Propagation of Reflected and Refracted Waves at an Interface

A propagating wave of given frequency (in Hz) is incident on an interface between two acoustic media, with varying propagation velocities. The incident angle is measured from the vertical, running from 0 to 90 degrees. The amplitudes of the reflected and refracted waves are computed for the specified incident angle. The reflected wave is added to the incident wave in the upper medium and the refracted wave is shown in the lower medium. At a certain incident angle that depends on the relative velocities, the incident wave will be totally reflected and the bottom half of the plot will become solid gray, indicating zero amplitude.


  • [Snapshot]
  • [Snapshot]
  • [Snapshot]


This Demonstration shows how a propagating plane wave interacts with an interface between two acoustic media. The incident wave impinges on the boundary from the top and causes a reflected wave that interferes with the incident wave. The incident wave is transmitted into the bottom layer as a refracted wave with its own propagation velocity. The incident and refracted wave angles (angle of the wave normal relative to vertical) are related by the well-known equation
where is velocity and the subscripts 1 and 2 denote the top and bottom media. The incident wave is taken to have amplitude 1. If denotes the reflection coefficient for the scattered wave amplitudes, then the wave motion in the top layer is described by (Towne, 1988)
while, with denoting the transmission coefficient, the motion in the bottom layer is described by
where and are the horizontal and vertical wavenumbers, with the subscripts 1 and 2 for the top and bottom layers, respectively, and is the frequency in radians. Here all lengths are in meters.
The coefficients and are computed for variable angles of incidence and velocities with
assuming, with no loss of generality, that the densities of the two media are equal. This definition puts in the range , while has the range . Because of wave interference, the top media can have amplitudes in the range . The overall amplitudes are scaled in the plots such that black is -2, white is +2, and medium gray is 0. Note that amplitudes appear to be continuous across the boundary. This physical fact is the basis for the equations for computation of and .
D. H. Towne, Wave Phenomena, New York: Dover Publications, 1988.
    • Share:

Embed Interactive Demonstration New!

Just copy and paste this snippet of JavaScript code into your website or blog to put the live Demonstration on your site. More details »

Files require Wolfram CDF Player or Mathematica.

Mathematica »
The #1 tool for creating Demonstrations
and anything technical.
Wolfram|Alpha »
Explore anything with the first
computational knowledge engine.
MathWorld »
The web's most extensive
mathematics resource.
Course Assistant Apps »
An app for every course—
right in the palm of your hand.
Wolfram Blog »
Read our views on math,
science, and technology.
Computable Document Format »
The format that makes Demonstrations
(and any information) easy to share and
interact with.
STEM Initiative »
Programs & resources for
educators, schools & students. »
Join the initiative for modernizing
math education.
Step-by-Step Solutions »
Walk through homework problems one step at a time, with hints to help along the way.
Wolfram Problem Generator »
Unlimited random practice problems and answers with built-in step-by-step solutions. Practice online or make a printable study sheet.
Wolfram Language »
Knowledge-based programming for everyone.
Powered by Wolfram Mathematica © 2018 Wolfram Demonstrations Project & Contributors  |  Terms of Use  |  Privacy Policy  |  RSS Give us your feedback
Note: To run this Demonstration you need Mathematica 7+ or the free Mathematica Player 7EX
Download or upgrade to Mathematica Player 7EX
I already have Mathematica Player or Mathematica 7+