Huygens' principle is usually stated in the following way: each point on the wavefront of a sound wave gives rise to a Huygens' wavelet. Each wavelet propagates outward as a spherical sound wave, and the superposition of all of the wavelets creates the total sound wave. This is a very useful concept for understanding diffraction and some other aspects of wave propagation. The principle applies to electromagnetic and other waves as well, but the form of the wavelet varies.
Equation G14 in the section on the General Solution of the wave equation is the mathematical expression of this principle. Actually equation G14 shows that a more general statement of the principle is that each point on any surface gives rise to a wavelet - not just an equiphase surface. However the equations are simpler for an equiphase surface, and this is the conventional definition, so that will be assumed here.
We consider the specific case of a plane wave propagating in the +z direction, and the wavefront at the plane z=0. The velocity u and pressure p (at a given instant in time), and the normal n on the wavefront are specified to be
The vector Rp extends from the origin to the evaluation point, and makes an angle θ with the z-axis. The form of a wavelet emanating from the point at the origin can be obtained directly from equation G14 for the pressure term, and by taking the gradient for the velocity term. The results are
There are several interesting things about these equations. First, the equation for velocity u(Rp) is actually a bit more complicated than the Huygens' wavelet for an electromagnetic wave. This is surprising since sound is generally viewed as a simpler process. Second, the waves only become truly spherical in the far-field, and are pretty complicated in the near field. In the far-field the overall directivity of the Huygens' wavelet is proportional to 1+cosθ, for both pressure and velocity, exactly the same as the electromagnetic case. This is only true if the ratio p0/Z0=u0. At a boundary between media of different acoustic impedances this is not true, and a reflected wave is created as well as a forward propagating wave.
To me the most startling aspect of these equations is that the velocity component u0 of the source at the origin radiates isotropically, whereas the pressure component p0 has a directional pattern! Physically, higher pressure means a higher molecular density and higher temperature, which is totally lacking in directionality. Conversely the velocity is directional.
This seems so counterintuitive that I felt compelled to double check the equations. For the incoming plane wave I considered a spherical surface centered at the origin. I used the values of velocity and pressure on the surface in equation G14, and in the equation derived from the gradient of G12, and numerically computed the sound pressure and velocity inside, and outside, the surface. Inside the surface I got the plane wave, and outside the surface I got zero, exactly as expected. Note that the wavelets in this case are more complicated than the above equations, since the sphere is not an equiphase surface, but the above equations are a special case of the more general wavelets.
So the equations are mathematically valid. Do they have a physical significance? Obviously sound cannot really emanate from a point, so that is one limit to any physical interpretation. Huygens' wavelets are a valid solution of the sound wave equations in all space for ρ > 0, and mathematical sources exist for the wavelets. However it is hard for me to imagine any set of real physical sources (i. e. molecules) resulting in this peculiar directivity behavior.
The numerical test case I ran is also counterintuitive; the superposition of the wavelets create a plane wave inside the sphere, but add up to zero everywhere outside the sphere. Mathematically this is the correct behavior, but again it is difficult for me to imagine real physical sources resulting in this situation.
Finally, I computed the power flow, which is shown here [44 kb]. The lines are parallel to the direction of power flow, which, except for the loops, is away from the origin. When I initially computed this pattern the flow was in the opposite z-direction, which is how I learned of an inconsistency regarding the direction of my ar unit vector. I get the correct numerical solution to the field inside the sphere with the original orientation, but obviously the power flow must be in the positive z-direction. The above equations in this section are for a unit vector in the opposite direction, which gives the power flow shown in the figure, but does not give the correct result inside the sphere. I have spent quite a few days on this and have not yet been able to resolve this problem.
The pattern indicates the 1+cosθ directional behavior in the far-field, but the near-field behavior is quite complex. The net outward flow of power through a sphere of any radius is constant, as it must be. For small radii the near-field is dominated by two terms that are inversely proportional to the fourth power of the radius. This creates the looping flow where power surges out from the origin in the positive z-direction, but then loops around and flows back into the origin from the negative z-direction. There are similar power flow loops for the electromagnetic Huygens' wavelet, although the pattern of the flow is somewhat different. Even ignoring the singularity at the origin this leads to huge power flows. Again it is difficult for me to connect this to any molecular behavior I can imagine.
So in summary, although equation G14 yields correct answers to real-world problems, at the level of Huygens' wavelets I don't know how to connect the mathematics with real-world molecules. If anyone has additional thoughts on this, send me an e-mail and I will append any comments I find interesting - with full credit of course.
To the list of physics of sound subsections
