## 1

The purpose of the objective lens is to increase the amount of light collected from the object in order to make a bright image at the focai point. The objective lens can collect a great deal more light than the eye's pupil. Let 0, be the angle subtended by

The latter follows from the small angle approximation. (The sign is a consequence of the inverted image.) Thus, for great magnification it is desirable to have a large f, and a small f2. The similarity to equation (4) had not escaped Tesla.

(8) * = pt - a»t - const and differentiating with respect to time

In a nondispersive medium, both spectra] components propagate with the same phase velocity, v+1 =Vj2, and in the electromagnetic case in free space both phase velocities are equal to the speed of light, c. The interference of the two waves in Equation (1) is seen most simply by examining Equation (5). The carrier propagates with velocity v t =©„/(}„=

The modulation envelope, m(r,t) also propagates. To find its velocity, we again we set the argument equal to a constant:

(10) = = const. Differentiating gives the modulation envelope velocity as w„ Af_

The interference pattern fringe spacing is simply the spatial distance between intensity maxima, (see Figure 13)

Figure 12 shows the relations between the various wave forms. In particular, notice that the intensity waveform oscillates with two "beats" per modulation waveform. Further, the spatial separation between maxima of the propagating modulation (envelope) wave is given by a ■

The human ear is a square law detector responding to sound intensity, EJ. The modulation frequency, corresponding to m(r,t), is 'A of the beat frequency a)m = Vi c)b. There are two beats per cycle of the modulation wave.

For the case of acoustic waves, Crawford points out that if f[ and f2 differ by more than about 6% from then one "hears" the sound as two separate tones, or oscillator?, with significantly different notes [Equation (1)]. In music, this is called a w

Figure 13. Free space DSB transmitter radiating the ward propagating beal pattern of Equation 1.5. (a) a spends to Figure 11(b). The while bands are the prop»«* ing envelope maxima and the white rings in each band respond to tfrepeafcs of the oscillating carrier. Tiie between the outward propagating beats is \ = c/&f.

Figure 13. Free space DSB transmitter radiating the ward propagating beal pattern of Equation 1.5. (a) a spends to Figure 11(b). The while bands are the prop»«* ing envelope maxima and the white rings in each band respond to tfrepeafcs of the oscillating carrier. Tiie between the outward propagating beats is \ = c/&f.

"chord". However, iff, and f2 are closer, then on "hears" the sound as a single oscillator of frequent) fp, modulated with a slowly varying amplitude m(l [Equation (5)].'11

Finally, we mention that in a dispersive medium e.g. a waveguide near cutoff, or a cavity resonatn near its resonant frequency, v and the ten don't propagate at the speed of the carrier wavr The modulation appears as a disturbance prop« gating along the carrier wave with velocity V( through space.

### Beats as Moving Interference Patterns

Several discussions of "beats", produced by Iwi sources of slightly differing frequencies, have beci published,78'79'80-81'82 Such phenomena may be dc scribed in terms of a traveling pattern o interference fringes. In the discussion above, kw have examined only the case where the two poin sources coincide. The above referenced article treat the more general case, which reduces to our discussion when the spectrally different sources are at the same point, and reduces to Young's two sli diffraction when the two sources are at the same frequency but different locations (as with a phasec array antenna). When both spatially separata sources and different spectra are employed, the resulting interference pattern can result in slowly propagating envelopes and in stationary (standing waves, i.e. waves for which the envelope velocity is zero, just as Tesla said.

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