Vibrations and Waves Vibrations All around us thing

Vibrations and Waves

Vibrations All around us thing are seen to jiggle and wiggle, or vibrate. A vibration in space/time is a wave. Any back and-forth motion, or oscillations. The form of these oscillation, commonly referred to as simple harmonic motion is a sine curve.

Wave forms Most information travels by waves: light, television, radio and cell phones all rely on electromagnetic waves, sound is a longitudinal wave, water waves Note that while energy is transferred matter is not; matter only moves locally and is not transmitted long distances. Examples include a stone in a pond, the wave moves out, not the You. Tube - Laird water. Sound waves reach you not Hamilton - The molecules from the source. greatest big wave surfer to have lived?

Simple Harmonic Motion If an object vibrates or oscillates back and forth over the same path, each cycle taking the same amount of time, the motion is called periodic. The mass and spring system is a useful model for a periodic system.

The Simple Pendulum In order to be in SHM, the restoring force must be proportional to the negative of the displacement. Here we have: which is proportional to sin θ and not to θ itself. However, if the angle is small, sin θ ≈ θ.

Simple Harmonic Motion Any vibrating system where the restoring force is proportional to the negative of the displacement is in simple harmonic motion (SHM), and is often called a simple harmonic oscillator.

Simple Harmonic Motion • Displacement is measured from the equilibrium point • Amplitude is the maximum displacement • A cycle is a full to-and-fro motion; this figure shows half a cycle • Period is the time required to complete one cycle • Frequency is the number of cycles completed per second

Energy in the Simple Harmonic Oscillator If the mass is at the limits of its motion, the energy is all potential. If the mass is at the equilibrium point, the energy is all kinetic. We know what the potential energy is at the turning points:

Wave Motion A wave travels along its medium, but the individual particles just move up-and-down or side-to-side.

Wave Motion All types of traveling waves transport energy. Study of a single wave pulse shows that it is begun with a vibration and transmitted through internal forces in the medium. Continuous waves start with vibrations too. If the vibration is SHM, then the wave will be sinusoidal.

Crest and Troughs The section of the wave that rises above the undisturbed position is called the crest. That section which lies below the undisturbed position is called the trough.

Amplitude The term amplitude can have slightly different meanings depending upon the context of the situation. Its most general definition is that the amplitude is the maximum positive displacement from the undisturbed position of the medium. In some discussions it is important to distinguish between positive and negative amplitudes, as shown above.

Amplitude Sometimes it is necessary to discuss an amplitude at a certain point along the wave. Several of these amplitudes are shown above.

Amplitude • Amplitude is the displacement of the medium from its normal position. • Usually amplitude refers to the maximum positive displacement. • Often, especially in discussions about interference, amplitude means the displacement of the medium from its normal position at certain points, and this displacement can be positive or negative.

Wavelength The wavelength of a wave is the distance between any two adjacent corresponding locations on the wave train. This distance is usually measured in one of three ways: crest to next crest, trough to next trough, or from the start of a wave cycle to the next starting point.

The wavelength of the wave in the diagram above A is given by letter ______. The amplitude of the wave in the diagram above is D given by letter _____.

Frequency • The number of back and forth vibrations (oscillations) that occur in a given time interval. Note it does not show up in the diagrams of waves showing amplitude and wavelength • Frequency is specified in Hertz (Hz), units are sec-1. • Radio frequencies are in k. Hz and MHz, 101 MHz = 101, 000 Hz

Frequency and Period • The wave frequency is the inverse of the period of the wave. Period is the time to complete one complete oscillation. T = 1/f f = 1/T

Wave speed • The speed of a wave depends on the medium. For example sound travels faster in water than air. • For any given medium there is a known relationship between speed, wavelength and frequency. • v = λf

Wave Motion Wave characteristics: • Amplitude, A • Wavelength, λ • Frequency f and period T • Wave velocity

Types of Waves: Transverse and Longitudinal The motion of particles in a wave can either be perpendicular to the wave direction (transverse) or parallel to it (longitudinal).

Transverse Waves • Transverse waves are when the vibration is perpendicular to the wave velocity, examples include vibrations of strings and electromagnetic waves (e. g. light , TV, Radio, microwaves…)

Longitudinal waves • Not all waves are transverse, waves with the vibration in the direction of motion are called longitudinal waves, sound waves are longitudinal waves

Types of Waves: Transverse and Longitudinal Sound waves are longitudinal waves:

Types of Waves: Transverse and Longitudinal Earthquakes produce both longitudinal and transverse waves. Both types can travel through solid material, but only longitudinal waves can propagate through a fluid – in the transverse direction, a fluid has no restoring force. Surface waves are waves that travel along the boundary between two media.

Water Waves • Water waves are actually surface waves, they act as if they are partially transverse and partially longitudinal

Energy Transported by Waves Just as with the oscillation that starts it, the energy transported by a wave is proportional to the square of the amplitude. Definition of intensity: The intensity is also proportional to the square of the amplitude: (11 -15)

Energy Transported by Waves If a wave is able to spread out threedimensionally from its source, and the medium is uniform, the wave is spherical. Just from geometrical considerations, as long as the power output is constant, we see:

Example • Two earthquake waves of the same frequency travel through the same portion of the Earth, but one is carrying twice the energy. What is the ratio of the amplitudes of the two waves? Intensity (or energy) is proportional to the square of the amplitude A 12/A 22 = 2 A 1 = √ 2 A 2

Reflection and Transmission of Waves A wave reaching the end of its medium, but where the medium is still free to move, will be reflected (b), and its reflection will be upright. A wave hitting an obstacle will be reflected (a), and its reflection will be inverted.

Reflection and Transmission of Waves A wave encountering a denser medium will be partly reflected and partly transmitted; if the wave speed is less in the denser medium, the wavelength will be shorter.

Reflection and Transmission of Waves Two- or three-dimensional waves can be represented by wave fronts, which are curves of surfaces where all the waves have the same phase. Lines perpendicular to the wave fronts are called rays; they point in the direction of propagation of the wave.

Reflection and Transmission of Waves The law of reflection: the angle of incidence equals the angle of reflection.

Multiple waves • Matter cannot share space, e. g. only one book can sit on a desk at a given location. . . multiple books cannot share the same space. • Waves can share a medium, there can be many waves in a single medium, e. g. multiple waves in water are formed when several objects are thrown in simultaneously. • The interaction of multiple waves is commonly referred to as interference.

Interference; Principle of Superposition The superposition principle says that when two waves pass through the same point, the displacement is the arithmetic sum of the individual displacements. In the figure below, (a) exhibits destructive interference and (b) exhibits constructive interference.

Interference; Principle of Superposition These figures show the sum of two waves. In (a) they add constructively; in (b) they add destructively; and in (c) they add partially destructively.

Interference Constructive Wave Interference Destructive Wave Interference 2 | Zona Land Education

Standing Waves; Resonance Standing waves occur when both ends of a string are fixed. In that case, only waves which are motionless at the ends of the string can persist. There are nodes, where the amplitude is always zero, and antinodes, where the amplitude varies from zero to the maximum value.

The animation below depicts two waves moving through a medium in opposite directions. The blue wave is moving to the right and the green wave is moving to the left.

In the standing wave shown, what is the amplitude? 10 cm In the standing wave shown, what is its wavelength? 1 meter In the standing wave shown, how many nodes are there? 6 nodes

Standing Waves; Resonance The frequencies of the standing waves on a particular string are called resonant frequencies. They are also referred to as the fundamental and harmonics.

Standing Waves; Resonance The wavelengths and frequencies of standing waves are: Standing Wave Diagrams 1 - Both Ends Fixed | Zona Land Education

Forced Vibrations; Resonance Forced vibrations occur when there is a periodic driving force. This force may or may not have the same period as the natural frequency of the system. If the frequency is the same as the natural frequency, the amplitude becomes quite large. This is called resonance.

Resonance • When the forced vibration matches the natural frequency of an object the result is resonance Tacoma Narrows Bridge Collapse "Gallopin' Gertie" - Bing video

Forced Vibrations; Resonance The sharpness of the resonant peak depends on the damping. If the damping is small (A), it can be quite sharp; if the damping is larger (B), it is less sharp. Like damping, resonance can be wanted or unwanted. Musical instruments and TV/radio receivers depend on it.

Standing waves Standing Waves, Medium Fixed At Both Ends

Refraction If the wave enters a medium where the wave speed is different, it will be refracted – its wave fronts and rays will change direction. We can calculate the angle of refraction, which depends on both wave speeds:

Refraction The law of refraction works both ways – a wave going from a slower medium to a faster one would follow the red line in the other direction.

Diffraction When waves encounter an obstacle, they bend around it, leaving a “shadow region. ” This is called diffraction.

Diffraction The amount of diffraction depends on the size of the obstacle compared to the wavelength. If the obstacle is much smaller than the wavelength, the wave is barely affected (a). If the object is comparable to, or larger than, the wavelength, diffraction is much more significant (b, c, d). Internet Archive: Details: Physics B Lesson 46: Interference and Diffraction

We locate earthquakes epicenters by following these 4 simple steps: 1. Measure the time difference between P and S wave arrivals on a seismogram.

2. Use a travel-time graph to get the distance from the earthquake to the seismic station.

3. Draw a circle on a map that represents this distance.

4. Three (or more) circles from different seismic stations should intersect at the earthquake’s location.

We calculate earthquake magnitudes by following these 3 simple steps 1. Measure the maximum amplitude of S wave arrivals on a seismogram

2. Calculate the distance from the earthquake to the station.

3. Use a nomogram to determine the magnitude based on distance and amplitude.

Tsunamis have a small amplitude (wave height) offshore, and a very long wavelength (often hundreds of kilometers long, whereas normal ocean waves have a wavelength of only 30 or 40 metres) which is why they generally pass unnoticed at sea, forming only a slight swell above the normal sea surface. They grow in height when they reach shallower water.

While everyday wind waves have a wavelength (from crest to crest) of about 100 meters and a height of roughly 2 meters, a tsunami in the deep ocean has a wavelength of about 200 kilometres (120 mi). Such a wave travels at well over 800 kilometres per hour (500 mph), but owing to the enormous wavelength the wave oscillation at any given point takes 20 or 30 minutes to complete a cycle and has an amplitude of only about 1 meter

A drawback occurs because the water propagates outwards with the trough of the wave at its front. Drawback begins before the wave arrives. Drawback can exceed hundreds of meters, and people unaware of the danger sometimes remain near the shore to satisfy their curiosity or to collect fish from the exposed seabed.

As the tsunami approaches the coast and the waters become shallow, wave shoaling compresses the wave and its velocity slows below 50 mph. Its wavelength diminishes to less than 12 miles and its amplitude grows enormously. Since the wave still has the same very long period, the tsunami may take minutes to reach full height.

Summary • A simple pendulum approximates SHM if its amplitude is not large. Its period in that case is: • When friction is present, the motion is damped. • If an oscillating force is applied to a SHO, its amplitude depends on how close to the natural frequency the driving frequency is. If it is close, the amplitude becomes quite large. This is called resonance.

Summary • Vibrating objects are sources of waves, which may be either a pulse or continuous. • Wavelength: distance between successive crests. • Frequency: number of crests that pass a given point per unit time. • Amplitude: maximum height of crest. • Wave velocity:

Summary of Chapter • Vibrating objects are sources of waves, which may be either a pulse or continuous. • Wavelength: distance between successive crests • Frequency: number of crests that pass a given point per unit time • Amplitude: maximum height of crest • Wave velocity:

Summary of Chapter 11 • Transverse wave: oscillations perpendicular to direction of wave motion. • Longitudinal wave: oscillations parallel to direction of wave motion. • Intensity: energy per unit time crossing unit area (W/m 2): • Angle of reflection is equal to angle of incidence.

Summary • When two waves pass through the same region of space, they interfere. Interference may be either constructive or destructive. • Standing waves can be produced on a string with both ends fixed. The waves that persist are at the resonant frequencies. • Nodes occur where there is no motion; antinodes where the amplitude is maximum. • Waves refract when entering a medium of different wave speed, and diffract around obstacles.

• Internet Archive: Details: Physics B Lesson 42: Wave Basics • Lesson 43: Properties of Traveling Waves • Internet Archive: Details: Physics B Lesson 44: Properties of Standing Waves • Internet Archive: Details: Physics B Lesson 45: Sound Waves and Doppler Shift • Internet Archive: Details: Physics B Lesson 46: Interference and Diffraction