Massachusetts Institute of Technology Physics 8 03 Fall 2004 Problem Set 5 Due Friday October 22 2004 at 4 PM Reading Assignment French pages 161 178 189 196 Beke Barrett pages 165 178 To do 5 1 and 5 6 you need access to a piano Problem 5 1 Piano galore For this problem you need a piano Most of the dormitories and fraternity houses have at least one and there are several in the student center Below is a picture labelling the keys to which we will refer Let 256 Hz equal one unit of frequency 1 The harmonics of this note are then 2 3 4 etc Middle C on the piano is C256 if the piano is tuned that way We call it C4 The subscript refers to the octave it increases by one at each higher octave of C Thus the fundamental of C3 called a subharmonic of C4 is 128 Hz 12 To avoid confusion I will always refer to the fundamental as the rst harmonic Thus the rst harmonic of C3 is 128 Hz and the second harmonic is 256 Hz Suppose that piano strings behave ideally Then the mode frequencies of a given string would consist of the harmonic sequence 1 2 1 3 1 etc The names and frequencies of the rst 16 harmonics of string C4 and also its rst two subharmonics 1 3 and 1 2 would be as follows we underline C4 and its octaves N ames F2 C3 C4 C5 G5 C6 E6 G6 Bb6 C7 D7 E7 F 7 G7 G 7 Bb7 B7 C8 13 12 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 We will start this experiment by determining whether your piano belongs in a bar or a concert hall Strike the notes from C3 up one at the time and listen for beats Many keys not all activate two or three identical strings simultaneously A Steinway grand piano has a total of 216 strings 88 keys If the three or two strings which make up one note are not properly tuned you will hear beats Suppose you hit C5 and you hear maximum sound at 1 second intervals a What then is the di erence in tension between the strings of C5 The tension in each string in the piano is about 250 Newtons b What is the approximate total force on the frame of the piano that holds all the strings 1 Steadily hold down various keys one at the time so as to lift their dampers without sounding the notes Then while you are still holding down a key of your choice strike the G5 sharply hold it for a few seconds and release it you still hold the other key down Listen carefully You clearly hear sound in case you had chosen C4 or G6 c What frequencies do you hear in these two cases d Which other notes might be excited by G5 G5 also produces higher harmonics Verify your predictions To demonstrate the presence of higher harmonics we will make you hear the 6th harmonic of C3 Hold the C3 key down keep your nger on it strike G5 hold it for a few seconds and let it go Now listen carefully to the sound produced by your C3 strings This sound is the 6th harmonic of C3 Clearly if your piano is out of tune things may sound quite di erent But even if it is in tune you may notice by listening carefully that the G5 does not sound exactly like the 6th harmonic of the C3 It seems that our piano strings do not behave as ideally as we earlier assumed e How would you explain that The lowest two notes on the piano are A0 27 5 and A 0 29 1 Their beat frequency is thus 1 6 Hz which is easily detectable Hit both notes together gently Once you think you hear beats let one key up but not the other f Do the beats go away 2 Problem 5 2 Holes in woodwind instruments A simpli ed ute as shown in the gure is open at D There is also a large opening at A near the mouth piece and there are two holes at B and C AB BD and BC CD The distance AD 37 cm The speed of sound is 340 m sec What frequency do you expect to hear when you blow and when you a hold both holes at B and C closed b hold only hole C closed c hold only hole B closed d do not close either one of the holes B or C Keep in mind that wherever the air inside the ute is in open contact with the air outside no pressure can build up pressure nodes Now read the pages 204 and 205 of Horns Strings and Harmony by Benade and reconsider your answers If you play any woodwind instrument we recommend that you read Chapter IX of Horns Strings and Harmony by Benade Very enjoyable Problem 5 3 Plucked string Do Problem 6 12 from French A P Vibrations and Waves New York N Y W W Norton and Company January 1 1971 ISBN 0393099369 A string of length L which is clamped at both ends and has a tension T is pulled aside a distance h at its center and released a What is the energy of the subsequent oscillations b How often will the shape shown in the gure reappear Assume that the tension remains unchanged by the small increase of length caused by the transverse displacements Hint In part a consider the work done against the tension in giving the string its initial deformation Problem 5 4 Fourier analysis a Find the Fourier series of the function shown in the gure of problem 5 3 b If the release takes place at t 0 What will the string look like f x t at time t c Make sketches of the string at t T 1 8 T 1 4 and at T 1 2 T1 is the period of the lowest frequency rst harmonic With Matlab though not required you can do a great job 3 Text removed due to copyright reasons Please see Benade Arthur H Horns Strings and Harmony NY Dover Publications 1992 ISBN 0486273318 Problem 5 5 Fourier series Do Problem 6 14 from French A P Vibrations and Waves New York N Y W W Norton and Company January 1 1971 ISBN 0393099369 Find the Fourier series for the following functions 0 x L a y x Ax L x b y x A sin x L A sin 2 x L c y x 0 0 x L 2 L 2 x L Problem 5 6 Pianos can talk back Revisit your piano it does not have to be in tune Open the cover so that you can see the strings Hold down the damper peda l Shout heyeyeyey hold it for a few seconds into the region of the strings and sounding board If you have a grand piano that would …