Example: pulse on a string speed of pulse = wave speed = v Chapter 12: Waves sec 1-6 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?

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

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

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

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

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

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

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?

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.

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

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

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?

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