1. Lecture 30 Wave Equation and solution (Chap.47)
Travelling wave
Superposition
Waves on string, derivation of wave equation
Solution of wave equation

2. Traveling wave
Consider a function of 𝑥−𝑣𝑡 𝑓 𝑥,𝑡 =𝑓(𝑥−𝑣𝑡) At t=0 𝑓 𝑥,𝑡=0 =𝑓(𝑥) At 𝑡, 𝑥 ′ =𝑥+𝑣𝑡 𝑓 𝑥+𝑣𝑡 ,𝑡 =𝑓(𝑥) i.e. 𝑓 𝑥,0 =𝑓(𝑥+𝑣𝑡,𝑡) 𝑓 represents the propagation of a wave from left to right

3. Wave Motion in Time and Space
𝑓 𝑥,𝑡 =𝑓 𝑥±𝑣𝑡
The particles oscillate around the equilibrium positions.
It is the disturbance which travels, but not the individual particles in the medium 3

4. Superposition of Waves
The principle of superposition may be applied to waves whenever two (or more) waves travelling through the same medium at the same time.
The waves pass through each other without being disturbed.
The net displacement of the medium at any point in space or time, is simply the sum of the individual wave displacements.
𝑓 𝑡𝑜𝑡𝑎𝑙 𝑥,𝑡 = 𝑓 1 𝑥,𝑡 + 𝑓 2 (𝑥,𝑡)
This is true of waves which are finite in length (wave pulses) or which are continuous sine waves.

5. Interaction Between Two Solitons

6. Generation of waves
A simple model to generate waves is the combination of oscillators

7. Waves on string 𝛽 For a piece of string 𝐹 1 𝑐𝑜𝑠𝛼= 𝐹 2 𝑐𝑜𝑠𝛽=𝐹
Vertical direction
𝐹 2 𝑠𝑖𝑛𝛽− 𝐹 1 𝑠𝑖𝑛𝛼=Δ𝑚𝑎≈𝜆Δ𝑥 𝜕 2 𝑦 𝜕𝑡 2
Divide by 𝐹
𝜆Δ𝑥 𝐹 𝜕 2 𝑦 𝜕𝑡 2 = 𝐹 2 𝑠𝑖𝑛𝛽 𝐹 2 𝑐𝑜𝑠𝛽 − 𝐹 1 𝑠𝑖𝑛𝛼 𝐹 1 𝑐𝑜𝑠𝛼 =𝑡𝑎𝑛𝛽−𝑡𝑎𝑛𝛼
𝑡𝑎𝑛𝛽−𝑡𝑎𝑛𝛼 Δ𝑥 = 1 Δ𝑥 𝜕𝑦 𝜕𝑥 ​ 𝑥+Δ𝑥 − 𝜕𝑦 𝜕𝑥 ​ 𝑥 = 𝜕 2 𝑦 𝜕 𝑥 2
Wave equation
𝜕 2 𝑦 𝜕 𝑥 2 = 𝜆 𝐹 𝜕 2 𝑦 𝜕𝑡 → 𝜕 2 𝑦 𝜕 𝑥 2 = 1 𝑣 2 𝜕 2 𝑦 𝜕𝑡 2 , 𝑣 2 = 𝐹 𝜆 𝛼

8. Solution of wave equation
𝜕 2 𝑦 𝜕 𝑥 2 = 1 𝑣 2 𝜕 2 𝑦 𝜕𝑡 2 If 𝑦 𝑥,𝑡 =𝑦(𝑥±𝑣𝑡), then 𝜕𝑦 𝜕𝑥 = 𝑑𝑦 𝑑 𝑥±𝑣𝑡 𝑑 𝑥±𝑣𝑡 𝑑𝑥 = 𝑑𝑦 𝑑 𝑥±𝑣𝑡 So 𝜕 2 𝑦 𝜕 𝑥 2 = 𝑑 2 𝑦 𝑑 𝑥±𝑣𝑡 2 , 𝑎𝑛𝑑 𝜕 2 𝑦 𝜕 𝑡 2 = 𝑣 2 𝑑 2 𝑦 𝑑 𝑥±𝑣𝑡 2 , 𝑦(𝑥±𝑣𝑡) satisfy the wave equation and 𝑣 is the wave velocity.

9. Velocity of wave on string
𝜕 2 𝑦 𝜕 𝑥 2 = 𝜆 𝐹 𝜕 2 𝑦 𝜕𝑡 → 𝜕 2 𝑦 𝜕 𝑥 2 = 1 𝑣 2 𝜕 2 𝑦 𝜕𝑡 2 ,
𝑣 2 = 𝐹 𝜆
𝐹 is the tension of the string
𝜆 is the linear density of the string

10. Superposition
𝜕 2 𝑦 𝜕 𝑥 2 = 1 𝑣 2 𝜕 2 𝑦 𝜕𝑡 2 The wave equation is a linear equation. If 𝑦 1 𝑥,𝑡 𝑎𝑛𝑑 𝑦 2 (𝑥,𝑡) both are the solutions of the equation, then 𝑦 𝑥,𝑡 = 𝑦 1 𝑥,𝑡 + 𝑦 2 (𝑥,𝑡) is also a solution 𝜕 2 𝑦 𝜕 𝑥 2 = 𝜕 2 𝑦 1 𝜕 𝑥 2 + 𝜕 2 𝑦 2 𝜕 𝑥 2 , 𝜕 2 𝑦 𝜕 𝑡 2 = 𝜕 2 𝑦 1 𝜕 𝑡 2 + 𝜕 2 𝑦 2 𝜕 𝑡 2

11. Sine wave
𝜕 2 𝑦 𝜕 𝑥 2 = 1 𝑣 2 𝜕 2 𝑦 𝜕𝑡 2 A solution of the wave equation 𝑦 𝑥,𝑡 =𝑦 𝑥±𝑣𝑡 =𝐴𝑠𝑖𝑛(𝜔𝑡±𝑘𝑥) Substitute it into the wave equation, we get 𝑘 2 = 𝜔 2 𝑣 2 𝑜𝑟 𝑣 2 = 𝜔 2 𝑘 2 The general solution is the sum of sine waves with different 𝐴, 𝜔 𝑎𝑛𝑑 𝑘.

12. Periodicity of 𝑥 Periodicity of 𝑡 𝐴 Amplitute 𝑘 Wave number 𝑘 = 𝐿 −1
𝜔 Angular frequency 𝜔 = 𝑇 −1
𝜙 0 Initial phase
Periodicity of 𝑥
𝜆 wavelength 𝜆 =𝐿
Periodicity of 𝑡
𝑇 Period 𝑇 =𝑇

13. Velocity of wave
Wave propagate along 𝑥
Define 13

14. Energy Oscillation velocity Kinetic energy of a piece of string
Around 𝑦=0
Maximum speed and kinetic energy
Maximum deformation and potential
Around 𝑦= 𝑦 0
Minimum speed and kinetic energy
Minimum deformation and potential

15. Energy transfer and power
Kinetic energy of a small piece of string
Change rate
We may consider the kinetic energy transports along the string at this rate
Average rate at which kinetic energy is transported
Elastic potential energy transports at the same average rate
Average power