Past exam of the mathematics course of the University of Cambridge / 2003 / iii / Paper 78 / 1 / Solution

At normal incidence the P wave and the two S wave polarizations decouple. In each uniform layer the relevant displacement satisfies $\rho u_{tt}=C{u}_{zz}$, with $C=\lambda +2\mu$ for a longitudinal wave and $C=\mu$ for a shear wave. Its speed is $v=\sqrt{C/\rho }$ and its seismic impedance is $Z=\rho v$. A welded interface preserves particle velocity and traction. For displacement or velocity amplitudes, incidence from medium 0 onto medium 1 therefore givesReflection reverses the displacement sign when the receiving seismic impedance is larger. The energy-flux fractions areThe impedance factor matters: a displacement transmission coefficient larger than one does not imply energy amplification. Equal impedance eliminates reflection even when the densities and speeds differ.

For a stack, every interface creates reflected waves which can return to all earlier interfaces. A seismic layer transfer matrix incorporates those multiple reflections exactly. Use harmonic time dependence $e^{-i\omega t}$, particle velocity $V$ and compressive traction $P=-{\sigma }_{zz}$ for a P-wave, or the negative relevant shear traction for an S-wave. Through layer $j$ of thickness $h_j$,This follows by solving $V_z=i\omega P/C_j$ and $P_z=i\omega {\rho }_{j}V$. Continuity means that the state passes unchanged across each welded interface. Put $M=M_N\cdots M_1=\left(\begin{array}{cc}a& b\\ c& d\end{array}\right)$, with $ad-bc=1$, and let the exterior impedances be $Z_L,Z_R$. A unit incident velocity gives top state $(1+r,Z_L(1-r))$ and bottom state $(t,Z_{R}t)$. Eliminating these states yieldsFor real moduli and no attenuation, conservation of elastic-wave energy flux gives $|r{|}^2+(Z_R/Z_L)|t{|}^2=1$. Reciprocal incidence gives equal energy transmission in the reverse direction; flux-normalized amplitudes express that reciprocity without unequal exterior impedances obscuring it.

The wave phase accumulated in each layer determines constructive or destructive interference. For one layer between identical exterior media, an integral number of half-wavelengths makes $M_j=\pm I$ and gives perfect transmission, regardless of the layer impedance. At quarter-wave phase thickness, a single matching layer with $Z_j=\sqrt{Z_L{Z}_R}$ also gives zero reflection: the layer transforms the terminal impedance to $Z_j^2/Z_R=Z_L$. Away from these matching conditions there are transmission peaks, reflection maxima and rapid frequency-dependent phase changes. In the time domain these are a direct transmitted pulse followed by delayed internal reverberations and reflected echoes. Long residence in a resonant stack produces narrow peaks and long ringing, without creating energy.

For a periodic two-layer stack, the half-trace of one cell isThis is the periodic seismic multilayer stop band criterion. When the absolute right side exceeds one, the cell eigenvalues are a reciprocal growing/decaying pair and a long finite stack has exponentially small transmission. The missing transmitted energy is reflected, not absorbed. Large impedance contrast and repeated quarter-wave layers strengthen such stop bands. In pass bands the cell modes propagate, and a finite stack superposes transmission resonances on that propagation. An irregular stack need not have equally spaced peaks: the individual travel times and impedance contrasts set its interference pattern.

At sufficiently low frequency a finite stack has $M\to I$, so its leading reflection is that of the two exterior media alone. For many thin repeated layers with wavelength much longer than a cell, their effective density is the volume-weighted density and their normal modulus is the harmonic mean, $C_{\mathrm{eff}}^{-1}=⟨C^{-1}⟩$. This follows by expanding each propagation matrix to first order and averaging its two coefficients. At high frequency, sharp interfaces still reflect: there is no general disappearance of reflection, only increasingly rapid interference fringes. Smooth impedance grading instead suppresses short-wavelength reflection when its variation scale is long compared with the wavelength.

These calculations provide a reproducible investigation: vary the layer travel times, impedances and number of cells; compute the matrix product; plot reflected/transmitted fluxes and check their sum. Adding attenuation makes the flux sum smaller than one and reduces reverberation peaks, while oblique incidence introduces Snell law for elastic waves and P-SV mode conversion absent from this one-dimensional setting. Impedance contrasts generate reflections, propagation phases organize their interference, and energy conservation separates reflection from genuine absorption.