Superposition and Interference Physics 111 Waves of small amplitude traveling through the Lecture 31 (Walker: 14.7-8) same medium combine, or superpose, by simple addition. Neither wave affects the other, Wave Superposition but they both affect the total disturbance. Wave Interference Standing Waves Physics of Musical Instruments Temperature

Nov. 30, 2009

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Superposition and Interference Interference; Principle of Superposition If two pulses combine to give a larger pulse, These figures show the sum of two waves. In have constructive interference (left). If pulses (a) they add constructively; combine to give smaller pulse, have destructive (b) they add destructively; interference (right). (c) they add partially destructively.

Lecture 31 3/28 Lecture 31 4/28 Wave Fronts Wave Phase Two- or three-dimensional waves can be •The phase of a wave (at a given point) is the represented by wave fronts, which are curves part of its cycle the wave is in. Often use the of surfaces where all the waves have the same corresponding angle or rotation: phase. Wave fronts are usually crests. Phase Phase Angle (°) Phase angle (rad) x=+A Crest 0 0 Lines perpendicular x=0 after crest 90° π/2 to wave fronts called x= -A Trough 180° π rays; they point in direction of x=0 after trough 270° 3π/2 propagation of the π x= +A rest 360° or 0° 2 or 0 wave. • Two waves with same phase at given place and time give constructive interference

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Interference of Waves from 2 Sources If sources “in phase”, points where distance to Interference sources differs by an equal number of of Sound Waves wavelengths (A or C) will interfere constructively; in between (B), interference will be destructive. Sound waves interfere in space in same way that other waves do.

Lecture 31 7/28 Standing Waves Standing Waves; Resonance A standing wave is When both ends of a fixed in location, but string are fixed, only oscillates with time. waves which are These waves are found motionless at ends of on strings with both string can persist. These ends fixed, such as in points of zero a musical instrument, disturbance are called and also in vibrating nodes. In between the columns of air. nodes are antinodes, Standing waves occur where disturbance when reflections from amplitude is maximum. the ends interfere The different standing constructively. wave patterns are Lecture 31 9/28 Lecturemodes. 31 10/28

Standing Waves Standing Waves; Resonance The fundamental, or lowest, frequency on a fixed Frequencies of string has wavelength twice the length of string. standing waves on a Higher frequency harmonics have shorter particular string are wavelengths. called resonant frequencies. The lowest resonant

frequency f1 is referred to as the fundamental and the higher

resonant frequencies fn th = nf1 as the n harmonics.

Lecture 31 11/28 Lecture 31 12/28 Standing Waves of Guitar Strings Standing Waves - Fundamental Frequency Thin string In order for different In a piano, the strings vary in both length and strings to have different density. This gives the sound box of a grand fundamental frequencies, piano its characteristic shape. they must differ in length and/or mass/unit length. Once the length and material of the string is decided, individual strings may be tuned to the A guitar has strings that exact desired frequencies by changing the are all the same length, but tension. the mass/unit length varies. Musical instruments are usually designed so We change the effective that the variation in tension between the length of a string by different strings is small; this helps prevent pressing it against the Thick string warping and other damage. fingerboard. Lecture 31 13/28 Lecture 31 14/28

Standing Waves Example • Guitar string of length 0.75 m has m/L = 2x10-4 kg/m. What tension is needed to get a 440 Hz Standing waves can also be fundamental? excited in columns of air, such as soda bottles, f1 = 440 Hz = vw/2L; need vw = 2(0.75m)(440 Hz)=660m/s woodwind instruments, or organ pipes. F v = As indicated in the drawing, w m / L the closed end is a node (N), F = (m/L)(660m/s)2 = (2x10-4kg/m)(4.36x105 and the open end is an m2/s2) antinode (A). = 87N

Lecture 31 15/28 Lecture 31 16/28 Standing Waves: Open/Closed Pipe In this case, the fundamental wavelength is four Open/Closed Tube Resonances times the length of the pipe, and only odd- numbered harmonics appear. • Standing waves in a half-closed column of air:

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Standing Waves: Open/Open Pipe Open/Open Tube Resonances If the tube is open at both ends, both ends are • Standing waves in a fully open column of antinodes, and the sequence of harmonics is the air: same as that on a string. All harmonics appear.

Note that these results are the same as for the fixed string.

f2 = 2f1 (Octave above fundamental)

f3 = 3f1 (Octave + fifth above fundamental)

f4 = 4f1 (Two octaves above fundamental) - Lecture 31 19/28 Lecture 31 20/28 Demo Temperature (T) Random kinetic energies of atoms at different • Temperature (T) is a measure of how “hot” or temperatures. “cold” something is • Temperature is a measure of the random kinetic energy of each particle in an object. – The greater the motion/vibration the greater the T – The smaller the motion/vibration the lower the T • SI Unit: kelvin (K) – E.g., room temperature is about 295K • Kelvin is the natural temperature scale – 0 K is lowest possible temperature – No negative temperatures – Random internal KE is zero at T = 0 K Lecture 31 21/28 Low T liquid. Lecture 31High T liquid. 22/28

The Three Basic Phases of Matter Other Temperature Scales The Celsius scale: Solid Liquid Gas Water freezes at 0° Celsius. Water boils at 100° Celsius. The Fahrenheit scale: Water freezes at 32° Fahrenheit . Water boils at 212° Fahrenheit .

Sequence of increasing molecule motion (and energy)

Lecture 31 23/28 Lecture 31 24/28 Kelvin Temperature Scale Temperature Scales Compared The Kelvin scale has the same step size (size of one degree) as the Celsius scale, but the Kelvin scale has its zero at absolute zero. Conversion between a Celsius temperature and a Kelvin temperature:

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End of Lecture 31

• For Wednesday, Dec. 2, read Walker 16.1-4.

• Homework Assignment 14c is due at 11:00 PM on Thursday, Dec. 3.

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