Читать книгу Introduction To Modern Planar Transmission Lines - Anand K. Verma - Страница 245

Doppler effect is related to the change in frequency of a source due to the relative motion of the source and receiver. If a source is moving away, i.e. receding, from the receiver in a DPS medium, the received frequency is less than the stationary frequency of the source. However, if the source is moving toward the receiver, the received frequency is increased.

Let us examine the receding case of a moving source. Figure (5.14a) shows that a source #S is moving away from receiver #R with velocity V_s along the positive x‐axis. The source occupies positions P₁ (x = d₁) and P₂(x = d₂) at time t₁ and t₂ (t₂ > t₁), respectively. The source S radiates frequency f_s and the radiated spherical wave propagates with the phase velocity V_p in the DPS medium. Figure (5.14a) shows the spherical wavefronts at time t_1,…,t₄. The phase of the received wave at time t₁ and t₂ could be written as

(5.5.38)

The received frequency f_R of the wave at the stationary receiver is the rate of change in phase, i.e. ω_R = 2πf_R = Lt_{Δt → 0}(Δϕ/Δt). The received frequency, from the above expression (5.5.38c) is obtained as

(5.5.39)