Past exam of the mathematics course of the University of Cambridge / 2015 / iii / Paper 76 / 1

Use the time-harmonic wave convention $\mathrm{Re}(\psi e^{-i\omega t})$, with a positive background wavenumber $k$. The scalar Helmholtz equation for a constant-density acoustic model isWriting $\psi =e^{ikx}E$ gives the exact envelope equationThe paraxial approximation neglects $E_{xx}$ relative to $2ik{E}_x$, giving the parabolic wave equationThe choice of carrier and the direction of propagation matter: this is a forward, slowly varying envelope approximation. Sufficient scale conditions are $|E_{xx}|\ll 2k|E_x|$, transverse spectral components $|p|\ll k$, and medium/envelope variation on longitudinal scales large compared with $1/k$. With the carrier fixed at the background $k$, small $|n-1|$ makes the refractive phase vary slowly too. Large-angle propagation, appreciable backscattering, or rapid longitudinal variation violates the approximation. Acoustic models with variable mass density can have additional gradient terms, so the scalar Helmholtz equation itself is a model assumption. For the Gaussian beam in part (b), useful initial conditions are $kD\gg 1$ and $D/|F|\ll 1$, with the second condition absent for an initially uncurved beam. A narrow angular spectrum, rather than merely the label “Gaussian”, justifies the paraxial approximation.

PutIn free space, the parabolic wave equation is $E_x=i{E}_{zz}/(2k)$. Under the Fourier transform convention $\hat{E}(p)=\int E(z)e^{-ipz}dz$, it becomesThe Gaussian integral is legitimate because $\mathrm{Re}A=1/D^2>0$. Invert the transform after multiplying by $e^{-i{p}^{2}x/(2k)}$. A second Gaussian integral, or equivalently one-dimensional transverse Fresnel propagation, givesChoose the square-root branch continuously from $q(0{)}^{-1/2}=1$; for real $x\ge 0$ there is no zero of $q$. This gives the correct incident field at $x=0$, and direct differentiation verifies the free parabolic wave equation.

For clarity, the squared envelope magnitude isThe Gaussian beam with one transverse coordinate remains Gaussian, with one-transverse-coordinate amplitude factor $q^{-1/2}$, not the $q^{-1}$ of a beam with two transverse coordinates. The negative initial quadratic phase produces focusing for $F>0$; diffraction prevents a singularity at $x=F$. These expressions describe the paraxial approximation to free propagation, rather than an exact unrestricted Helmholtz equation beam.

No: the first moment generally depends on the transverse coordinate. At entry to the random medium, the initial data are deterministic, sowhich is a nonconstant Gaussian beam profile. Statistical homogeneity of the medium says that shifting both the medium and the incident data gives correspondingly shifted field statistics. It does not make the response to fixed, localized incident data translation invariant.

In particular, Gaussian coherent-field propagation in the Markov approximation, derived in part (d), givesfor the linear weak-index model. A homogeneous attenuation factor multiplies the varying beam profile. Thus spatially homogeneous medium statistics can coexist with a transversely inhomogeneous coherent field. The first moment is not the mean wave intensity, and disappearance of the coherent component is not itself disappearance of the total wave energy.

Write $m=⟨E⟩$ and $L_0=i{\partial }_z^2/(2k)$. Substituting $n=1+\mu W$ into part (a), without prematurely replacing $n^2$ by its linear part, givesTaking the expectation yields the exact mean equation within the paraxial approximation:The correlation term cannot be discarded just because $⟨W⟩=0$: the field depends on the same random medium as $W$. Nor can $⟨W^{2}E⟩$ be factored exactly from the given variance alone. The printed statistics do not specify the autocorrelation function of a random field, so they do not determine a unique local closed equation for $m$.

A useful covariance-dependent equation follows to second order in $\mu$. Defineand letbe the Fresnel propagator kernel for one transverse coordinate. The Duhamel principle applied to the linear fluctuation term giveswhere $E^{(0)}=E_{\mathrm{free}}$. Multiply by $W(x,z)$ and average. Also $⟨W(x,z{)}^{2}E(x,z)⟩=E^{(0)}(x,z)+O(\mu )$. Since $m=E^{(0)}+O({\mu }^2)$, replacing $E^{(0)}$ by $m$ in terms already multiplied by ${\mu }^2$ preserves second-order accuracy. We obtain the weak-fluctuation memory equation for the coherent fieldThis perturbation statement is for fixed propagation intervals with sufficient covariance regularity and moment bounds. The kernel is understood as an oscillatory integral. If the field is jointly Gaussian, odd perturbative contributions vanish and the remainder can be improved under the corresponding expansion hypotheses. Joint Gaussianity itself still leaves the covariance unspecified. The term $ik{\mu }^{2}m/2$ is the quadratic refractive-index shift in the coherent field; it comes from the mean of $W^2$.

For the usual local answer, impose the additional Markov approximation for a random medium. Its longitudinal correlation length must be much shorter than the distance on which the envelope varies, and diffraction across that correlation distance must not resolve substantial transverse covariance variation. Away from the entry layer, replace the slowly varying factors in the memory integral by their local values and extend the longitudinal integral to infinity. Define the integrated covarianceFor an integrable real stationary covariance, $C(s,0)$ is even, and $B(0)\ge 0$ is its zero-longitudinal-frequency spectral density. The conventional leading weak-index model keeps only the linear random potential $ik\mu W$; in that model the local first-moment equation isThe factor $1/2$ can be checked independently in the white-noise model. If ${\mathcal{B}}_x(z)$ is a Brownian field with $⟨d{\mathcal{B}}_x(z)d{\mathcal{B}}_x(z^{\prime })⟩=B(z-z^{\prime })dx$, writeConverting the Stratonovich integral to an Itô integral adds $-k^2{\mu }^{2}B(0)Edx/2$. The remaining stochastic integral has zero mean, giving the boxed equation andIf the finite-correlation model retains $n^2$ consistently through order ${\mu }^2$, the local second-order expansion also has $+ik{\mu }^{2}m/2$, adding the phase factor $e^{ik{\mu }^2(x-x_0)/2}$. This phase is computed before the white-noise idealization: squaring ideal Gaussian white noise is not the finite-variance operation $W^2$.

The need for a correlation assumption can be seen without any closure argument. The stationary Gaussian random field $W(x,z)=Z$, with a single standard normal variable $Z$, has all the printed one-point statistics. In the linear weak-index model it givesIts attenuation is quadratic in propagation distance, whereas the Markov model gives linear-distance exponential attenuation. The local attenuation equation requires the additional correlation/Markov assumption; it does not follow from statistical homogeneity and unit variance alone.