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The tympanic membrane and ossicular coupling system that transfers airborne vibration toward the fluid-filled inner ear. Its mechanical transformation helps address an acoustic impedance mismatch; it is not an ideal, lossless, frequency-independent amplifier.

The normal acoustic impedance of a progressive acoustic wave at a planar boundary is the pressure amplitude divided by the normal-velocity amplitude. For propagation angle $\theta$ from the normal,The distinction from the acoustic impedance $\rho c$ matters at oblique incidence: interface reflection depends on the ratio of normal impedances. For an evanescent outgoing wave $k_n$ is imaginary and the normal impedance is reactive, giving no mean normal acoustic energy flux.

Fix a positive normal from medium 1 toward medium 2, and define all velocity amplitudes in that coordinate. A right-going linear sound wave satisfies $p^{\prime }=Z{u}^{\prime }$ and a left-going wave satisfies $p^{\prime }=-Z{u}^{\prime }$, with acoustic impedance $Z=\rho c_s$. Continuity gives $u_i+u_r=u_t$ and $Z_1(u_i-u_r)=Z_2{u}_t$. Eliminating $u_t$ gives the displayed reflected-to-incident velocity ratio. The pressure reflection coefficient is its negative. A surface mass, surface restoring force or oblique incidence changes the boundary calculation.

Let all velocity amplitudes refer to the same positive normal direction, from medium 1 into medium 2. For a right-going linear sound wave, its pressure amplitude is $p^{\prime }=Z{u}^{\prime }$ with acoustic impedance $Z=\rho c_s$; for a left-going wave it is $p^{\prime }=-Z{u}^{\prime }$. Continuity of normal velocity and pressure at the interface givesEliminating $u_t$ yields the normal-incidence acoustic velocity reflection coefficientThe common adiabatic index and mean particle mass $\bar{m}$ give $c_{s,j}=\sqrt{\gamma k_B{T}_j/\bar{m}}$, soA negative coefficient reverses the reflected velocity perturbation's phase in this fixed coordinate system. The reflected pressure coefficient has the opposite sign. Pressure balance also gives ${\rho }_1{T}_1={\rho }_2{T}_2$, so the impedance ratio can equivalently be written $Z_2/Z_1=\sqrt{{\rho }_2/{\rho }_1}$. These are normal-incidence, small-amplitude results without a surface mass or tension.

Take incidence from A into B and first specify pressure amplitude. For a plane acoustic wave traveling normally to the interface, the acoustic impedance is $Z=\rho c_s$. A right-moving wave has pressure amplitude $P=ZU$, where $U$ is its fluid-velocity amplitude; a left-moving reflected wave has $P=-ZU$.

Let $P_i,P_r,P_t$ denote incident, reflected, and transmitted pressure amplitudes. Continuity of pressure and normal velocity at the interface givesSolving the two linear equations gives the acoustic reflection and transmission at an interface coefficientsFor the same perfect gas, take a common adiabatic index $\gamma$. The adiabatic sound speed satisfies $c_s^2=\gamma p/\rho$. The equal equilibrium pressure therefore implies $Z_B/Z_A=\sqrt{{\rho }_B/{\rho }_A}=\sqrt{\delta }$, givingThe word amplitude is otherwise ambiguous. For fluid velocity, or displacement at the same frequency, the corresponding answer isThese are consistent because pressure and velocity amplitudes differ by the impedance. As an energy check, the time-averaged acoustic intensity is $|P{|}^2/(2Z)$, so its transmission coefficient is $\mathcal{T}=4\sqrt{\delta }/(1+\sqrt{\delta }{)}^2\le 1$, even when the transmitted pressure amplitude exceeds the incident one.

Equal mean molecular weight alone does not logically guarantee equal $\gamma$. If the gases have different values of the adiabatic index, replace $\sqrt{\delta }$ in these formulas by $\sqrt{{\gamma }_B\delta /{\gamma }_A}$; a density-only answer requires the common-$\gamma$ assumption.

The linear acoustic momentum and continuity equations are $\rho u_t=-p_x$ and $p_t=-\rho c^2{u}_x$. For a right-moving plane wave depending on $x-ct$, integration gives $\boxed{p=\rho cu}$, with zero perturbation constants. For a left-moving wave the sign is reversed. The factor $Z=\rho c$ is the acoustic impedance. Proportionality applies to a traveling wave; a superposition of opposite directions generally has no single such constant.

Use time dependence $e^{-i\omega t}$ and set $Z_j={\rho }_j{c}_j$. In the outer regions write pressures $I{e}^{i{k}_{0}x}+R{e}^{-i{k}_{0}x}$ and $TI{e}^{i{k}_{0}x}$, taking $I=1$ for convenience. The corresponding velocities are $(e^{i{k}_{0}x}-R{e}^{-i{k}_{0}x})/Z_0$ and $T{e}^{i{k}_{0}x}/Z_0$. Pressure and normal velocity are continuous at both layer boundaries. Within the cooled layer a decomposition into right and left waves gives, with $\delta =k_{1}L$,This matrix follows by writing $p=A{e}^{i{k}_{1}x}+B{e}^{-i{k}_{1}x}$ and $Z_{1}u=A{e}^{i{k}_{1}x}-B{e}^{-i{k}_{1}x}$. Applying its inverse and using $p(0)+Z_{0}u(0)=2$ eliminates $R$, givingConsequentlyFor $\lambda \gg 1$, transmission is weak except near $k_{1}L=n\pi$, where $|T|=1$. The cooled layer acts as a resonant filter: an integer number of half-wavelengths fits within it at each resonance. The half-power width in $\delta$ is asymptotically $2/\lambda$ on either side. Since the two outer impedances agree, the transmitted power fraction is $|T{|}^2$.

For a progressive plane wave, writeThe momentum equation gives ${\rho }_0\omega U=kP$, while the dispersion relation is $\omega =c_{0}k$. Therefore the pressure and longitudinal velocity amplitudes obeyThe constant $Z$ is the acoustic impedance.

Let $Z_j={\rho }_j{c}_j$, $k_j=\omega /c_j$, and $\varphi =k_{1}L$. Use the harmonic convention $e^{-i\omega t}$ and write the pressure and velocity amplitudes asContinuity of pressure and normal velocity at both interfaces is equivalently the layer transfer relationSubstituting $p(0)=1+R$, $u(0)=(1-R)/Z_0$, $p(L)=T$, and $u(L)=T/Z_0$, then adding the two resulting equations, givesHence the acoustic transmission through a uniform layer isFor either $\lambda \ll 1$ or $\lambda \gg 1$, the strong acoustic impedance mismatch makes transmission small except near $k_{1}L=n\pi$, where $|T|=1$. Away from those resonances,At the antiresonant points $k_{1}L=(n+\frac{1}{2})\pi$, the minimum is $2/(\lambda +{\lambda }^{-1})$.

No reflection requiresUsing $\lambda =\rho c_P^2$, ${\lambda }^{\prime }={\rho }^{\prime }(c_P^{\prime }{)}^2$, and
$\sin {\theta }^{\prime }=(c_P^{\prime }/c_P)\sin \theta$, this reduces to the normal acoustic impedance matching conditionAfter squaring and eliminating ${\theta }^{\prime }$, the required incidence angle satisfiesSuch a no-reflection angle exists only when the right-hand side lies in $[0,1]$ and the transmitted wave is propagating.

Hearing requires estimating sound-producing events from noisy mixtures at two ears. It involves mechanical detection, neural coding, source segregation and recognition. A frequency analysis alone cannot identify who or what produced the sound.

The physical input is an acoustic pressure fluctuation. At one ear a simplified room model iswhere source signals $s_j$ are filtered by propagation and reflection paths $h_j$, and $n$ is noise. The convolution model assumes approximately linear, time-invariant propagation over the analysis interval; a moving source or listener can violate that simplification. This is a many-to-one inverse problem: different source configurations can produce similar ear signals. Distance, reverberation and head orientation alter a source's waveform. Pressure level is logarithmic: sound pressure level is $L_p=20{\log }_{10}(p_{\mathrm{rms}}/p_0)$, with reference $p_0=20\mu \mathrm{Pa}$ in air. A 100 dB difference corresponds to a pressure ratio $10^{100/20}=10^5$. The system must remain sensitive to weak signals while avoiding saturation over a large operating range.

The middle ear helps transfer airborne vibrations into cochlear fluids despite an acoustic impedance mismatch. Within the cochlea, the graded mechanics of the basilar membrane produce frequency-dependent traveling-wave responses: high frequencies peak nearer the base, low frequencies farther toward the apex. This tonotopy provides a distributed spectral representation. Deflection of sensory hair cells drives mechanotransduction; inner hair cells provide most afferent signaling, while outer hair cells contribute to mechanical amplification and tuning. Detection therefore depends on an active biological filter, not merely a passive bank of resonators.

A revealing experiment by Mario A. Ruggero and colleagues measured chinchilla basilar-membrane vibrations using the Mössbauer technique or Doppler laser velocimetry before and after a reversible intervention that reduced cochlear transduction currents. Responses near the characteristic frequency were reduced particularly strongly for weak sounds, with smaller changes at high levels and away from that frequency.^[ref] This supports feedback-dependent cochlear gain. The intervention also affects other cochlear processes, so it does not by itself identify every cellular contribution or directly measure human loudness. The important methodological contrast is controlled sound input plus measured mechanical output and a reversible physiological perturbation.

The resulting neural code must preserve several attributes simultaneously. Place of excitation and neural phase locking can provide complementary frequency and temporal information; firing rates, adaptation and population responses encode level and envelopes. Individual neurons have limited dynamic ranges, and temporal precision has limits. Pitch is not simply the location of the strongest spectral component, and timbre depends on spectral shape and temporal development. Even physically, a signal containing harmonics at 200 and 300 Hz repeats every 10 ms although it contains no 100 Hz component: both components complete integer cycles in that interval. Recovering source periodicity is therefore different from selecting a component in a frequency spectrum. It does not follow that every such signal has one obligatory perceptual pitch.

Localization adds ambiguity. Ear separation yields interaural time differences; head shadow yields interaural level differences; outer-ear filtering supplies direction-dependent spectral cues. In an ideal far-field point-receiver model, $\Delta t\simeq d\sin \theta /c$, obtained by dividing the path-length difference by sound speed. Real head diffraction modifies this approximation. One time difference can be compatible with several spatial directions, and phase is ambiguous when several cycles fit the delay. Head-related transfer functions, head movements and environmental context help resolve these ambiguities. Reverberation also requires distinguishing useful early information from later reflections.

Competing sources create both energetic masking, in which the target is poorly represented against overlapping acoustic energy, and informational masking, involving source confusion or selection despite available acoustic cues. Common onsets, harmonic relations, coherent modulation, timbre and spatial cues help group components into auditory objects. The grouping must remain stable when streams overlap, but also update when a source changes. These are the problems of auditory scene analysis.

E. Colin Cherry's classic experiments used simultaneous spoken messages and separate messages delivered to the two ears. Asking listeners to repeat one message while ignoring the other made source selection experimentally measurable.^[ref] Ear and voice differences can aid selection; the result does not show that unattended information is never processed, nor that real rooms provide the clean separation of headphone channels.

Finally, meaningful recognition requires tolerance of speaker, rate and context variation. Richard M. Warren removed a speech segment and substituted a cough-like sound. Listeners could hear the utterance as continuous and had difficulty locating the replacement, whereas a silent interruption was more readily detected.^[ref] Phonemic restoration shows that recognition uses constraints beyond the momentary waveform; it does not mean missing acoustic information has been physically recovered. Successful auditory perception combines peripheral sensitivity with grouping and context-sensitive inference, while retaining the possibility of masking, ambiguity and systematic error.

For a plane sound wave, acoustic impedance is $Z=p/u=\rho c$, with pressure amplitude $p$ and particle-velocity amplitude $u$. Continuity of pressure and normal velocity at normal incidence on a lossless interface givesMatching $Z_1=Z_2$ eliminates reflection. A quarter-wave intermediate layer with $Z_m=\sqrt{Z_1{Z}_2}$ transforms the load to the source impedance at its design frequency; its bandwidth is limited. For strings, the analogous mechanical impedance is transverse force divided by velocity, $Z=T{v}^{-1}=\sqrt{T\mu }$; continuity of displacement and transverse force gives the corresponding matching condition. These amplitudes use different force/velocity conventions from acoustic pressure, so reflection signs must be stated consistently.

At normal incidence between lossless media with real positive acoustic impedances $Z_1,Z_2$, continuity of pressure and normal particle velocity gives the pressure transmission amplitude $t_p=2Z_2/(Z_1+Z_2)$. Since the mean acoustic intensity is $|p{|}^2/(2Z)$, the fraction of incident energy transmitted isThis is acoustic reflection and transmission at an interface. With the supplied acoustic impedances,A thin water-based gel layer excludes air gaps and approximately matches the body's acoustic impedance. About $99.90\%$ of incident energy would otherwise reflect at the air-body interface. The coupling is also needed on the return journey, so it strongly improves the detectable echo. These figures concern interface transmission, before absorption within tissue; the gel is approximated by the stated water value.

A characteristic wave impedance relates an effort-like quantity to its conjugate flow-like quantity in a single travelling wave: for a mechanical wave these are force and velocity, and their product is power. The sign depends on the propagation and force conventions. Use the transverse force transmitted towards increasing $x$, $F=-T{\Psi }_x$. A rightgoing travelling wave $\Psi =f(x-vt)$ has transverse velocity $u={\Psi }_t=-v{\Psi }_x$, soThis is the positive characteristic impedance of a stretched string. A leftgoing component has $F/u=-Z$ with the same force orientation. Its units differ from pressure-based acoustic impedance because it uses force, rather than force per area.

The normal-incidence acoustic impedance is $Z=\rho V_P$. ThereforeThese are $2.25$ and $7.00$ MRayl. They combine the material’s density and wave speed; using $V_S$ would give shear rather than acoustic P impedance.

Choose using target depth, scale, expected contrasts, overburden, noise and required result. A seismic reflection survey uses reflected travel times and amplitudes to image interfaces; it needs useful acoustic impedance contrasts, sufficient bandwidth, suitable offsets and dense coverage for moveout correction, stacking and seismic migration. A large clay-cap/igneous contrast gives a potentially strong reflection here, but fractures, scattering and attenuation can obscure coherent events. At $100$ Hz the nominal quarter-wavelength vertical seismic resolution is about $3.75$ m in the claystone and $8.75$ m in the igneous rock, if that frequency reaches the target.

A seismic refraction survey uses first arrivals and critically refracted waves to estimate velocities and interface structure. The increase from $1500$ to $3500\mathrm{m}{\mathrm{s}}^{-1}$ is favorable for a cap-base head-wave branch, provided the spread reaches sufficient offsets. Refraction/tomography may better constrain lateral velocity variations and fractures, but simple head-wave inversion can miss a low-velocity layer or a thin hidden layer and often has poorer boundary detail.

Use expected reflector continuity, velocity inversions, required aperture, access, near-surface complexity and cost to choose or combine the surveys. A pilot gather and travel-time modelling can test which phases will be measurable; the supplied velocities alone do not specify the reservoir depth or the best acquisition geometry.

The normal-incidence acoustic impedance is $Z=\rho V_p$, with units $\mathrm{kg}{\mathrm{m}}^{-2}{\mathrm{s}}^{-1}$. ThusDefine the pressure-amplitude reflection coefficient and pressure-amplitude transmission coefficient for incidence downwards from layer $i$ into layer $j$ byHere $p$ denotes compressional normal-stress amplitude; in the water it is the acoustic pressure. Continuity of normal traction and normal particle velocity gives these expressions. They describe each interface separately, before any layer layer reverberation.

For a lossless normal-incidence plane wave, intensity is proportional to $p^2/Z$, soA transmitted pressure amplitude exceeding unity is therefore compatible with conservation of energy. If instead the coefficient is defined for particle velocity measured along the same downward coordinate, $r_v=-r_p$ and $t_v=2Z_i/(Z_i+Z_j)=1-r_p$: the two transmitted velocity coefficients are $0.8000$ and $18/29=0.62069$. Upward incidence reverses the sign of $r_p$, interchanges the transmitted pressure and velocity coefficients, and leaves $R_E,T_E$ unchanged. At normal incidence between these ideal isotropic layers there is no P-to-S seismic mode conversion.

Take horizontal interfaces, water depth $h_w=50\mathrm{m}$, and sediment thickness $h_s=20\mathrm{m}$. Both source and receiver are just below the sea surface. Every angle below is measured from the vertical interface normal. Snell's law preserves the horizontal seismic ray parameterA basement seismic head wave of speed $V_h$ has horizontal propagation at the basement interface, so $p=1/V_h$. The two possibilities are:

The water supports only P waves. At oblique incidence the seabed can convert a P wave into an SV-wave in the sediment, and the return SV-wave can convert back to a P wave in the water. Further seismic mode conversion can occur at the basement. Thus the down-going and up-going sediment legs can independently be P or S. The S entries refer to SV-waves; a fluid P wave source in this two-dimensional isotropic geometry does not excite the decoupled SH-wave sector.

Let $L_h\ge 0$ be the distance travelled along the basement interface. For sediment legs $a,b\in \{P,S\}$,The first-appearance distance is the limiting value at $L_h=0$:

For example, the unconverted P branch gives $2(50+20)\tan 36.870^{\circ }=105\mathrm{m}$. The earliest geometrically permitted basement P seismic head wave is the doubly converted path with sediment S legs, at approximately $90\mathrm{m}$. The corresponding basement S family first appears at approximately $297\mathrm{m}$. These limiting distances describe existence of ray branches; a finite detectable wave amplitude also depends on conversion efficiency and acquisition noise.

There is no separate finite-distance P head wave at the water–sediment interface: $V_{p,w}=V_{p,s}=1500\mathrm{m}{\mathrm{s}}^{-1}$, so the nominal critical angle is $90^{\circ }$, and a ray from a source 50 m above the interface cannot reach it at finite horizontal distance. Nor can the slower sediment S speed, $900\mathrm{m}{\mathrm{s}}^{-1}$, support a seismic head wave launched from water: it would require $\sin {\theta }_w=1500/900>1$. A seismic reflection still exists at the seabed because the densities, hence the acoustic impedances, differ.

First appearance and first arrival differ. For each permitted branch the travel time isThe all-P basement P branch has $t_h=x/2500+0.0746667\mathrm{s}$. It overtakes the direct water P wave, $t_d=x/1500$, only at $x=280\mathrm{m}$. The converted P-head branches have larger seismic intercept times, so even where they exist at shorter offsets they are later arrivals. The S-head branches are likewise later than the all-P P-head branch wherever they exist in this model. Reflections and layer reverberations can supply additional later phases.

Use a high-resolution multichannel seismic reflection survey, with closely spaced crossing lines or a suitable small three-dimensional survey over the turbine locations. It should map the seabed, the sediment–sandstone boundary, local faults, buried channels, and lateral changes relevant to foundations. Seismic reflection responds to acoustic impedance contrasts and can image an irregular boundary without requiring it to be a first-arrival refractor. Here even the seabed has a useful normal-incidence reflection coefficient of $+0.20$ despite its zero P-velocity contrast.

A seismic refraction survey can provide complementary seismic velocities and geotechnical information, particularly using seabed multicomponent receivers for S waves. Its limitations as the principal imaging method here include the missing water–sediment P refractor, required offsets of hundreds of metres to obtain useful basement first arrivals, and weaker constraints on thin layers or velocity inversions. Variable bathymetry and sediment thickness further complicate simple refraction interpretations.

A shallow-towed boomer seismic source or small sparker seismic source is a sensible initial seismic source for penetrating about 20 m of sediment beneath a 50 m water column. A chirp sub-bottom profiler can add finer imaging of the upper cover where penetration permits; penetration depends on grain size, consolidation, gas, and seismic attenuation. A pilot line should establish the minimum useful source energy and bandwidth. Large low-frequency air-gun arrays are unnecessary for this shallow target. Small coastal vessels, sea state, tow stability and acoustic background constrain usable frequencies and operations. Marine mammals and other local receptors motivate a low-energy effective source, monitoring and appropriate operational mitigation.^[ref]

The expected normal-incidence two-way travel times are $2(50)/1500\simeq 0.0667\mathrm{s}$ for the seabed and $2(50+20)/1500\simeq 0.0933\mathrm{s}$ for basement, separated by $26.7\mathrm{ms}$. Source duration, recording sample interval and bandwidth must distinguish them. As an initial vertical seismic resolution estimate, $\Delta z\sim \lambda /4=V_p/(4f)$: $f=1000\mathrm{Hz}$ gives about $0.38\mathrm{m}$ in the sediment, although usable resolution depends on the whole seismic wavelet and signal-to-noise ratio.

Choose shot and receiver spacing from the shortest useful wavelength, target size, maximum reflector dip, seismic migration needs and achievable navigation accuracy. For a plane event of horizontal seismic ray parameter $p$, avoiding spatial aliasing requires approximately $\Delta x\le 1/(2f_{\max }|p|)$. Source spacing and receiver spacing also set midpoint coverage and seismic acquisition fold; adequate offsets enable moveout correction and velocity estimation. For nearly horizontal shallow reflectors a few-metre shot interval can be practical, whereas steep events at high frequency require denser spatial sampling. These values must be designed together rather than imposing one spacing for every target. Cross-line spacing must resolve lateral features between turbine sites, not merely along each track.

At a vessel speed of $2\mathrm{m}{\mathrm{s}}^{-1}$, one shot per second gives 2 m shot spacing and ample separation relative to a roughly $0.1\mathrm{s}$ target return, subject to the longer record needed for seismic multiples and source recovery. Shallow towing limits seismic source ghosts and receiver ghosts, but requires stable tow depth. Accurate positioning, tides, measured water sound speed, bathymetry and checks for seabed seismic multiples are essential for correct target depths.

For incidence from cap rock (1) into reservoir (2), use the conventional normal-incidence stress-amplitude reflection coefficientHere acoustic impedance is the P wave impedance; $\rho V_s$ is the corresponding shear-wave impedance. Positive $R$ preserves the chosen stress-wave seismic reflection polarity; negative $R$ reverses it. These are amplitude coefficients, not reflected energy fractions. At normal incidence in the ideal isotropic model there is no P-to-S mode conversion. The cap-rock impedances are

The following are approximate readings of the supplied Figure 1 PNG, checked against the matching source PDF, rather than exact tabulated parameters. At zero gas, use $\rho \simeq 1.75\mathrm{g}\mathrm{c}{\mathrm{m}}^{-3}$, $V_p\simeq 1.75\mathrm{km}{\mathrm{s}}^{-1}$, and $V_s\simeq 0.60\mathrm{km}{\mathrm{s}}^{-1}$; at full gas use $\rho \simeq 1.25\mathrm{g}\mathrm{c}{\mathrm{m}}^{-3}$, $V_p\simeq 1.15\mathrm{km}{\mathrm{s}}^{-1}$, and $V_s\simeq 0.65\mathrm{km}{\mathrm{s}}^{-1}$. The left-hand $V_p$ endpoint is crowded against the axis and the density curve; readings differing by roughly $0.05$ in the plotted units are reasonable. The resulting coefficients deserve about two significant figures, not the apparent precision of the intermediate arithmetic.

Both end members change seismic reflection polarity, but the P wave change is much larger: approximately $+0.15$ to $-0.22$, compared with $+0.08$ to $-0.05$ for S waves. Thus a calibrated P wave survey is better for distinguishing these two saturation end members and should give a stronger reflection from the fully gas-filled reservoir. A negative P wave reflection relative to the water-filled prediction is a useful gas indicator in this stipulated model. In a real seismic reflection survey, lithology, porosity, effective pressure, and seismic wavelet phase also affect the response, so a bright or reversed reflection alone does not prove the presence of gas.

The reflected-to-incident compressional pressure or normal-stress amplitude ratio at a planar interface. For normal incidence between lossless homogeneous media, the displayed expression uses acoustic impedances $Z_j={\rho }_j{V}_{pj}$. The reflected energy fraction is $r_p^2$, while a same-coordinate particle-velocity reflection coefficient has the opposite sign.