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Lecture 2: Introduction to wave theory (II)

Lecture 2: Introduction to wave theory (II). Mathematical description of waves (Y&F 15.3):. Phase velocity : the velocity (speed) at which we would have to move to keep up with a point of constant phase on the wave. Right moving wave: Derivative with respect to t is zero:.

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Lecture 2: Introduction to wave theory (II)

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  1. Lecture 2: Introduction to wave theory (II) Mathematical description of waves (Y&F 15.3): • Phase velocity: the velocity (speed) at which we would have to move to keep up with a point of constant phase on the wave. • Right moving wave: • Derivative with respect to t is zero: Phase velocity: is the same as speed of wave: • Transverse speed of wave: P1X: Optics, Waves and Lasers Lectures, 2005-06.

  2. y A x l/4 l/2 3l/4 l - A vy Aw x l/4 l/2 3l/4 l - Aw • Transverse speed maximum: • Transverse speed minimum: Offset by 90o P1X: Optics, Waves and Lasers Lectures, 2005-06.

  3. Example: 15-2 from Y&F (page 556) Find the maximum transverse speed of the example shown in lecture 1. What is the velocity at t=0, at the end of the clothes-line and at 3.0 m from the end. At x=t=0, velocity is maximum transverse speed = +0.94 ms-1 At x=3.0 m and t=0: P1X: Optics, Waves and Lasers Lectures, 2005-06.

  4. Simple Harmonic Motion (Y&F 13.1-2, 13.4-5): • Definition: • Simple Harmonic Motion (SHM) is motion in which a particle is acted on by a force proportional to its displacement from a fixed (equilibrium) position and is in the opposite direction to the displacement: • Examples: • Mass vibrating on a spring. • Simple pendulum (only when displacement is small). P1X: Optics, Waves and Lasers Lectures, 2005-06.

  5. Simple pendulum: • Vertical: • Horizontal: • If q is small then when and therefore: and: The same as the restoring force of a spring but with: P1X: Optics, Waves and Lasers Lectures, 2005-06.

  6. x A T/2 T/4 3T/4 T t - A • Solution: • What function satisfies ? • Try with the angular frequency (rad/s): • For the case of the spring: • For the case of the pendulum: P1X: Optics, Waves and Lasers Lectures, 2005-06.

  7. Definitions: a) Amplitude A is maximum displacement (m). b) Frequency f: number of oscillations per second. (Units: 1 Hertz = 1 cycle/s = 1 s-1) c) Period T: time (s) between oscillations d) Phase constant (f ): gives position of oscillation at t=0. P1X: Optics, Waves and Lasers Lectures, 2005-06.

  8. Example: 13-2 from Y&F A spring is mounted horizontally. A force of 6.0 N causes a displacement of 0.030 m. If we attach an object of 0.50 kg to the end and pull it a distance of 0.020 m and watch it oscillate in SHM, find (a) the force constant of the spring, (b) the angular frequency, frequency and period of oscillation. (a) At x = 0.030 m, F=-6.0 N (b) m=0.50 kg, k=200 N/m: The frequency: The period: P1X: Optics, Waves and Lasers Lectures, 2005-06.

  9. Example: 13-8 from Y&F Find the frequency and period of a simple pendulum that is 1.0 m long (assume g=9.80 m/s). The angular frequency: The frequency: The period: P1X: Optics, Waves and Lasers Lectures, 2005-06.

  10. Example: Vertical SHM Vertical oscillations from a spring hanging vertically. 1) At rest: Spring is stretched by Dl such that: 2) x above equilibrium: 3) x below equilibrium: Same SHM as in vertical case, oscillations with angular frequency: Fnet Equilibrium is at stretched position Dl instead of x=0 P1X: Optics, Waves and Lasers Lectures, 2005-06.

  11. Example: 13-6 from Y&F Shock absorbers of an old car with mass 1000 kg are worn out. When a person weighing 100 kg climbs into the car, it sinks by 2.8 cm. When the car is in motion and hits a bump it oscillates. What is the frequency and period of oscillation? The spring constant: The angular frequency: The frequency: The period: P1X: Optics, Waves and Lasers Lectures, 2005-06.

  12. Simple Harmonic Motion initiates sinusoidal waves and sets the boundary conditions for wave motion • For example, a string attached to a vertical spring • A radio transmitting antenna causes electromagnetic waves by oscillating molecules P1X: Optics, Waves and Lasers Lectures, 2005-06.

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