Reflection or Transmission Coefficient Tool

Physics Oscillations and Waves • Waves Properties and Equations

Written by STEM Calculators Team Published March 6, 2026 Updated March 6, 2026

When a wave hits a boundary between two media with impedances \(Z_1\) and \(Z_2\), the amplitude reflection and transmission coefficients (normal incidence) are \[ r=\frac{Z_2-Z_1}{Z_2+Z_1},\qquad t=\frac{2Z_2}{Z_2+Z_1}. \] For many 1D waves, impedance can be written as \(Z=\rho v\). If \(r<0\), the reflected wave is phase-inverted. This tool also checks an energy-flux relation: \[ R=r^2,\qquad T=\frac{Z_1}{Z_2}\,t^2,\qquad R+T\approx 1. \] Includes an interactive coefficient plot (zoom/pan, axis units) and a slow boundary “bounce” animation.

Inputs

Input mode:

Impedance \(Z_1\) (rayl or N·s/m³): Impedance \(Z_2\) (rayl or N·s/m³):

Visualization

Plot type: Plot max ratio \(\eta_{\max}\) (unitless): Plot resolution (samples):

Show grid Show axes Show marker Show plot Show animation

Animation speed 0.25× —

Note: r,t are amplitude coefficients. Energy uses \(R=r^2\) and \(T=(Z_1/Z_2)t^2\) (normal incidence).

Ready

Boundary “bounce” animation

Shows an incident wave hitting a boundary at \(x=0\), producing reflected and transmitted waves with amplitudes \(rA\) and \(tA\). If \(r<0\), the reflected wave flips phase.

Axes: x (m, schematic), y (relative amplitude). Slow by default.

Interactive coefficient plot

Plots coefficients vs impedance ratio \(\eta=Z_2/Z_1\) (unitless). y-axis is coefficient (unitless) or energy fraction (unitless). Includes numeric ticks + units. Zoom with wheel; pan by dragging.

Tip: on narrow screens, scroll horizontally to see the full plot.

Enter values and click “Calculate”.

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Theory

When a wave traveling in medium 1 reaches a boundary with medium 2, part of the wave is reflected back into medium 1 and part is transmitted into medium 2. For one-dimensional waves at normal incidence, the behavior is governed by the wave impedance \(Z\), which measures how strongly the medium “resists” motion due to a wave. For many linear waves, \[ Z=\rho v, \] where \(\rho\) is density and \(v\) is wave speed. (For strings, the relevant impedance is often expressed differently, but the same reflection formula structure applies.)

Amplitude coefficients. Let the incident wave have amplitude \(A_i\), reflected amplitude \(A_r\), and transmitted amplitude \(A_t\). The amplitude reflection coefficient and amplitude transmission coefficient are defined by \[ r=\frac{A_r}{A_i},\qquad t=\frac{A_t}{A_i}. \] Applying boundary conditions (continuity of displacement/pressure and force/particle velocity) yields \[ r=\frac{Z_2-Z_1}{Z_2+Z_1},\qquad t=\frac{2Z_2}{Z_2+Z_1}. \] If \(Z_2 > Z_1\), then \(r>0\) and the reflected wave is not inverted. If \(Z_2 < Z_1\), then \(r<0\) and the reflected wave is phase inverted (a \(\pi\) phase shift).

Energy (intensity) check. Amplitude coefficients do not directly represent energy fractions. For normal incidence, the reflected energy fraction (reflectance) is \[ R=r^2, \] while the transmitted energy fraction (transmittance) is \[ T=\frac{Z_1}{Z_2}\,t^2. \] Under the ideal assumptions (no losses), energy is conserved so \(R+T=1\). In real materials, absorption and scattering reduce the transmitted energy and the sum can be less than 1.

Why plot vs ratio? Because \(r\) and \(t\) depend only on the ratio \(\eta=Z_2/Z_1\), it is common to plot coefficients against \(\eta\) to see trends: as \(\eta\to\infty\), \(r\to 1\) (almost total reflection); as \(\eta\to 0\), \(r\to -1\) (total reflection with inversion); and when \(\eta=1\), \(r=0\) and there is no reflection.

University-level extensions include oblique incidence, polarization, and multilayer interference (thin films), where coefficients depend on angle, frequency, and phase accumulation across layers.

Frequently Asked Questions

What specifically triggers waves to reflect immediately at a junction?

Waves inherently reflect backward practically whenever they suddenly hit a strict change natively in physical medium impedance roughly.

What is fundamental acoustic impedance?

Acoustic impedance mathematically embodies merely how thoroughly a specific physical substance vigorously naturally broadly resists wave pressure.

Does a profoundly negative reflection physically make structural sense?

Yes, a negative reflection coefficient means exactly that the physically bouncing wave effectively completely inverted its phase entirely upside down fundamentally.

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Frequently Asked Questions

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