Wave Reflections and “Echo”

• If a traveling wave encounters a change in the physical properties of the medium through which it propagates, the wave will be perturbed where the change occurs. This perturbation generally involves some level of reflection, absorption and transmission at the boundary.

• For example, if a wavefront impinges on an ideally rigid surface, all of the wave energy will be reflected from the surface. However, if the surface is instead covered with a layer of absorbing material, only a portion of the wave energy will be reflected, with the remainder being trapped and damped within the material.

• The extent of such reflection can be characterized by a reflection coefficient ($R$), which specifies the ratio of reflected to incident wave energy. Materials that are very reflective will have a value of $R$ close to 1, while $R$ will be close to zero for materials that are very absorptive. In general, the reflection coefficient will be frequency dependent.

• As previously mentioned, wave reflection from surfaces will also depend on the shape of the surface. If an acoustic wave encounters a rigid wall that is flat over at least several wavelengths in all directions, the wavefront will experience specular reflection from that surface (equal angles of incidence and reflection). On the other hand, a wall that is very uneven will reflect a wavefront in many directions (“diffuse” scattering).

• Figure 4 illustrates a source-listener arrangement with “multipath” wave propagation. If the time delay between the arrival of the direct and reflected waves is greater than about 50 milliseconds, the reflected sound will be perceived as an echo.

  Figure 4: A source-listener arrangement with an “echo” or “floor bounce” propagation delay.

  

• The system of Fig. 5 (top) provides a signal processing block diagram to simulate the sound wave propagation of Fig. 4 using digital delay lines. The scale factors $g_d$ and $g_r$ account for losses over the respective direct and reflected paths due to the combined effects of air absorption and spherical spreading. If the floor had a reflection coefficient less than one, this could also be included in the $g_r$ factor.

  Figure 5: Floor reflection block diagrams.

  

• The delay common to the two paths can be pulled out and implemented separately, as illustrated in the lower part of Fig. 5. In this case, the length of the delay line for the reflected path must be adjusted by subtracting from it the common delay length and its attenuation factor appropriately scaled.

• $M_r - M_d$ is found as:
  $$M_r - M_d = \frac{2 a - d}{c T_s} = \frac{2\sqrt{h^2 + (d/2)^2} - d}{c T_s}.$$

• If the gain factors were only used to simulate spherical propagation scaling, the scaling for the reflected path in the lower plot of Fig. 5 would be calculated relative to $d$ as:
  $$g_r/g_d = \frac{1 / 2 a}{1 / d} = \frac{d}{2 a} = \frac{1}{\sqrt{1 + (2h/d)^2}}.$$

• The portion of the signal processing block diagram within the dashed lines at the bottom of Fig. 5 is a feedforward comb filter. This simulated system results in a feedforward filter structure because none of the propagated sound returns to the listener.

• Depending on the distance between the direct and reflected paths, certain frequency components in the sound will be destructively cancelled at the listener position, corresponding to the notches in the frequency response of the feedforward comb filter. In this way, the feedforward comb filter is a computational physical model of a source-listener arrangement involving a direct and single reflected path.

• If the source-listener arrangement involved two parallel walls with the source and listener located in between, the corresponding wave simulation block diagram would include a feedback comb filter, which would model a series of echoes, exponentially decaying and uniformly spaced in time.