Match sound at a material interface

Let the incident wave travel to the right from medium one toward medium two. Define all normal particle velocities using the same rightward coordinate. At a massless interface with no surface tension contribution to the linear normal stress, total pressure perturbation is continuous. Normal particle velocity is also continuous when the interface moves with both fluids. The reflected wave travels left, so its velocity amplitude enters with the opposite sign from its pressure amplitude. This sign is the key algebraic detail.

Denote positive real characteristic impedances by Z₁ and Z₂. At the boundary, incident plus reflected pressure equals transmitted pressure, while incident minus reflected pressure divided by Z₁ equals transmitted pressure divided by Z₂. Solving these two linear equations gives a pressure reflection coefficient r and a pressure transmission coefficient t. They refer to amplitudes at the interface and assume planar waves, normal incidence, no dissipation, and no additional interface dynamics or mode conversion.

Each progressive wave carries mean intensity proportional to squared pressure amplitude divided by its medium’s impedance. The reflected energy fraction is therefore r², but the transmitted fraction is t²Z₁/Z₂. The impedance ratio cannot be omitted. A high-impedance second medium can support a larger transmitted pressure with a smaller transmitted velocity. Because energy flux is their product, a large pressure alone does not indicate a large transported power or a violation of conservation.

Insert the amplitude expressions and simplify. Reflection plus transmission equals one for this lossless ideal interface. This check catches reversed impedance ratios, forgotten velocity signs, and the mistaken rule that transmission must equal one minus the pressure reflection coefficient. When the second impedance is much smaller, the reflected pressure changes sign. When the impedances match, reflection disappears even if density and sound speed individually differ. Matching their product is what this normal-incidence model requires.

The rigid-wall limit is approached as the second impedance becomes arbitrarily large: reflected pressure nearly equals incident pressure, while total normal velocity at the boundary tends to zero. The pressure-release limit gives a reflected pressure of nearly opposite sign. Real boundaries can have frequency-dependent complex impedance, damping, compliance, or mass. Oblique incidence also changes the matching problem and may involve refraction. The simple two-real-impedance formulas should not be reused without checking those changes in assumptions.

Use the waves lab only to visualize addition of incident-size and reflected-size pressure signals at a chosen boundary point. Predict the sum when the amplitude ratio is one and phase is either zero or 180°. Its components both propagate in the same direction, so the spatial display does not represent an incident/reflected pair. It cannot compute interface motion or transmitted flux. The useful connection is the algebra of adding boundary pressures; the physical matching and energy accounting remain calculations in this lesson.