Physics 111 Lecture 31 (Walker: 14.7-8) Wave Superposition Wave Interference Physics of Musical Instruments Temperature Superposition and Interference Waves of small amplitude traveling through the same medium combine, or superpose, by simple addition. Neither wave affects the other, but they both affect the total disturbance. Nov. 30, 2009 Lecture 31 1/28 Lecture 31 2/28 Superposition and Interference If two pulses combine to give a larger pulse, have constructive interference (left). If pulses combine to give smaller pulse, have destructive interference (right). Interference; Principle of Superposition These figures show the sum of two waves. In (a) they add constructively; (b) they add destructively; (c) they add partially destructively. Lecture 31 3/28 Lecture 31 4/28
Wave Phase The phase of a wave (at a given point) is the part of its cycle the wave is in. Often use the corresponding angle or rotation: Phase Phase Angle ( ) Phase angle (rad) x=+a Crest 0 0 x=0 after crest 90 π/2 x= -A Trough 180 π x=0 after trough 270 3π/2 x= +A rest 360 or 0 2π or 0 Two waves with same phase at given place and time give constructive interference Wave Fronts Two- or three-dimensional waves can be represented by wave fronts, which are curves of surfaces where all the waves have the same phase. Wave fronts are usually crests. Lines perpendicular to wave fronts called rays; they point in direction of propagation of the wave. Lecture 31 5/28 Lecture 31 6/28 Interference of Waves from 2 Sources If sources in phase, points where distance to sources differs by an equal number of wavelengths (A or C) will interfere constructively; in between (B), interference will be destructive. Interference of Sound Waves Sound waves interfere in space in same way that other waves do. Lecture 31 7/28
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. Standing waves occur when reflections from the ends interfere constructively. Lecture 31 9/28 ; Resonance When both ends of a string are fixed, only waves which are motionless at ends of string can persist. These points of zero disturbance are called nodes. In between the nodes are antinodes, where disturbance amplitude is maximum. The different standing wave patterns are Lecture modes. 31 10/28 ; Resonance Frequencies of standing waves on a particular string are called resonant frequencies. The lowest resonant frequency f 1 is referred to as the fundamental and the higher resonant frequencies f n = nf 1 as the n th harmonics. Lecture 31 11/28 The fundamental, or lowest, frequency on a fixed string has wavelength twice the length of string. Higher frequency harmonics have shorter wavelengths. Lecture 31 12/28
of Guitar Strings Thin string In order for different strings to have different fundamental frequencies, they must differ in length and/or mass/unit length. A guitar has strings that are all the same length, but the mass/unit length varies. We change the effective length of a string by pressing it against the Thick string fingerboard. Lecture 31 13/28 - Fundamental Frequency 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. Lecture 31 14/28 Example Guitar string of length 0.75 m has m/l = 2x10-4 kg/m. What tension is needed to get a 440 Hz fundamental? f 1 = 440 Hz = v w /2L; need v w = 2(0.75m)(440 Hz)=660m/s v w = F m / L F = (m/l)(660m/s) 2 = (2x10-4 kg/m)(4.36x10 5 m 2 /s 2 ) = 87N Standing waves can also be excited in columns of air, such as soda bottles, woodwind instruments, or organ pipes. As indicated in the drawing, the closed end is a node (N), and the open end is an antinode (A). Lecture 31 15/28 Lecture 31 16/28
: Open/Closed Pipe In this case, the fundamental wavelength is four times the length of the pipe, and only oddnumbered harmonics appear. Open/Closed Tube Resonances Standing waves in a half-closed column of air: Lecture 31 17/28 Lecture 31 18/28 : Open/Open Pipe 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. All harmonics appear. Open/Open Tube Resonances Standing waves in a fully open column of air: Lecture 31 19/28 Note that these results are the same as for the fixed string. f 2 = 2f 1 (Octave above fundamental) f 3 = 3f 1 (Octave + fifth above fundamental) f 4 = 4f 1 (Two octaves above fundamental) - Demo Lecture 31 20/28
Temperature (T) Temperature (T) is a measure of how hot or 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 Random kinetic energies of atoms at different temperatures. Low T liquid. High T liquid. Lecture 31 22/28 The Three Basic Phases of Matter Solid Liquid Gas Other Temperature Scales The Celsius scale: 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 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: Temperature Scales Compared Lecture 31 25/28 Lecture 31 26/28 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. Lecture 31 27/28