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Unformatted text preview: Fourier theory says that for a wave packet Δ f Δ t ≈ 1 . We don’t have a sharp equality, because the relationship of duration and frequency range depend somewhat on the shape of the wave packet. Also, notice that the arrival time of a packet is not unambiguous: it can be the beginning of the packet, the middle, the end, or any point in between. We will find this discussion, here in the context of classical wave theory, to be very useful when we discuss the Heisenberg uncertainty principle in quantum mechanics. We also see an important application to digital information transmission. Each bit of data is a short pulse. To have a larger rate of transmission of information we want shorter pulses, which means that the frequency range needs to be larger. This is the need for large bandwidth in communications. In a similar way to the group velocity in the case of beats, the general expression for group velocity is v group = ∂ω ∂k v phase = ω k so that v group = v phase only if ∂ω ∂k = ω k , which is a separable equation whose solution is ω ∝ k . Problem : What is the most general dispersion relation for which v phase × v group = c 2 ? Solution : v group = ∂ω/ ∂k , and v phase = ω/k . The desired relation is c 2 = v phase × v group = ω k ∂ω ∂k ω dω = c 2 k dk ω 2 = c 2 k 2 + ω 2 ω ( k ) = c 2 k 2 + ω 2 where ω 2 / 2 is an integration constant. Doppler Effect [Doppler effect Physlet Illustration 18.4] The detected frequency is the rate at which the detector intercepts wavefronts (or individual wavelengths). In the case that both the source and the detector are stationary, this rate is 23 the same as the wave frequency f . f = vt/λ t = v λ = f where f is the detected frequency and f is the emitted frequency. vt = distance wavefronts are traveling vt λ = number of wavelengths that fit in this distance If the detector is moving toward the source: vt = distance wavefronts are traveling v D t = distance detector is traveling ( v + v D ) t = distance relative to D wavefronts are traveling f = ( v + v D ) t/λ t = v + v D λ = v λ (1 + v D v ) = f (1 + v D v ) therefore f > f blue shift. We add the velocities because the relative motion is toward each other: effective relative v is the sum. If the detector is moving away from source: We may take the previous result, and change v D → - v D , so that f = ( v- v D ) t/λ t = f (1- v D v ) therefore f < f red shift. Important: v D is w.r.t. the medium in which the waves propagate! We’ll talk about that in great detail when we discuss EM waves in the context of Special Relativity....
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- Spring '09
- Physics, spherical charge distribution, Amp`re law