Chapter 19 Vibrations and Waves

Vibration: A disturbance “wiggle” in time.

Oscillatory Motion The to-and-fro vibratory motion, such as that of a pendulum.

Simple Pendulum For small displacements, the period of the simple pendulum is related to its length (L) and the acceleration due to gravity (g) by the following:

Simple Harmonic Motion is a type of oscillatory motion in which the motion repeats itself. This motion is caused by a “restoring force” that acts in the opposite direction of the displacement.

Simple Pendulum Under small displacements, the simple pendulum behaves as a harmonic oscillator. For a pendulum, the “restoring force” is usually exerted by GRAVITY

Amplitude The maximum displacement from some equilibrium (mid point) position. (Applies to both vibrations and waves.)

Mass-Spring System is Another Example of a Simple Harmonic Oscillator Live Demonstration

Wavelength The distance between successive crests, troughs, or identical parts of a wave. Common symbol used for wavelength is the Greek letter - pronounced “lambda”

Sine Curve Amplitude A Wavelength Crest Trough

Frequency: The number of vibrations per unit time. Common symbols are f and the Greek letter - pronounced ”nu” Period: The time in which a vibration is completed. Common symbols are T and the Greek letter  - pronounced “Tau”

More on frequency We can talk about the frequency of a vibration or of a wave. Frequency is measured in inverse seconds, or Hertz (Hz). E.g.. f = 10 cycles/sec = 10sec -1 = 10 Hz.

Frequency and Period are related Frequency equals inverse Period. Period equals inverse Frequency.

In symbols, this means... f = 1/T or = 1/  and T = 1/f or  = 1/

Examples AM radio frequencies are measured in KiloHerts - (KHz). The period is 1/1,000Hz = 1x10 -3 sec = 1millisecond (ms) Kilo = one thousand = 1,000 = 1x10 3.

FM radio frequencies are measured in MegaHertz (MHz) The period is (1/1,000,000 Hz)= 1x10 -6 sec = 1 microsecond (  s). Mega = one million = 1,000,000 = 1x10 6

More Examples Water waves might have a frequency of 2Hz (i.e. 2 cycles per second). The corresponding period is equal to: 1/f = 1/2Hz = 0.5 seconds

The AM and FM radio waves are examples of Light is another example of an electromagnetic wave Electromagnetic Waves

The water waves are examples of Mechanical Waves Mechanical waves require a medium in which to propagate. Electromagnetic waves do not.

Wave Speed The speed with which waves pass a particular point. Common symbol used for wavespeed is the letter v.

Wavespeed = wavelength / period v = f but, since we already know that frequency is the same as inverse period ( f = 1/T), then we can also write this as v = /T In symbols, this is:

A note as to why we use “v” The letter v is used for velocity in general. Velocity is speed in a specific direction. Velocity and speed are closely related.

For Example If I tell you I’m traveling at 55 miles/hour due north, I have told you my velocity If I tell you I’m traveling at 55 miles/hour, I have told you my speed.

Types of Waves There are two types of waves 2)Longitudinal Waves. 1) Transverse Waves.

1) Transverse Wave: A wave in which the vibration is in a direction perpendicular (transverse) to the direction in which the wave travels. e.g. Light waves. Waves on a string. Seismic “S”-waves.

2) Longitudinal Wave: A wave in which the medium vibrates in a direction parallel (longitudinal) to the direction in which the wave travels. e.g. Sound. Seismic P-waves.

In a longitudinal wave, the medium has regions of compression and expansion which are along the direction of wave propagation. Regions of expansion are also called (rarefactions)

Interference A number of different waves can add, constructively or destructively. The superposition of two or more waves results in interference. This is known as superposition.

Destructive Interference: Exactly out of Phase Cancellation + Zero displacement

Constructive Interference: In Phase Maximum Displacement Reinforcement +

Interference Pattern The pattern formed by superposition of different sets of waves that produce mutual reinforcement in some places and cancellation in others. Superposition Principle of Wave

Standing Wave A stationary wave pattern formed in a medium when two sets of identical waves pass through the medium in opposite directions. lecture demos

Standing Wave V V V V Incident Wave Reflected Wave

Beats Sometimes, two waves with slightly different frequencies but the same amplitude can form the phenomenon known as beats Beats

Blue colored wave + green colored wave ==> red colored wave. Two waves with same amplitudes but slightly different frequencies.

Doppler Effect The shift in received frequency due to motion of a vibrating source toward or away from a receiver The Doppler Effect

Bow Wave The V-shaped wave made by an object moving across a liquid surface at a speed greater than the wave speed. (Since the source is moving faster than the wave speed, the wavefronts pile up.)

Shock Wave The cone-shaped wave made by an object moving at supersonic speed through a fluid. (Here, the source is moving faster than the wave speed, which is the speed of sound!!) (Super-sonic speed)

Sonic Boom The loud sound resulting from the incidence of a shock wave. (This is the result of the pile up of many wave fronts which produces a sonic boom)

Sonic Boom Piled up wave fronts produce a shock wave Plane

Twice the speed of sound - Mach 2 Plane 2 units 1 unit Wave front

The End of Chapter 19