Wave Motion Think of yourself in the ocean… When you are far out beyond the breaking of the waves, which way do you move when a wave goes by and you are floating? The velocity of a wave, and particles within the wave, are different in magnitude and direction. Waves will move along the surface of the water, the water particles will move up and down in the same spot.

What are waves? Waves carry energy from one place to another. Example, the energy given by my hand to the long spring causes a wave to carry that energy to the other end. A rock thrown into a still pond will give energy to the water and cause a wave to be formed.

Wave “pulse” With a quick upward movement of a hand on the end of a rope (or spring)… Causes the rope to go upwards, it will flow along the rope, but the end where the movement began goes back to the same position. pulse simulations

Continuous or Periodic wave Instead of having one disturbance on the end of the rope, what if I have a continuous disturbance? The disturbances are called vibrations. If the vibration is in SHM, then the wave itself will be sinusoidal in both space and in time. space: the wave will always look like either a sine or cosine function. Time: the wave will look like SHM over a long period of time.

Describing waves Amplitude: The maximum height of the crest, or depth of a trough, from an equilibrium point. Wavelength: The distance between two successive crests or two successive troughs (λ – lambda) Frequency: the number of crests that pass a given point per unit of time. (a full cycle). This is the inverse of the period (time for one cycle).

Parts of a wave

Wave velocity The velocity of a wave is the distance of one wavelength in one period. Thus the wave velocity is lambda/T. Since 1/T = f… v = λf

Types of waves Transverse wave: The particles of the medium in which the wave travels through move perpendicular (transverse) to the motion of the wave. Longitudinal waves: The vibrations of the particles in the medium are along the same direction as the motion of the wave. The wave compresses and expands.

demonstration

Energy transported by waves Remember from SHM, E = ½ k A 2 Therefore, energy transported by a wave is proportional to the square of the amplitude.

Intensity of a wave Intensity is defined as Power/Area. So… I = (energy/time)/area, and energy is proportional to the amplitude squared… Intensity is proportional to the amplitude squared.

Intensity continued… We will be talking about waves that are spherical, so surface area of a spherical wave (from geometry) is 4πr 2 So if I = Power/Area, or I = Power/ 4πr 2, Then Intensity is in proportion to the inverse square of the distance. (think butter gun)

Sound Waves Three things to know about sound waves: 1) There must be a source for a sound wave, that source will be a vibrating object. 2) The energy transferred from the source is longitudinal. 3) The sound is detected by an ear or an instrument.

Characteristics of Sound waves Sound can travel in different materials besides for air. In air, the speed of sound is 343 m/s. Does the temperature of air effect the speed of sound? Which type of material do you think will have the greatest speed of sound? Why would someone put their ear to the ground to determine if someone is coming towards them?

More characteristics Pitch: high or low sounds (like a flute compared to a tuba). The lower the frequency the lower the pitch. The audible range is between 20Hz and 20,000Hz for healthy hearing. As a person gets older, the high-frequency limit lowers to about 10,000Hz. Frequencies above 20,000 Hz is called ultrasonic (different from supersonic)

Ultrasonic Many animals can hear ultrasonic frequencies. Dogs can hear up to 50,000 Hz, and bats 100,000 Hz. Autofocus cameras emit a pulse of ultrasonic sound that travels to the object being photographed and back to the camera. A sensor times the reflected sound to know how far the object is.

Infrasonic Sound waves that are below the audible range (20Hz) Earthquakes, thunder, volcanoes, and waves produced by vibrating heavy machinery can produce infrasonic waves. Infrasonic waves, like ones that can be produced by heavy machinery can harm the human body.

Characteristics of sound Loudness: This is the intensity of the sound wave. Intensity varies with the inverse square of the distance. (think butter gun) The human ear can detect sounds with an intensity as low as W/m^2 and as loud as 1 W/m^2 (larger will cause pain) This is a huge range…

Alexander Graham Bell Used a logarithmic scale to measure the intensity of a sound. We call this unit of measurement a Bel (or more commonly a decibel, 10 dB = 1 bel). Beta is measured in dB, and I0 is a reference level, which will be the lowest intensity we can hear (1x ) Example…

Doppler Effect

Doppler effect As an object that emits a sound is moving towards an observer, the frequency of the sound increases. As an object that emits sound is moving away from an observer, the frequency of the sound decreases. The Doppler effect has applications with sound, but will also have applications next week when we discuss light waves in more detail.

Sheldon Cooper Explains the Doppler Effect Big Bang Theory Clip

Sonic Booms When an object is moving faster than the speed of sound, it is said to have reached supersonic speed. An object moving faster than the speed of sound has “outrun” its sound waves.

Doppler effect

Day 2: Reflection and Interference

Reflection

If the end of a cord is free to move, the pulse will reflect on the same side of the cord as it is sent. If the end of the cord is fixed, then the pulse will comeback inverted from the way it is sent. This is due to Newton’s Third Law

Law of Reflection The angle of reflection equals the angle of incidence.

Interference When two waves pass through the same region of space at the same time. Principle of Superposition The region where waves overlap, the resultant is the algebraic sum of their separate displacements. This could be constructive or destructive interference.

Interference

Phases and interference for continuous waves For constructive interference to occur, waves are said to be “in phase”. For destructive interference to occur, waves are said to be “out of phase”

wave interference simulation

Sound “beats” and interference When two sounds (or more) of different frequencies are played at the same time, there is both constructive and destructive interference. This causes a “beat”. And the best for last…

Standing waves If you have a fixed end of a cord and you can vibrate it at a certain frequency so it just looks like it is oscillating up and down without traveling down the cord, this is called a standing wave. Places where there is complete destructive interference are called nodes, and places where there is constructive interference are called anti-nodes.

Resonance Frequencies at which standing waves are produced are called “natural frequencies” or “resonant frequencies”. Resonance occurs because everything in nature has a natural frequency. In vibrating objects, there is only one resonant frequency. If this frequency is hit, then it causes the amplitude of the wave to increase… sometimes catastrophically.

Resonance in cords Cords are different because they have many natural resonant frequencies. Each of which is a whole- number multiple of the lowest resonant frequency.

Harmonics First, notice that the different resonant frequency depends on the length of the cord. The lowest frequency, the fundamental frequency, corresponds to one half of a wavelength, L = 1/2λ1. When a frequency is an integral multiple of the fundamental frequency, they are called harmonics. The fundamental frequency is the first harmonic

Other harmonics The second harmonic is now one full wavelength, or L = 1λ. The third harmonic is now 1.5 wavelengths, or L = 3/2λ In total… Or solving for lambda And since f = v/λ…

Sounds by air columns First, let us examine an instrument such as a flute, also known as an “open tube”. Just like the string situation, we will look at the number of wavelengths for each overtone.

Closed tube An example of a closed tube could be a clarinet, there is always a displacement node at the closed end, because the air is not free to move, and an anti-node at the open end. We will see in a moment, that this means that the fundamental frequency frequency will be L = ¼ λ. We will also see, that there is no way for the even harmonics to exist, but only odd harmonics.

harmonics demonstration