= 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 gives
$$
1+r=t,\qquad Z_0(1-r)=Z_1t,
\qquad\boxed{r=\frac{Z_0-Z_1}{Z_0+Z_1},\quad t=\frac{2Z_0}{Z_0+Z_1}}.
$$
Reflection reverses the displacement sign when the receiving <seismic impedance> is larger. The energy-flux fractions are
$$
\boxed{\mathcal R=|r|^2,\qquad\mathcal T=\frac{Z_1}{Z_0}|t|^2,\qquad\mathcal R+\mathcal T=1}.
$$
The 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 $h_j$,
$$
\binom{V}{P}_{\!j,\mathrm{bottom}}=
M_j\binom{V}{P}_{\!j,\mathrm{top}},\qquad
M_j=\begin{pmatrix}\cos\delta_j&i\sin\delta_j/Z_j\\ iZ_j\sin\delta_j&\cos\delta_j\end{pmatrix},
\qquad\delta_j=\frac{\omega h_j}{v_j}.
$$
This follows by solving $V_z=i\omega P/C_j$ and $P_z=i\omega\rho_jV$. Continuity means that the state passes unchanged across each welded interface. Put $M=M_N\cdots M_1=\begin{pmatrix}a&b\\c&d\end{pmatrix}$, 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 $(t,Z_Rt)$. Eliminating these states yields
$$
D=Z_Ra+Z_Ld-c-Z_LZ_Rb,
\qquad\boxed{r=\frac{c+Z_Ld-Z_Ra-Z_LZ_Rb}{D},\qquad t=\frac{2Z_L}{D}}.
$$
For 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 $Z_j=\sqrt{Z_LZ_R}$ 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 is
$$
\cos(Kd)=\cos\delta_1\cos\delta_2
-\frac12\left(\frac{Z_1}{Z_2}+\frac{Z_2}{Z_1}\right)\sin\delta_1\sin\delta_2.
$$
This 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, $C_{\rm eff}^{-1}=\langle C^{-1}\rangle$. 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. \b[Impedance contrasts generate reflections, propagation phases organize their interference, and energy conservation separates reflection from genuine absorption].
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