Combining results gives us the case where both observer and source are moving:

Sample Problem: A bus approaching a bus stop at 24 m/s blows its horn Sample Problem: A bus approaching a bus stop at 24 m/s blows its horn. What the perceived frequency that you hear, if the horn’s true frequency is 150 Hz? f = 150 hz, vs = 340 m/s, uo = 24 m/s f ’ = ? f ’ = (1 - u/v)f (source moving toward) f ’ = [1/(1 - 24/340)]150 f ’ = 161.39 Hz

Ex: A car moving at 18 m/s sounds its 550- Hz horn Ex: A car moving at 18 m/s sounds its 550- Hz horn. A bicyclist, traveling with a speed of 7.2 m/s, moves toward the approaching. What frequency is heard by the bicyclist, if the speed of sound is 343 m/s? f = 550 Hz, v = 343 m/s, uo = 7.2 m/s, us = 18 m/s f ’ = ? Both are moving toward each other. f ’ = [(1 + uo/v)/(1 – us/v)]f f ’ = [(1 + 7.2/343)/(1 – 18/343)]550 f ’ = 590 Hz

Doppler Effect Stationary source Moving source http://www.kettering.edu/~drussell/Demos/doppler/mach1.mpg Animations courtesy of Dr. Dan Russell, Kettering University Supersonic source http://www.lon-capa.org/~mmp/applist/doppler/d.htm

The Doppler effect has many practical applications: weather radar, speed radar, medical diagnostics, astronomical measurements. At left, a Doppler radar shows the hook echo characteristic of tornado formation. At right, a medical technician is using a Doppler blood flow meter.

Principle of Superposition When two or more waves pass a particular point in a medium simultaneously, the resulting displacement at that point in the medium is the sum of the displacements due to each individual wave. The waves interfere with each other.

Types of interference. If the waves are “in phase”, that is crests and troughs are aligned, the amplitude is increased. This is called constructive interference. If the waves are “out of phase”, that is crests and troughs are completely misaligned, the amplitude is decreased and can even be zero. This is called destructive interference.

Constructive Interference crests aligned with crest waves are “in phase”

Constructive Interference

Destructive Interference crests aligned with troughs waves are “out of phase”

Destructive Interference

Sample Problem: Draw the waveform from its two components.

Sample Problem: Draw the waveform from its two components.

Two-dimensional waves exhibit interference as well Two-dimensional waves exhibit interference as well. This is an example of an interference pattern.b

Here is another example of an interference pattern, this one from two sources. If the sources are in phase, points where the distance to the sources differs by an equal number of wavelengths will interfere constructively; in between the interference will be destructive.

Standing Wave A standing wave is a wave which is reflected back and forth between fixed ends (of a string or pipe, for example). Reflection may be fixed or open-ended. Superposition of the wave upon itself results in a pattern of constructive and destructive interference and an enhanced wave

A standing wave is fixed in location, but oscillates with time A standing wave is fixed in location, but oscillates with time. These waves are found on strings with both ends fixed, such as in a musical instrument, and also in vibrating columns of air.

The fundamental, or lowest, frequency on a fixed string has a wavelength twice the length of the string. Higher frequencies are called harmonics.

There must be an integral number of half-wavelengths on the string; this means that only certain frequencies are possible. Points on the string which never move are called nodes; those which have the maximum movement are called antinodes.

In a piano, the strings vary in both length and density In a piano, the strings vary in both length and density. This gives the sound box of a grand piano its characteristic shape. Once the length and material of the string is decided, individual strings may be tuned to the exact desired frequencies by changing the tension. Musical instruments are usually designed so that the variation in tension between the different strings is small; this helps prevent warping and other damage.

Standing waves can also be excited in columns of air, such as soda bottles, woodwind instruments, or organ pipes. As indicated in the drawing, one end is a node (N), and the other is an antinode (A).

Fixed-end standing waves (violin string) 1st harmonic 2nd harmonic Animation available at: 3rd harmonic http://id.mind.net/~zona/mstm/physics/waves/standingWaves/standingWaves1/StandingWaves1.html

Fixed-end standing waves (violin string) Fundamental First harmonic  = 2L First Overtone Second harmonic  = L Second Overtone Third harmonic  = 2L/3

If the tube is open at both ends, both ends are antinodes, and the sequence of harmonics is the same as that on a string.

Open-end standing waves (organ pipes) L Fundamental First harmonic  = 2L First Overtone Second harmonic  = L 2nd Overtone Third harmonic  = 2L/3

In this case, the fundamental wavelength is four times the length of the pipe, and only odd-numbered harmonics appear.

Mixed standing waves (some organ pipes) L First harmonic  = 4L Second harmonic  = (4/3)L Third harmonic  = (4/5)L

Sample Problem How long do you need to make an organ pipe that produces a fundamental frequency of middle C (256 Hz)? The speed of the sound in air is 340 m/s. A) Draw the standing wave for the first harmonic B) Calculate the pipe length. C) What is the wavelength and frequency of the 2nd harmonic? Draw the standing wave

Sample Problem How long do you need to make an organ pipe whose fundamental frequency is a middle C (256 Hz)? The pipe is closed on one end, and the speed of sound in air is 340 m/s. A) Draw the situation. B) Calculate the pipe length. C) What is the wavelength and frequency of the 2nd harmonic?

Resonance Resonance occurs when a vibration from one oscillator occurs at a natural frequency for another oscillator. The first oscillator will cause the second to vibrate. Demonstration.

Beats “Beats is the word physicists use to describe the characteristic loud-soft pattern that characterizes two nearly (but not exactly) matched frequencies. Musicians call this “being out of tune”. Let’s hear (and see) a demo of this phenomenon.

Beats are an interference pattern in time, rather than in space Beats are an interference pattern in time, rather than in space. If two sounds are very close in frequency, their sum also has a periodic time dependence, although with a much lower frequency.

What word best describes this to physicists? Amplitude Answer: beats

What word best describes this to musicians? Amplitude Answer: bad intonation (being out of tune)

Diffraction Diffraction is defined as the bending of a wave around a barrier. Diffraction of waves combined with interference of the diffracted waves causes “diffraction patterns”. Let’s look at the diffraction phenomenon using a “ripple tank”. http://www.falstad.com/ripple/

Double-slit or multi-slit diffraction 

Double slit diffraction nl = d sinq n: bright band number (n = 0 for central) l: wavelength (m) d: space between slits (m) q: angle defined by central band, slit, and band n This also works for diffraction gratings consisting of many, many slits that allow the light to pass through. Each slit acts as a separate light source.

Single slit diffraction nl = s sinq n: dark band number l: wavelength (m) s: slit width (m) q: angle defined by central band, slit, and dark band

Sample Problem Light of wavelength 360 nm is passed through a diffraction grating that has 10,000 slits per cm. If the screen is 2.0 m from the grating, how far from the central bright band is the first order bright band?

Sample Problem Light of wavelength 560 nm is passed through two slits. It is found that, on a screen 1.0 m from the slits, a bright spot is formed at x = 0, and another is formed at x = 0.03 m? What is the spacing between the slits?

Sample Problem Light is passed through a single slit of width 2.1 x 10-6 m. How far from the central bright band do the first and second order dark bands appear if the screen is 3.0 meters away from the slit?