CHAPTER 5 -- OSCILLATIONS AND WAVES
Oscillations describe repetitive motion back and forth around an equilibrium position. Waves describe the propagation of a disturbance through space, often transporting energy without transporting matter.
5.1 HARMONIC OSCILLATORS
Simple Harmonic Motion (SHM) occurs when a restoring force is directly proportional to the displacement from equilibrium and acts in the opposite direction.
F = -k * x (Hooke's Law)
By Newton's second law:
m * d^2x/dt^2 = -k * x
d^2x/dt^2 + (k/m) * x = 0
The solution is a sinusoidal function:
x(t) = A * cos(omega * t + phi)
where:
- A is Amplitude (maximum displacement).
- omega is angular frequency (sqrt(k/m)).
- phi is the phase constant (determined by initial conditions).
5.2 FREQUENCY, WAVELENGTH, PERIOD
Oscillation Parameters:
- Period (T): Time for one complete cycle. T = 2*pi / omega.
- Frequency (f): Number of cycles per unit time. f = 1 / T = omega / (2*pi). Unit: Hertz (Hz).
Wave Parameters:
For a traveling wave (e.g., y(x,t) = A * cos(k_w * x - omega * t)):
- Wavelength (lambda): The spatial distance between consecutive peaks.
- Wave Number (k_w): Spatial frequency. k_w = 2*pi / lambda.
- Wave Speed (v): The speed at which the wave peaks move. v = lambda * f = omega / k_w.
5.3 RESONANCE
Resonance is a phenomenon where a system responds with maximum amplitude to an external driving force at a specific frequency, known as the resonant frequency.
- Driven Damped Oscillator: When the driving frequency matches the system's natural frequency (omega_0), energy transfer is most efficient.
- Examples: Pushing a swing, shattering a glass with sound, tuning a radio circuit.
- If damping is low, the resonance peak is sharp and high. If damping is high, the peak is broad and lower.
5.4 STANDING WAVES
Standing waves are formed by the superposition (interference) of two waves traveling in opposite directions with the same frequency and amplitude. Unlike traveling waves, standing waves do not propagate; they oscillate in place.
Nodes and Antinodes:
- Nodes: Points of zero amplitude (destructive interference).
- Antinodes: Points of maximum amplitude (constructive interference).
Boundary Conditions:
- String fixed at both ends: Length L must be an integer multiple of half-wavelengths.
L = n * (lambda / 2), where n = 1, 2, 3...
Allowed frequencies: f_n = n * (v / 2L).
These discrete frequencies are called harmonics or overtones.
- Open/Closed Pipes: Similar rules apply to sound waves in tubes, depending on whether the ends are open (pressure node) or closed (pressure antinode).
