THE SCIENCE AND APPLICATIONS OF ACOUSTICS - H. H. Arnold ...

3.7 Reflection 41Figure 3.7. Construction for a spherical wave from point S incident upon plane surfaceα–α ′ . Point S ′ is an imaginary point that is the mirror image of point S on the other sideof plane α–α ′ .intersects any normal to the wavefront, a scissorlike effect occurs, not unlike oceanwaves breaking obliquely along a beach. The intersection of these waves alongthe normals constitutes a projection of the incident and reflected waves. From theconcept of wave motion, the distance between crests along the normal may be establishedas a projected wavelength λ ′ , which relates to the incident wave as follows:λ ′ =λ = λ sec ϑ 1 (3.12)cos ϑ 1In obeying the laws of reflection, the reflected wave also scissors back alongthe normal in the opposite direction, producing a traveling wave with a projectedwavelength also equal to λ ′ . Hence, there occurs along any normal line the superimpositionof two waves traveling in opposite directions with wavelength λ ′ .From the concept of standing waves it can be inferred that nodes and antinodesoccur along the normal line, and, moreover, the spacing between the nodes andantinodes needs only to be modified by the factor sec θ 1 of Equation (3.12).Consider a complex periodic wave that impinges upon a fully reflective planesurface. A standing wave sound field will exist. The distance d ′ between peaksalong the normal ensues from Equation (3.12) in the following manner:d ′ = λ′2 = λn sec θ1where λ n is the wavelength of the nth harmonic and θ i the angle of incidence ofthe propagating wavefronts. From the last equation it will be noted for the specialcase of θ i = 0 (normal incidence), the nodal spacing reduces to λ/2, accordingto Equation (3.12). As the angle of incidence increases, the spacing between thenodes likewise increases, and in the limit θ i = π/2, there is no reflected wave, andthus the standing wave field vanishes.