Rotation of a Rigid Object

Wave MotionA wave is a rhythmic (periodic) disturbance that transports energy through matter or space. Mechanical waves travel through a medium. The medium is the matter through which a wave transfers energy.

Ocean waves carry energy through water.Earthquakes carry energy through the Earth.Sound waves carry sound through solid, liquid or gaseous materials.

Light waves carry light through a space.Slinky WaveLets use a slinky wave as an example.When the slinky is stretched from end to end and is held at rest, it assumes a natural position known as the equilibrium or rest position.To introduce a wave we must first create a disturbance.We must move a particle away from its rest position.

Slinky WaveOne way to do this is to jerk the slinky to the side.The beginning of the slinky moves away from its equilibrium position and then back.The disturbance continues down the slinky.This disturbance that moves down the slinky is called a pulse.If we keep pulsing the slinky back and forth, we could get a repeating disturbance.Wave TypesTransverse waves the wave travels perpendicular to the motion of the medium. The direction of the wave is the direction of energy transfer. Earthquake wave

Wave TypesLongitudinal waves the wave travels parallel to the motion of the medium. The direction of the wave and medium is the direction of energy transfer.Sound wave

Transverse WavesThe differences between the two can be seen

Wave Characteristics

Transverse Waves:High points are called crests.Low points are called troughs.Longitudinal Waves:Compression- place on a longitudinal wave where particles of matter are closest together.Rarefaction- place on a longitudinal wave where particles of matter are farthest apart.Wave Characteristics

For all wave types:Wavelength is the distance between a point on one wave and the identical point on the next wave (crest to crest, trough to trough, and compression to compression). Amplitude is the distance from the crest (or trough or compression) to the rest position or the medium. The amplitude corresponds to the amount of energy carried by the wave. The greater the energy the larger the amplitude.Frequency is the number of wave crests (or compressions) that pass one point each second. Frequency is expressed in units of Hertz (Hz). One hertz is the same as one wave per second.Wave VelocityWave velocity is how fast the wave travels through the medium. (Not the speed of the particles, speed the crests move.) Wave velocity = frequency x wavelengthv = velocity of the wave (m/s)f = frequency (Hz or 1/s) = wavelength (m)

Wave VelocityExample 1: A wave is measured to have a frequency of 60Hz. If its wavelength is 24cm, determine how fast it is moving.

Example 2: The speed of light is always 3.00x108 m/s. Determine the frequency of red light which has a wavelength of 700nm.

Wave BehaviorAll waves share common behaviors. Velocity/Medium dependency

Frequency/Wavelength relationship

Reflection/Transmission at medium boundary

Wave BehaviorAll waves share common behaviors. Velocity/Medium dependency: The speed with which a wave travels depends only on properties of the medium through which it travels.Frequency/Wavelength relationship: The lower the frequency the larger the wavelength. The higher the frequency the shorter the wavelength. Reflection/Transmission at medium boundary: A wave when it reaches a barrier will bounce off and reverse direction (reflect). Depending on the medium beyond the barrier, some of the wave energy may transmit through the barrier into the new medium.Frequency/Wavelength RelationshipThe lower the frequency the larger the wavelength. The higher the frequency the shorter the wavelength.

Wave BehaviorWe know that waves travel through mediums.But what happens when that medium runs out?Boundary BehaviorThe behavior of a wave when it reaches the end of its medium is called the waves BOUNDARY BEHAVIOR.When one medium ends and another begins, that is called a boundary.Fixed EndHere the incident pulse is an upward pulse.The reflected pulse is upside-down. It is inverted.The reflected pulse has the same speed, wavelength, and frequency as the incident pulse.

Wave ReflectionA wave when it reaches a barrier will bounce off and reverse direction.

Characteristics of the reflected pulse include:the speed of the reflected pulse is the same as the speed of the incident pulse the wavelength and frequency of the reflected pulse is the same as the wavelength and frequency of the incident pulse the amplitude of the reflected pulse may be less than the amplitude of the incident pulse

Free EndAnother boundary type is when a waves medium is attached to a stationary object as a free end.In this situation, the end of the medium is allowed to slide up and down.What would happen in this case?

Free EndHere the reflected pulse is not inverted.It is identical to the incident pulse, except it is moving in the opposite direction.The speed, wavelength, and frequency are the same as the incident pulse.

Free End Animation

20What Do You Think?What happens at the point where two waves meet if:

the waves are traveling on opposite sides of a slinky and one wave has an amplitude of 10 cm while the second wave has an amplitude of 20 cm? the waves are traveling on the same side of a slinky and one wave has an amplitude of 10 cm while the second wave had an amplitude of 20 cm?

What happens after they meet?Wave BehaviorAll waves share common behaviors. When two waves meet while traveling along the same medium it is called Interference.Constructive Interference: Two or more waves combine to form a wave with greater amplitude.Destructive Interference: Two or more waves combine to form a wave with a smaller amplitude.Wave InterferencePrinciple of SuperpositionWhen two waves meet their amplitudes ADD together at the point at which they meet and then they continue on their way as before.

Constructive InterferenceDestructive Interference

Check Your UnderstandingWhich points will produce constructive interference and which will produce destructive interference?

ConstructiveG, J, M, NDestructiveH, I, K, L, OStanding Waves

Nodes and Antinodes

Guitar VideoclipVibrating StringThe fundamental vibration mode of a stretched string is such that the wavelength is twice the length of the string.

Applying the basic wave relationship gives an expression for the fundamental frequency:

Since the wave velocity is given by , the frequency expression

can be put in the form:

HarmonicsAn ideal vibrating string will vibrate with its fundamental frequency and all harmonics of that frequency.

If you have a string withstarting pitch: 100 Hzand change to the pitchwill bedouble the length 50 Hzfour times the tension 200 Hzfour times the mass 50 Hz

HarmonicsHarmonicPattern# of LoopsLength-Wavelength Relationship

1st 1L = 1 / 2 2nd 2L = 2 / 2 3rd 3L = 3 / 2

Example: The string at the right is 1.5 meters long and is vibrating as the first harmonic. The string vibrates up and down with 33 cycles in 10 seconds. Determine the frequency, period, wavelength and speed for this wave.

Harmonics Given: L = 1.5 m, 33 cycles in 10 seconds

The frequency refers to how often a point on the medium undergoes back-and-forth vibrations; it is measured as the number of cycles per unit of time. f = (33 cycles) / (10 seconds) = 3.3 Hz

The period is the reciprocal of the frequency. T = 1 / (3.3 Hz) = 0.303 seconds The wavelength of the wave is related to the length of the rope. For the first harmonic as pictured in this problem, the length of the rope is equivalent to one-half of a wavelength. That is, L = 0.5 where is the wavelength. = 2 L = 2 (1.5 m) = 3.0 m The speed of a wave can be calculated from its wavelength and frequency using the wave equation:v = f = (3.3 Hz) (3. 0 m) = 9.9 m/s

Nodes and Antinodes

ResonanceResonance is a special type of constructive interference. Tie off a piece of rope to a wall, and then stretch it out. Standing at the far end flick your wrist to send a wave pulse to the other end. When the pulse hits the wall most of it will be reflected back towards you as an inverted wave. If you held tightly with your hand, the wave would hit your hand and most would be reflected away from you But if you flick your wrist again at exactly the instant that the first wave hits. the two waves are now both traveling away from you. Their amplitudes will add together to make a bigger wave!

Keep doing this over and over again and the wave keeps getting bigger.

In this example the frequency of your wrist matches the frequency of the wave.

ResonanceIn this example the frequency of your wrist matches the frequency of the wave itself. Constructive interference is what causes the amplitude to increase.The frequency of the wave itself is equal to the frequency of new waves being created. We would say that the new waves are being created at the resonant frequency

ResonanceTacoma Narrows Bridge CollapseA famous example of resonance is the destruction of the Tacoma Narrows Bridge in Washington. Soon after the bridge was built in 1940, it began to vibrate due to wind.

ResonanceThe driving vibration frequencies from the wind matched the natural resonant frequency of the bridge. The amplitude of oscillations got larger and larger. Within hours, the entire bridge had broken apart and collapsed. It was a spectacular failure. The failure was not due to extraordinarily strong winds, but rather it was the matching of natural vibration frequency that took it down.