Introduction to physics

Lecture 32 Beats (Chap. 48, 49) Beats Phase speed and group speed

Interference of Waves: Beats Superposition of two waves with the different frequencies

For fix 𝑥 such as 𝑥=0 𝑦= 𝐴 1 𝑒 𝑖𝜔 1 𝑡 + 𝐴 2 𝑒 𝑖 𝜔 2 𝑡 The intensity is 𝐼= 𝐴 1 2 + 𝐴 2 2 +2 𝐴 1 𝐴 2 cos 𝜔 1 − 𝜔 2 𝑡 Which swells and falls at a frequency 𝜔 1 − 𝜔 2

Side bands: information propagation Modulated wave 𝑆= 1+𝑏𝑐𝑜𝑠 𝜔 𝑚 𝑡 𝑐𝑜𝑠 𝜔 𝑐 𝑡 Where 𝜔 𝑐 is the frequency of the carrier and 𝜔 𝑚 is the frequency of the audio tone. 𝑆=𝑐𝑜𝑠 𝜔 𝑐 𝑡+ 1 2 𝑏𝑐𝑜𝑠 ( 𝜔 𝑐 +𝜔 𝑚 )𝑡+ 1 2 𝑏𝑐𝑜𝑠 ( 𝜔 𝑐 −𝜔 𝑚 )𝑡 The output wave consists of three waves. 𝜔 𝑐 ± 𝜔 𝑚 are called side bands. In order to transport information effectively, 𝜔 𝑚 should be much less than 𝜔 𝑐

Phase speed and group speed If 𝜔 1 𝑘 1 = 𝜔 2 𝑘 2 , the modulation and the carrier transport with the same speed. The ratio 𝜔/𝑘 is called phase speed, which is the speed at which the phase move along 𝑣 𝑝 = 𝜔 𝑘

Phase speed and group speed If 𝜔 1 𝑘 1 ≠ 𝜔 2 𝑘 2 , but the differences between 𝜔 1 , 𝜔 2 and 𝑘 1 , 𝑘 2 are small The speed of fast oscillation 𝜔 1 + 𝜔 2 2 / 𝑘 1 + 𝑘 2 2 ≈ 𝜔 𝑘 The speed of modulation wave 𝑣 𝑚 = 𝜔 1 − 𝜔 2 𝑘 1 − 𝑘 2 is called group speed, which is speed of information 𝑣 𝑔 = 𝑑𝜔 𝑑𝑘