1. Example: pulse on a string speed of pulse = wave speed = v
Chapter 12: Waves
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Example: pulse on a string
speed of pulse = wave speed = v
depends upon tension T and inertia (mass per length )
actual motion of string
motion of “pulse”
Example: The stainless steel forestay of a racing sailboat is 20m long, and had a mass of 12 kg. To find the tension, the stay is struck by a hammer at the lower end and the return pulse is timed. If the interval is 0.20 s, what is the tension in the stay?

2. Reflections at a boundary
fixed end = “hard” boundary
Pulse is inverted

3. Reflections at a boundary
free end = “soft” boundary
Pulse is not inverted

4. Reflections at an interface
light string to heavy string = “hard” boundary
faster medium to slower medium
heavy string to light string = “soft” boundary
slower medium to faster medium

5. When pulses pass the same point, add the two displacements
Principle of Superposition: When Waves Collide!
When pulses pass the same point, add the two displacements

6. Periodic Waves
a.k.a. Harmonic Waves, Sine Waves ...

7. Important characteristics of periodic waves
wave speed v: the speed of the wave, which depends upon the medium only.
wavelength : : (greek lambda) the distance over which the wave repeats, it is also the distance between crests or troughs.
frequency f : the number of waves which pass a given point per second. The period of the wave is related to the frequency by T = 1/f.
Wavelength, speed and frequency are related by:
v =  f

8. Example: A radar operating at a frequency of 9
Example: A radar operating at a frequency of 9.40 GHz emits groups of waves 0.08 ms in duration. Radio waves are Electromagnetic waves, and so they travel at the speed of light: v = c = 3.00 x108 m/s.
(a) What is the wavelength of these waves?
(b) What is the length of each wave group?
(c) What is the number of waves in each group?

9. Important characteristics of periodic waves (cont’d)
Amplitude A: the maximum displacement from equilibrium. The amplitude does not affect the wave speed, frequency, wavelength, etc. A
The maximum displacement and maximum speed of the string depend upon the Amplitude.
=> Elastic Potential Energy and Kinetic Energy associated with energy depend upon Amplitude.
Energy per time (power) carried by a wave is proportional to the square of the amplitude.
“Loudness” depends upon amplitude.

10. Types of waves
Transverse Waves: “disturbance” is perpendicular to wave velocity, such as for waves on a string. (disturbance is a shear stress)
Longitudinal Waves: “disturbance” is parallel to wave velocity, such as the compression waves on the slinky.
Water surface waves: mixture of longitudinal and transverse

11. vibrations in fixed patterns
Standing Waves
vibrations in fixed patterns
effectively produced by the superposition of two traveling waves
constructive interference: waves add
destructive interference: waves cancel
 = 2L
 = 2L
 = 2L
 = 2L
node
antinode
antinode

12. Example: The A string on a violin has a linear density of 0
Example: The A string on a violin has a linear density of 0.60 g/m and an effective length of 330 MM. (a) Find the Tension in the string if its fundamental frequency is to be 440 Hz. (b) where would the string be pressed for a fundamental frequency of 495 Hz?

13. Resonance
When a system is subjected to a periodic force with a frequency equal to one of its natural frequencies, energy is rapidly transferred to the system.
musical instruments with fundamental or overtones
mechanical vibrations