Introduction to physics

Lecture 31 Interference (Chap. 29) Superposition principle How to add two waves? Interference of Waves from two different sources with same amplitude Interference of Waves travelling in different direction

Superposition Wave equations are linear Two solutions can be added to form a new solution The form of the new solution can be quite interesting

Interference of Waves I Superposition of two waves with phase shift

Interference of waves from two point sources. constructive interference 相长干涉 destructive interference 相消干涉

Young double slits experiment light is a wave

Phasor (相矢量) Use of complex number a phasor is a vector that has a magnitude equal to the amplitude of the wave and that rotates around an origin; the angular speed of the phasor is equal to the angular frequency 𝜔 of the wave.

Using phasor to add two waves

Interference of Waves: standing wave Superposition of two waves with the same amplitude, same frequency and same wavelength, but propagate along difference direction

Standing wave The wave patterns do not move left or right; the locations of the maximum and minimum do not change The nodes or antinodes are separated by 𝜆/2 波腹 波节

Reflection of waves Fix one end of a string 𝑦 𝑥=0,𝑡 =0 If we have two wave propagate along opposite directions 𝑦=𝐹 𝑥−𝑣𝑡 +𝐺(−𝑥−𝑣𝑡) We have 𝐹 −𝑣𝑡 +𝐺 −𝑣𝑡 =0 → 𝐺 −𝑣𝑡 =−𝐹(−𝑣𝑡) 𝑦=𝐹 𝑥−𝑣𝑡 −𝐹(−𝑥−𝑣𝑡)

Reflection of waves (Open ends) No strain at the open ends 𝑑𝑦 𝑥=0,𝑡 𝑑𝑥 =0 If we have two wave propagate along opposite directions 𝑦=𝐹 𝑥−𝑣𝑡 +𝐺(−𝑥−𝑣𝑡) We have 𝐹 ′ −𝑣𝑡 −𝐺′ −𝑣𝑡 =0 → 𝐺′ −𝑣𝑡 =𝐹′ −𝑣𝑡 𝑦=𝐹 𝑥−𝑣𝑡 +𝐹(−𝑥−𝑣𝑡)

Standing Waves Due To Reflections from Hard and Soft Boundaries As the reflected wave superposes with the incident wave, a standing wave is quickly formed.

Fix both ends of a string (length L) The wave length should satisfy the condition Fix one end and free the other end

Harmonic waves (谐波） Fundamental frequency is defined as the lowest frequency of a periodic waveform 𝜆 1 =2𝐿, 𝑓 1 = 𝑣 2𝐿 nst harmonic 𝜆 𝑛 = 𝜆 1 /𝑛, 𝑓 𝑛 =𝑛 𝑓 1 𝑛=1,2,3,… 基波 二次谐波 三次谐波

Fix one end Fundamental wave 𝜆 1 =4𝐿, 𝑓 1 = 𝑣 4𝐿 No even harmonic 𝜆 1 =4𝐿, 𝑓 1 = 𝑣 4𝐿 No even harmonic Harmonic 𝜆 𝑛 = 4𝐿 𝑛 , 𝑓 𝑛 = 𝑛𝑣 4𝐿 𝑛=1,3,5….