfies this divergence condition, an estimate $|b(t,x)-b(t,y)|\le A(t)\mu (|x-y|)$ with locally integrable $A$ gives uniqueness of characteristic curves. Apply a regularized integral comparison to their separation. For the log-Lipschitz modulus, ${\delta }^{\prime }(t)\le A(t)\delta (1-\log \delta )$ gives $\delta (t)\le \exp (1-(1-\log \delta (0))\exp (-{\int }_0^{t}A))$ while $\delta \le 1$. Zero initial separation stays zero. Unlike the usual linear Gronwall inequality, this criterion allows non-Lipschitz vector fields.

Take a dense lower layer and write $g^{\prime }=g({\rho }_1-{\rho }_0)/{\rho }_0>0$. In the Boussinesq approximation the fractional density difference is small; use the reference density in inertia and retain the difference in buoyancy. The one-layer model additionally treats the ambient as much deeper, with negligible distributed horizontal inertia away from the head. The shallow water approximation requires depth small relative to the horizontal variation length, small vertical accelerations, nearly hydrostatic pressure, and an approximately depth-uniform horizontal velocity. Neglect mixing, viscosity, bed friction and rotation. A slowly varying rectangular channel allows cross-sectional averages without large lateral separation.

A slice of width $b(x)$ contains volume $bhdx$. Its outgoing volume flux is $bhu$, so volume conservation gives $(bh{)}_t+(bhu{)}_x=0$. The integrated hydrostatic excess pressure is ${\rho }_0{g}^{\prime }b{h}^2/2$. Pressure on the sloping side walls contributes ${\rho }_0{g}^{\prime }{h}^2{b}^{\prime }/2$ to the longitudinal force. The conservative momentum equation is thereforeUse the volume equation to simplify it. The resulting shallow water equations areOmitting the wall force would incorrectly introduce a width term into the material acceleration equation.

For $(h,u)$ the principal matrix is $\left(\begin{array}{cc}u& h\\ g^{\prime }& u\end{array}\right)$. Its eigenvalues are $u\pm c$, where $c=\sqrt{g^{\prime }h}$. For $h>0$ they are real and distinct, so the system is strictly hyperbolic. It degenerates at a dry front. From the volume equation, $c_t+u{c}_x+(c/2)u_x=-(cu/2)b^{\prime }/b$. Combine this with the momentum equation to obtain the variable-width shallow-water characteristics:In a constant-width channel $u\pm 2c$ are Riemann invariants along the corresponding characteristic curves; with varying width they satisfy compatibility equations with a source term.

The volume alone does not specify the early release geometry. For a concrete sketch, use a rectangular lock initially occupying $0<x<{\ell }_0$ with depth $h_0=V/{\ell }_0$, at rest, and remove its gate at $t=0$. Let $c_0=\sqrt{g^{\prime }{h}_0}$ and impose a gravity-current front condition $u_f=F\sqrt{g^{\prime }{h}_f}$, with fixed $F$. The Benjamin deep-ambient front condition gives $F=\sqrt{2}$ for the usual deep-ambient idealization.

A rarefaction wave propagates back from the release. Before information returns from the closed wall, its invariant is $u+2c=2c_0$. With $s=(x-{\ell }_0)/t$, the fan has $u=2(c_0+s)/3$, $c=(2c_0-s)/3$. The front-adjacent shelf hasThe fan extends from $s=-c_0$ to $s=u_f-c_f$; a uniform shelf connects its tail to the finite-depth head at $x={\ell }_0+u_{f}t$. A small nonhydrostatic head displaces ambient fluid; the hydrostatic interior model does not resolve its overturning structure.

The fan head first reaches the rear wall at $t_r={\ell }_0/c_0$. The wall condition is $u=0$, and the returned positive characteristic curve begins to alter the interior. In the still-undisturbed fan it solves $dx/dt=(4c_0+(x-{\ell }_0)/t)/3$, henceIt reaches the fan's uniform shelf at $t_{\ast }=t_r[(F+2)/2{]}^{3/2}$, then propagates with speed $u_f+c_f$ to the head at $t_{\mathrm{hit}}=2t_{\ast }$. These times refer to this particular initial lock and front closure. After that first arrival, the finite supply influences the front: the constant-speed slumping stage gives way to deceleration and eventually the finite-volume inertial gravity current with $L\propto t^{2/3}$, depth proportional to $t^{-2/3}$ and speed proportional to $t^{-1/3}$. The initial fan and reflected signal are shown below; later characteristic trajectories change as the current adjusts.

Use material coordinates $X$ and write the placement as $y=X+u$. The strain energy density $W$ in the printed convention is per unit reference mass. It is convenient to use the conventional first Piola-Kirchhoff stress tensor $P=s^{\mathsf{T}}$, soAssume the undeformed reference state is stress-free. With $H=A-I$ and infinitesimal strain tensor $e=(H+H^{\mathsf{T}})/2$, material isotropy and material frame indifference give the quadratic expansionConsequently the Lamé parameters from hyperelastic energy Hessian areIndeed, differentiating the quadratic expression twice givesthe elastic stiffness tensor of linear elasticity. A prestressed reference state requires the corresponding initial-stress corrections; the displayed formula concerns the natural state.

For a plane elastic wave, $u=u(X_1,t)$, put $r=e_1+u_{,X_1}$ and $\Psi (r)=W([r,e_2,e_3])$. Balance of momentum without body force becomesThis is a material-coordinate equation: the reference density cancels between inertia and nominal stress. If $C{a}_{\alpha }=c_{\alpha }^2{a}_{\alpha }$, the first-order system has characteristic speeds $\pm c_{\alpha }$ and right eigenvectors $(a_{\alpha },\mp c_{\alpha }{a}_{\alpha })$. Along a characteristic curve with $d{X}_1/dt=s=\pm c_{\alpha }$, its differential compatibility relation isThis follows by multiplying the system by the left eigenvector $(-s{a}_{\alpha }^{\mathsf{T}},a_{\alpha }^{\mathsf{T}})$. It does not assert the existence of globally integrable Riemann invariants for an arbitrary three-component constitutive law.

There is a necessary qualification to the finite-amplitude transverse request. Set $u=(0,w,0)$ and $q=w_{,X_1}$, so $r=(1,q,0)$. The longitudinal momentum equation requiresThus arbitrary nonuniform pure transverse profiles are possible only if the compatibility condition for pure nonlinear transverse waves holds:The third component vanishes by the reflection symmetry implied by isotropy and objectivity. The shear equation then closes:

General compressible hyperelastic material assumptions alone do not imply the required compatibility. For a concrete counterexample, take the Saint Venant–Kirchhoff hyperelastic energyFor $A=[(a,q,0{)}^{\mathsf{T}},e_2,e_3]$, this becomesAt $a=1$, $P_{11}=(\lambda +2\mu )q^2/2$. A varying shear therefore produces a longitudinal force $(\lambda +2\mu )q{q}_{,X_1}$ and cannot preserve $u_1=0$. Infinitesimal transverse waves do exist, because this coupling is second order, but a general pure finite-amplitude transverse wave does not.

The compatible nonlinear shear energy class is nonempty and can be genuinely nonlinear. Define $J=\det A$, $S=\mathrm{tr}(A^{\mathsf{T}}A)-2J$, and, for $h\ge 0$, takeThis is objective and isotropic. Near $A=I$, $S-1=2\mathrm{tr}(e^2)-(\mathrm{tr}e{)}^2+O(|H{|}^3)$, so it has the specified Lamé parameters. On the plane deformation above, $S-1=(a-1{)}^2+q^2$ and $J=a$. Hence $P_{11}=0$ at $a=1$, whileFor positive $\lambda ,\mu$, the relevant incremental plane moduli are positive. This supplies an explicit stable pure transverse family and, when $h>0$, amplitude-dependent wave speed.

For any compatible family, letThe equations $q_t=v_{,X_1}$ and $v_t=c(q{)}^2{q}_{,X_1}$ give the Riemann invariantsConsider the initially undisturbed half-space $X_1>0$. A right-going simple wave has the incoming invariant $v+H=0$, so $v=-H(q)$. Denote the prescribed boundary nominal stress by $T(t)=P_{21}(0,t)=\tau (q_b(t))$. The physical boundary traction, whose outward normal is $-e_1$, is $-T(t)e_2$. Invert the monotone law $\tau$ to obtain $q_b$. The wave state emitted at boundary time $s$ is transported alongAhead of the leading front $X_1=c(0)t$ the state is zero. Before characteristic crossing, these equations determine the unique emission time $s$ for each disturbed point. The displacement itself is also determined, rather than just its derivatives:Differentiating the implicit emission relation verifies $w_{,X_1}=q_b(s)$ and $w_t=-H(q_b(s))$, so this is the complete pre-shock simple wave solution.

Write $c_b(s)=c(q_b(s))$. The boundary-generated characteristic curves lose invertibility whenThus the characteristic crossing in a boundary-generated elastic simple wave criterion isHere ${\dot{c}}_b=c^{\prime }(q_b)T^{\prime }/({\rho }_0{c}_b^2)$. A smoothly increasing boundary stress therefore steepens into a shock wave only when later states travel faster: $c^{\prime }(q_b)T^{\prime }>0$. Increasing stress alone is insufficient. For the compatible example with $h>0$ and $q_b>0$, $c^{\prime }(q_b)=3h{q}_b/({\rho }_0{c}_b)>0$, so a nonzero increasing ramp produces finite-time crossing. For $h=0$ all rays are parallel and no such shock forms. This criterion locates the first gradient singularity; subsequent discontinuous evolution additionally requires the jump conditions and an admissibility condition.

For a general vector simple wave, replace the pure-shear line by an integral curveSubstitution gives $a_t+c_{\alpha }(a)a_{,X_1}=0$, with the same emission construction and crossing criterion. Boundary tractions must lie on this one-family curve. Generic finite-amplitude quasi-transverse waves may contain longitudinal displacement; arbitrary vector boundary forcing need not generate a single simple wave.

On a region with smooth positive area, set $q=G(Z)Q(\theta ,\zeta )$, $\zeta =h(Z)$. Substitution into the simple wave equation givesRemove the last term by $G^{\prime }/G=-A^{\prime }/(2A)$, and choose $h^{\prime }=G$. Starting at a regular section $Z_0$, one convenient normalized area transformation of a nonlinear acoustic simple wave isIn the notation $g(\zeta )$, set $g(\zeta )=G(h^{-1}(\zeta ))$. The remaining equation is $Q_{\zeta }-Q{Q}_{\theta }=0$. With initial value $f$ at $Z_0$, the method of characteristics givesThe map from the initial label loses invertibility when $1-f^{\prime }(\xi )\zeta =0$. For smooth data with bounded derivative and $M=\sup f^{\prime }>0$, the first forward characteristic crossing is at ${\zeta }_s=1/M$; if $M\le 0$, no forward shock occurs.

For spherical spreading, write physical radius as $R>0$ and specify data at an emitting radius $R_0>0$. Since $A(R)=R^2$,The shock distance for a spherical simple wave launched at finite radius isIf the axial distance is measured from the emitting surface, $Z=R-R_0$, then $A(Z)=(R_0+Z{)}^2$ andNo finite shock distance exists in this model when $M\le 0$.

The printed combination $A=Z^2$ with general bounded data at $Z=0$ is singular and does not define the intended outgoing spherical initial-value problem. Along a spherical characteristic, $d\theta /dR=-q$ and $dq/dR=-q/R$, so $Rq$ is constant. A bounded trace at $R=0$ forces that constant to be zero for every characteristic curve emanating from the origin; arbitrary nonzero bounded $f$ cannot be launched there. For example, the ostensibly admissible data $f\equiv 1$ would demand both $Rq=0$ and a nonzero trace, a contradiction. The logarithmic transformed distance is also divergent at the origin. Thus an unqualified numerical shock formula from those literal data would be unjustified. The finite-radius formulas above give the physical interpretation once the launching radius, or the shifted area law, is supplied.

For $\alpha =1$, $f=V_0(s^2-s)$, so the characteristic speed is $a=V_0(2s-1)$ and the conservation law flux is strictly convex. Let $\tau =V_{0}t$ and let $\xi$ be the initial vertical position of a characteristic curve.

For the increasing ramp, the characteristic curves within its sloping part satisfy $z=\xi (1+0.6\tau )$. They spread apart and never cross. The outer plateau characteristic speeds are $-0.6V_0$ and $0.6V_0$, so the complete entropy solution isThe transition widens linearly and its gradient decreases. At large time it approaches a rarefaction wave, with $s\simeq (1+z/\tau )/2$ inside its expanding transition.

For the decreasing ramp, $z=\xi (1-0.6\tau )$ and the characteristic curves converge. Before they meet,All the ramp characteristic curves meet at $z=0$ atThereafter a stationary sedimentation shock separates $s_L=0.8$ below from $s_R=0.2$ above. Indeed the Rankine-Hugoniot condition givesand the incoming characteristic speeds satisfy $a(0.8)=0.6V_0>0>a(0.2)=-0.6V_0$, which makes it an admissible entropy shock. The discontinuity transports no net excess particle flux because both plateau fluxes equal $-0.16V_0$. These are local solutions before a finite container's boundaries affect the transition.

The dependence on $\alpha$ is controlled by the hindered-settling flux inflection:For any smooth initial profile, the characteristic flow map has derivativeThis formula gives the first characteristic crossing: minimize the positive values of $-1/[f^{\prime \prime }(s_0)s_0^{\prime }]$. At $\alpha =0$ neither ramp changes shape; both move down at $V_0$. For $0<\alpha \le 3/2$, the flux is convex throughout $0.2\le s\le 0.8$: the increasing ramp spreads and the decreasing ramp steepens, although its entire ramp generally no longer focuses simultaneously. For $\alpha \ge 9$, the flux is concave throughout that interval and these roles reverse. At the endpoint values the curvature vanishes at one plateau; it does not reverse sign inside the interval.

For $3/2<\alpha <9$, the inflection $s_i=2/(\alpha +1)$ lies between the plateau concentrations. Below $s_i$ the increasing ramp spreads, while above it that ramp compresses; the decreasing ramp has the opposite local behavior. After characteristic crossing, the entropy solution can contain both a shock wave and a rarefaction wave. Quantitatively, an increasing step is selected by the lower convex envelope of $f$ on the interval between its states: curved portions supply rarefaction waves, and straight portions supply shock waves. A decreasing step uses the upper concave envelope. A join between a fan and a shock wave at a state $s_{\ast }$ satisfies the tangent-chord relation $f^{\prime }(s_{\ast })=[f(s_b)-f(s_{\ast })]/(s_b-s_{\ast })$, where $s_b$ is the other shock state. Thus curvature, not simply the magnitude of $\alpha$, decides the wave pattern; the envelope may reduce to one shock if the tangent lies outside the state interval.