1. Many coupled oscillators From oscillators to waves:
PHY238Y Lecture 4
Coupled oscillators
Many coupled oscillators
From oscillators to waves:
Transverse and longitudinal waves
Wavelength and frequency of a wave
Speed of a traveling wave
References:
Lecture notes;
Halliday/Resnick/Walker: Fundamentals of Physics, 6th ed., Chapter 17 (17.1 – 17.5)
Thanks to dr. R. Nave for the permission to use some of the pictures from Hyper Physics:

2. Parallel oscillation:
PHY238Y Lecture 4
Coupled oscillators
Parallel oscillation:
Symmetric oscillation
See Lecture Notes

3. PHY238Y Lecture 4 Example for Lecture 4
Infrared spectroscopic data show that sodium
chloride (NaCl) has an inter-atomic oscillation
frequency of 1.4  1013Hz.
Find the inter-atomic force constant (k),
assuming a model of two masses connected by a
spring.
We take: mCl = 351.67 kg;
mNa = 231.67 kg
The model: two masses connected by an ideal spring

4. Many coupled oscillators: the mass-spring transmission line:
PHY238Y Lecture 4
Many coupled oscillators: the mass-spring transmission line:
See Lecture notes

5. PHY238Y Lecture 4
The mass-spring transmission line: very many masses and springs.
Displacement of mass number “n” from equilibrium position is ξn :

6. PHY238Y Lecture 4
The Wave Equation
The wave equation for a plane wave traveling in the x direction is
where v is the phase velocity of the wave and y represents the variable which is changing as the wave passes. This is the form of the wave equation which applies to a stretched string or a plane electromagnetic wave.

7. PHY238Y Lecture 4
Transverse waves: displacement is perpendicular to the direction of travel
Longitudinal waves: displacement is in the same direction as the direction of travel

8. Wavelength and frequency:
PHY238Y Lecture 4
Wavelength and frequency:
Periodic motion:
Period: the time required to complete a full cycle, T in seconds/cycle
Frequency: the number of cycles per second, f in 1/seconds or Hertz (Hz)
Amplitude: the maximum displacement from equilibrium A
For a traveling wave, one
needs also:
Velocity of propagation: v
Wavelength: repeat distance of wave