Phy 212: General Physics II

1. Phy 212: General Physics II Chapter 16: Waves I Lecture Notes

2. Types of Waves Types of waves: • Mechanical waves • Electromagnetic waves • Matter waves Classifying Waves: • Transverse Waves: oscillate perpendicular to the direction of wave propagation (examples: Light & E-M waves) • Longitudinal Waves: oscillate parallel to the direction of wave propagation (example: Sound)

3. vwave T l Position (m) 1 cycle Time (s) General Character of Traveling Waves All traveling waves exhibit fundamental properties that are useful for describing their behavior: • Wave speed (vwave) refers to the rate of travel of the wave as it propagates through a medium • Wavelength (l) refers to the displacement traveled by a position along the wave during 1 period of oscillation (T) • Frequency (f) is the time rate of complete “vibration cycles” as a wave passes through a given point in a medium vwave= lf

4. Traveling Waves General form for the disturbance producing a traveling wave: Transverse Velocity: Velocity Amplitude: Acceleration: Acceleration Amplitude:

5. FT FT vwave 2FT-y y(t) Simple Model: A Traveling Wave in a String A stretched string, with constant linear mass density (m) and string tension (FT), provides a simple demonstration of the motion of a traveling wave

6. Wave speed on Stretched String • When a transverse disturbance travels along a taut string, the motion (transverse to the string) of any given segment of mass along the string oscillates in simple harmonic motion. • The tension (FT ) in the string functions as an “elastic” force resulting in a traveling wave • The linear mass density (m, in kg/m) of the string acts as the inertial element for the string which restricts the wave propagation. • The wave speed along a string is therefore expressed as: Derivation:

7. Energy Transmission (Power) in Waves As a wave travels along the string, the total energy (dEtot) for a string mass segment of length, dx, is the combination of both its K and U: where: The average energy is: where: dK(t)avg = dU(t)avg The average power is:

8. The Wave Equation • The motion of a traveling wave is dependent on both time and position: • This equation (and all equations) that describe traveling wave motion, must satisfy the wave equation:

9. Superposition of Waves & Interference Waves of the same type that cross paths do not alter the other. • They do appear to combine to produce a resulting wave that is a superposition of each wave. • For 2 identical waves (y1 and y2) that are out of phase, the resultant wave is given by: • The resultant wave is itself sinusoidal and its magnitude is depends on the phase difference between y1 and y2

10. y3 y’ y2 y1 Phasors Phasors are analogous to vectors and are useful for analyzing the superposition of same frequency traveling waves: • A phasor is a vector representation of the magnitude (length) and phase (angular direction) for a wave • When 2 or more waves undergo interference, the magnitude and phase of the resultant (superimposed) wave can be determined by the vector addition of the corresponding phasors • The resultant phasor magnitude is the magnitude of resultant wave • The resultant phasor angle is the phase angle of resultant wave

11. n=1 n=2 n=3 Standing Waves in a String • Standing waves are self-reinforcing waves produced by the constructive interference of a wave with its reflection. • Consider a wave traveling along a string (fixed at each end). When the wave reaches the end of the string it reflects back in the opposite direction (180o out of phase). • The wave will produce a standing wave if the nodes for the resultant wave are stationary Condition for a standing wave: where: L is length of string n is the number of ½ wavelengths (l)

12. Standing Wave Superposition