From: Subject: No Title Date: Wed, 4 Mar 2009 15:37:05 -0600 MIME-Version: 1.0 Content-Type: multipart/related; type="text/html"; boundary="----=_NextPart_000_0000_01C99CDF.12652B70" X-MimeOLE: Produced By Microsoft MimeOLE V6.00.2900.3350 This is a multi-part message in MIME format. ------=_NextPart_000_0000_01C99CDF.12652B70 Content-Type: text/html; charset="Windows-1252" Content-Transfer-Encoding: quoted-printable Content-Location: http://www.physics.purdue.edu/people/faculty/ng/tmp/book_strings/book_strings.html No Title THE PHYSICS OF VIBRATING STRINGS=20

Vibrating strings are key components of many musical instruments, = such as=20 guitars, violins, and pianos. Hence, if we want to appreciate the = physics of=20 these instruments, we must begin with the physics of a vibrating string. = To a=20 first approximation, all strings are created equal, as all are described = by the=20 wave equation=20

  3Dequation7=20=20

where y is the displacement of the string from its equilibrium = position, which is a function of position along the string, x, = and time,=20 t. But since guitars sound different from violins, and both sound = different from pianos, we know that there must be more to the story than = (1).=20 This section of the course begins with a discussion of the wave equation = and how=20 to attack it numerically. This brings out some nice lessons in numerical = stability. It turns out that the Courant condition, which concerns how = to best=20 choose the temporal and spatial grid sizes in a finite difference = treatment, can=20 be easily understood from the physics of the wave equation. We then = consider=20 waves moving on strings in several different contexts, including plucked = guitar=20 strings, and piano strings as they are struck by a (piano) ``hammer.''=20

To construct a numerical scheme for attacking the wave equation, we=20 discretize x and t in units of 3Dtex2html_wrap_inline114=20 and 3Dtex2html_wrap_inline116=20 , so that 3Dtex2html_wrap_inline118=20 , and write the derivatives in finite = difference form=20

  3Dequation15=20=20

The 3Dtex2html_wrap_inline120=20 symbol is used here to emphasize that this is = only an=20 approximation; there are correction terms which can be important. = Rearranging=20 (2)=20 gives=20

  3Dequation23=20=20

where 3Dtex2html_wrap_inline122=20 . Thus, if we know the string configuration = at time=20 steps n and n-1, we can calculate the configuration at = step=20 n+1. The initial conditions will depend on how the string is = excited, and=20 we will now consider two cases.=20

Some results obtained with the algorithm (3)=20 are shown in Fig. 1.=20 Here the initial string profile (shown at the top of the figure) was = chosen to=20 be triangular, with the string at rest, as would be appropriate for a = plucked=20 guitar string. The kink associated with this pluck is seen to split into = two=20 separate kinks, one propagating to the left and one to the right, which = reflect=20 from the ends of the string. In our simulation we kept the ends of the = string=20 fixed, so the reflections are inverted.=20

   3Dfigure28=20
Figure 1: Waves = propagating on=20 a string with fixed ends. The string had a length of 0.65 m with=20 c=3D200 m/s, as would be appropriate for a guitar string. The = simulation used=20 the values 3Dtex2html_wrap_inline130=20 , and 3Dtex2html_wrap_inline102=20 . The initial string profile is given at the = top, and=20 successive traces (moving from top to bottom) show the string at = progressively=20 later times. For clarity, each trace is shifted downwards from the = previous one.=20

The astute reader will recognize that (3)=20 bears a strong resemblance to a no-frills Euler algorithm. While the = Euler=20 method is a very simple and useful approach for many problems, it is = known to=20 fail miserably for some situations that involve oscillatory motion, such = as a=20 simple harmonic oscillator or planetary motion. Wave motion is also a = type of=20 oscillation, so one might have expected that an Euler approach would = fail here=20 too. However, the results in Fig. 1=20 suggest that it works quite nicely. Understanding why it works so well = provides=20 a useful lesson in numerical methods, and (as was noted above) leads to = a=20 derivation of the Courant condition for the stability of the algorithm.=20

Once we have obtained the solution for the string displacement as a = function=20 of time, as in the above figure, we can then discuss the sound which is = produced=20 and its associated spectrum. This brings us to several important issues = and=20 techniques, including spectrum analysis, the Fourier transform, and the = Fast=20 Fourier Transform.=20

We next proceed to the case of a piano. A piano string is set into = motion by=20 the blow from a ``hammer,'' which is actually just a wooden mallet = covered with=20 a compressible layer of felt. While our first urge might be to treat the = felt as=20 a simple spring describable by Hooke's law, it turns out that life is = not this=20 simple. Experiments have found that the restoring force for felt depends = on the=20 rate at which it is compressed, and the preceding compression history. = Because=20 of this hysteresis, a full, general treatment of the piano hammer = problem has=20 not yet been worked out. However, it is not hard to give a fairly good=20 approximate approach, which is basically equivalent to the best that has = been=20 done to date in the research literature.=20

We can add the hammer to the simulation by treating it as a mass = which=20 strikes our numerical string at a certain location, with the = hammer-string=20 interaction force given by=20

  3Dequation36=20=20

where z is the amount which the felt is compressed, the = exponent describes how the stiffness varies with = compression, and=20 K is a force constant. Here g(x) is a function = which=20 describes how the hammer force is distributed along the string. This can = be=20 accommodated by letting g(x) be a gaussian function = centered at=20 point where the middle of the hammer meets the string, with a full width = which=20 corresponds to the width of the hammer. (Note that (4) has been obtained = from=20 direct experimental measurements.) In a typical case, the hammer meets = the=20 string a distance 1/8 from one end, and has a width of about 1 cm. = The=20 strategy then is to begin at t=3D0 with the string in its = undisplaced state=20 (y=3D0 everywhere) and give the hammer some initial velocity. = When the two=20 meet, the hammer felt is compressed by an amount equal to the difference = between=20 the hammer position and the string position at the strike point, = resulting in an=20 interaction force which acts on both the string and the hammer. This = causes the=20 string to move, and the hammer to eventually rebound. Some results from = such a=20 simulation are given in Fig. 2,=20 which shows the string profile during the initial hammer-string impact=20 (0.3 ms), just after the hammer falls away from the string = (2.1 ms),=20 and while the string is vibrating freely (3.0 ms). Here we have = used string=20 and hammer parameters appropriate for the note middle C on a piano. The=20 calculated hammer-string contact time is 2.1 ms, which is quite = close to=20 typical measured values.=20

   3Dfigure41=20
Figure 2: = Simulation of a piano=20 string. The string and hammer parameters we used are appropriate for a = string=20 near middle C. The string profile is shown at several different times = after=20 impact with the hammer.

These results can also be compared to those for the plucked guitar = string,=20 and differences in the spectra of guitar and piano tones can be = identified and=20 understood. Other topics of interest can also be addressed, including = damping,=20 and the effect of string stiffness.=20


Nick Giordano
Mon Sep 8 10:19:18 EST 1997=20
------=_NextPart_000_0000_01C99CDF.12652B70 Content-Type: image/gif Content-Transfer-Encoding: base64 Content-Location: http://www.physics.purdue.edu/people/faculty/ng/tmp/book_strings/img3.gif R0lGODlh9AErAIAAAAAAAL+/vyH5BAEAAAEALAAAAAD0ASsAAAL+jI+py+0GwJu02vsi3rz7D4bi SJbmiaYZpLagFMDuTNf2jed6Hskxu6v1EDBf8IhMKpfMJvCXMDpDMql0is1qt9yk8dv1gInhsvmM Tm98V3WF7Y7L5/RltX654/f8vt/SI3H3pRcXCPVDeND21+j42MRWNLSIyBgmCUHJUgjp+Ql6o9gg WOk2ylD6FMra6vqicPkku2U10fmaq7u7YLuyiubrgMtbbMw69ovIpHEbRUF8LD3tlzys6VTkTLbd TP0NjhfdC6ykvb18nR7O3n62Ce09RatJvyjvni9Nb1+r1d8BoL6Bj/oJzPbPxUGCDPm0eRjM1LwU 4xpaLBgLn0T+M/AQLryl8aLIRoxKjuzV8aRKfSZTqUy5MiZLdS5FVpSJk9qlncgC+QxJKqdQgjyD 6viJNKlSHB9jKH0KdajULEVrNmw6NSukquQsYtUKthrNrqCgGgyLthjXWGmA2oDz1p7btHQDjmVr Zy6Ur1RgAtLLjR/fuoTX1QM6WMTNc3JuBn5TOPKHhYmpPNuIue1lyNAke17DobIJuJvF5eH8OXXn 01QzG55zdrXq2XfRmRsi6JBrNYcywRNNW3VsO0SIqTJU3FSh48Gbh5ZtLgou3aekK88I2Ll2OsKA Ey9dbrv4T93/CNs9Pr350t6jcwOvPr7Y93sdwrckP799YMwUTa+aFJ5+AvKmUXYEWgHggAp6UAAA Ow== ------=_NextPart_000_0000_01C99CDF.12652B70 Content-Type: image/gif Content-Transfer-Encoding: base64 Content-Location: http://www.physics.purdue.edu/people/faculty/ng/tmp/book_strings/img4.gif R0lGODlhGQAOAIAAAAAAAL+/vyH5BAEAAAEALAAAAAAZAA4AAAIyjA+nywvfYntJ2pDolRXuaXhT JmJKiYSpilQOezLo+jpuHY8gbu8mBQx2QkOMUNgiHgoAOw== ------=_NextPart_000_0000_01C99CDF.12652B70 Content-Type: image/gif Content-Transfer-Encoding: base64 Content-Location: http://www.physics.purdue.edu/people/faculty/ng/tmp/book_strings/img5.gif R0lGODlhFAAOAIAAAAAAAL+/vyH5BAEAAAEALAAAAAAUAA4AAAIrjA+ny+AMlUsxUGRfvIbqBmbN dHGftJgcxkrrO6YodJqbnZH6hCX7j6AUAAA7 ------=_NextPart_000_0000_01C99CDF.12652B70 Content-Type: image/gif Content-Transfer-Encoding: base64 Content-Location: http://www.physics.purdue.edu/people/faculty/ng/tmp/book_strings/img6.gif R0lGODlhDQEgAIAAAAAAAL+/vyH5BAEAAAEALAAAAAANASAAAAL+jI8ZoO3Onpz0xYqz3ufyT3mg ImLliJzoWKqGu5obHGttTZM1tKN5Agj2ZkLbEPRjGXG6IyfZKTpD0sZtusRCoSErVmYMcncMscX7 7aYX4JhqnHLPXnD54s1D2tMXqcclZGYhdlL3QtURh4ZWBVTWSBIhyNhCuPiEeJh4OUgHyNan2BQq 8alXyqapGleY0poK6+V3CRkLpIRKupqY87ipU/Zru7l1hgQTvFqIF8Uo80f7ymmDDAsdWZG8Syw8 LKqWd4y6y3zrGJ6t+01njv5UTe7euc0db5w9Hkm4z38vUr7ujrx83iq1m+aJn0JTtgASNKXOWzeC TcT5OzhRYi36LtcCsssI0gS8hgjRQfwY0toESSXTrURJTxs9gRV71ZtZhWXFdy9pZpR5saCmK4qK /LnhMIsxoq4cLRQkSSdMZbGeWjqUNIPNYcvOHQT6a9LPqF6bDmxTClJXV1b7kDW7Lmdbt/cwUWHa DG63WiR3ogVn9+/ML4b2BMbnt+4cjxIp9lyTlc8axVrOJtaq0rJmjEciMq48+XIPzylL++M7uKVq HKgbOym8Ana6kZQxKx0dGnHu3aZzy64NmAnvx8Mn/x5ynHPx5cybO38OPbr06dSrW7+OPbv27dy7 e/8OPrz48eTLmz+PPr369ezbu38PP778+fTr229fAAA7 ------=_NextPart_000_0000_01C99CDF.12652B70 Content-Type: image/gif Content-Transfer-Encoding: base64 Content-Location: http://www.physics.purdue.edu/people/faculty/ng/tmp/book_strings/img7.gif R0lGODlh9QIyAIAAAAAAAL+/vyH5BAEAAAEALAAAAAD1AjIAAAL+jI+py+0Po5y02ouz3rz7DoTA R5bmiabqaoojC8fyTNf2jatvzvf+D9TsgiCiY/hDGhHKJaO5gTofUkiVeLVlgVtnlyL9ThdKcc28 Ko/RPjWHvYS7vcF5fNywZ6DwtZ+c9KfQdyEipIMnwUYIRsL48ajoOGO4hziYaBF5UonZ5hXyJLPT tymaSVXI0ln0mRYKOXpQeplgGkgjFqZZcsuZBzMEW+ELiApMfDjhi8YsO7nKtBhteyyZ23iaHSvo +cqkmmatvax8zb3tUYyMriMcXjs7HjD8bkz1wnqU7n2+TAru3pF8w6CRi8EHoDyBeQjumhfQlkID fPxZUeSQncX+CPUiVmuBkWLBj5SeTKR30KTIhwuPjTjZ5OXFjhfLaeSYDCagLC9PqttHKSZKih7l 8aTnsyXEoeB0YtqiSxLNfjgbOa1WrEtPpUVXCb3adGZSol3xyCSLVhpOe6lspmxbDuwsqEzr/rxJ 7anSmHTZklyqRq7IjUBnlrVbM+5eMuu01q1YcufiQX0PI1knLi1izQ254i3MMLHby28Ze25VGsXI iKRDq+R82ppQzZBF2x4YezPoc62p3uU4FnaK1S17/8Ut3C7mWsYP90voIvqVKtOlR6+qPPXfMNaJ H/9OuTsx6lxrw9WNHtXs7K6Rdvee+/lnae/p0k5d/7ZGlgP+xf8r3Fx8HvGXHh33+bYfYfMlp502 ARY44C8LggQgeF/4BR4O8L3l03r6TbgdiAjC9SBmFzq33IjFWegWeg+aNVhZZ5HIYI0Z2ngjMiN9 hV0+vew2XGwEsWjfUB7y4GOLC+0o2YcNsgabGc3EeNqM7dC4WYpQGsOkXokleZyWEupBZZlpWSmi cxDmeAqZSEkkX1ZASigcmFVeZgeaAkYWlSdubqimb6u9Yx52Nv3ZGGVxPomabnZGieeiDImZJl+I whlmkfjosyebJl2qVqaOuMBoFNI1BOpcVHlHaTJKzpXqef0VNCimKpYqUawgkKqqqO2Net2nCO6o B6ueqvb+apQKwhMFLxgcmSY2RkCL660a8sOiJc0+6yyzncbjKha/HptSqxYS5+aa6hpqTrLsfmuu uzkUGu24N0g5HyGP0BJuv4Hm1S0XpfL7L50rfhTcmvG+S6GLDM8rcHKaYtvDlFvam1nAyEoMsbcd O+ykrDj+uJ0bnP678MO9bHiUuFywHLLIVnAaiZ5Anvytwf6qBvO92ibR85yEpVwGhuse7QouLkFk oiZDFhzeSn1qnEjKQk9h9RsbW9vwnFNXPe3SgHXdYiWA3kMw1ViHrd5SSKu8ME/Uuk133Xbr7J7F Wt/Nd99+Uwxue/UNTnjhhh+OeOKKL854444/DnnkkhP+feXkll+Oeeaab85555NDfaWiGP9Neumm t7vt6aqvzrS0r43OeuyyY0Ez4B7Pjnvu10amXda6/w58r10uq3bwxh//G0LszI18887/7J6Z5+n7 fPXWy/v67Y/Bfn3xxZ/dfZC5NoMz9uHvbZCt54eOD+pSn+w77uRh++j6wMf/+/we128/9Mp73z/b RY8f/Avg7PCXO+qA7yAFNKAAt/ZAB8oMVjerlgT7hkD59cd/Obsg1/A2QQ+GECXlo1YGRfgMFO6s KQt0zQlZZzV8qRB7TxtRumZIuhfCECgtFBQOIwjBEf5QQDWEFbpAN0S36VB105GequCHxB/6bmJJ HJfmMJhHqCqabomn09+b/OSZLGqxXmNkGzmO6KDylXFsa7zVFbkEIDVqkYttdFLRckTHOiZPjwT6 orXyGDtA6vFXd1STIAfJwTb20VhXQ+QhESmk7HkRknd7JN/o5cQmUvKDm/xYhmilrE5WkpKY9OPF ojhHUa4tPPkhSilVKRtSMmZwZIHFKx0Jyzuw8j294l4udYnLXfrni/Cx5BZ/aUaSIbN1sswYLI0J yaYtk5nRBJgqoVlNZU5zHNi0W4q6mcNtRix94mzbJr+5THAO8hbqFCevgkm2Tr6znBXbIz3vOU72 4ZMIBQAAOw== ------=_NextPart_000_0000_01C99CDF.12652B70 Content-Type: image/gif Content-Transfer-Encoding: base64 Content-Location: http://www.physics.purdue.edu/people/faculty/ng/tmp/book_strings/img8.gif R0lGODlhDAAKAIAAAAAAAL+/vyH5BAEAAAEALAAAAAAMAAoAAAIXDA6hi3rrEnuGQiSh3jzaKX3V RTUaFhQAOw== ------=_NextPart_000_0000_01C99CDF.12652B70 Content-Type: image/gif Content-Transfer-Encoding: base64 Content-Location: http://www.physics.purdue.edu/people/faculty/ng/tmp/book_strings/img9.gif R0lGODlhyQIYAIAAAAAAAL+/vyH5BAEAAAEALAAAAADJAhgAAAL+jI+py+0Po5y02usAwLz7D4bi p43miabqyrbuC8dytAX1jOf6Wt/7DwwKh8SicdY7KpdIg48JjUqnjCfTWtRgib1SZ/vNgY9jVVlb KXN16qEVLW23Fe+rPeqbV+kgvah+F3NmAajk98e3hHUoNnHIKLgn5rWwuIYA6cQTGZiQiaH2+PJZ KUNKceqZ2OkEp+poSPnapFmKKftTSLgJk2qGe+ArIZfW4tpnagwcNqvYjKsnjDotydlq25xrgwlq PIrXQOyxMQjL4SddC5Ga/tD+1H7ydgM8F79Ojd2bpx8spMXvwr1sZsBVyzDuWD93zIaZsEei10OC WdwVohRwVYb+LhDNHcRhSaOyawyr0OO2sBLHcCEwohRpstXAMaFodBHo6aS/lDw/hlvps+etneoo RrQpkyVRMvhKkVtqqabLpg6VTko5cI+Xcnx0FvUn1avRYmKj4ps60apallt3gYXqlEY+hmjHvmQL t6s8h3XvZkVmEszTsirlZkToUegtgIwzbQn5LZrWvDkN373cjTAdYocRrVWsiRQ/zZXlJkbYeeln vUUX3YOYujWrzZLEbgNN8jZV00FHPcY9Yllv3fCGl9YNs9vX30i/ejbuPOa5WcWBo3NkGzNu28zT Nkf+bDbtRNyhU87ZGGBJ8ymEg4/+M73k0y+rJ9cneLN8xOD+u69/v1h6u8Ekn3rs8aeOfUYVqNB9 +K3GoHDl9cTggAjmp1pFFtkyoV3h8QaigyzQBNxzIdZH0XzwAYjTciXGBph5HU2HonYqnqgWhv9Z pWCGLSWmI4v/FJYbSjfZKGSNNdGnnUoV5gNZkwUxWQs0cVGVhIjUWMnaelkGh+CBp1THpSriVMXb l1KuuVOZ3MSjoprnGeFaMCHJCSA0ujxlYZgeohBblN7M9Youdh4Hl3/MGFqkDXsmSaOMOBZjZHLu /YlZPW+yJw6jjgI6nl8swoioRBx212CdQ2VzqY9OSmhKY/0w9g2h6FkKj6GwtqRpqI32946BWkK6 In2pEtnN56t56KqomL2WSoKwn7Lqnxx/rZbUR4Ji2+JRdFLhKpKTjhspm+HucOaIQYnmrVvJrquN cuea21638M3YbkMagssVpsRyK6+/1wYMMJj3sTtOuX5iO/COhOJrzTTZzVssSM6Am2e46doqb5AF W3wJcR/z4q63Hidjb8UUT+lRg+Y2TClTGD926ZIpf1Hzu+ju6yiJjRCckM8g37wxylSOHDIZrbqx 9MIxmsovnU2vnDTUU+QKc8JocmwIxl7fnK/VBn09rBtsUJF1vbsVAAA7 ------=_NextPart_000_0000_01C99CDF.12652B70 Content-Type: image/gif Content-Transfer-Encoding: base64 Content-Location: http://www.physics.purdue.edu/people/faculty/ng/tmp/book_strings/img10.gif R0lGODlhZgAeAIAAAAAAAL+/vyH5BAEAAAEALAAAAABmAB4AAALAjI+py+2/AJy0WiYvylH77wDi xyliCabeGTboNqryJbGYe9vz/mSxibsFfpMYMXFKKiG6g++1BAqPsOGo5uNtnFvnaxiUGqjgstWs 5QrH3y8aWZWiyDv3+9zt6ONidU8JGNbHBrdWyHfolzZmiGjXhgGZuOhWWZKVN+m1V7OY+AQYBQYV ERr1xHgXaUrH9sOatDnoCivb6ulph7vLo8v7m+ILPGwhTHzMhKy8zNzs/AwdLT1NXW19jZ2tvc3d jVAAADs= ------=_NextPart_000_0000_01C99CDF.12652B70 Content-Type: image/gif Content-Transfer-Encoding: base64 Content-Location: http://www.physics.purdue.edu/people/faculty/ng/tmp/book_strings/img12.gif R0lGODlhTgEpAYAAAAAAAL+/vyH5BAEAAAEALAAAAABOASkBAAL+jI+py+APo5y02ouz3rz7D4bi KDLmaQLoyrbuq6iBA9f2/cr4zu+6/es1hMTi0KB6GJfMIPPZc+agCCn1ivpprdhuzAt2cVljYjl8 DTqV6Pa57X2nunK4eZ2q2337/oyPpecHdEMzeIg4KEiYuLSYwNYoOfn0CGNJCVlkmNnpeUn3WdjE KWp6+heIKpaGufqaFgp74ipWOosLV0uWu7B7+dsrzKs6jLR3a6wsFEyr3FyYvDwNWiwMzSNNve1s nYvNrM09fuw9C24kTk6Nrvl9qr5u3F713icYKT9Nf8BP5Z8HQo14+l4BBNgETQRIBGM0LPjpICyE MwTSevQQIiX+iav8LcyRsV9IjYc4wms10iEQiiSzyTKFzuKmlClbKnwZMV0+KDRr2jRXCeYdliJX /mxkslMtmch6Ej3qC+dGHDsR1fQJNYpUSYuYZqq6AmzWf1sT1fEKD+PTn0nNvhFrsOdYoAm5XoTb y+ncoHTvNcC77OpaeW39WMH6Te9erX11/c0qeDGgWCWPzAXsa3Dgsm4eS644EDHUwp2jEpYb+jMx ysgscxOINoti1aZZO669LzZmlalpu7N9E/ewj7K7zqZNmrNmR7szG+2tOjnwftdihzZuPDpnssJR Saik9vpi6Xy/TBRtKzv00dvrmk+LPpp4YMsVtU/n2lNzOqj+gbG9b0Z+k1hXEnbrFURegN1ZtZ8i 4dEHUYKMvedWfT70B5I+Ek72m4PxMTgfSBZyN90mArrRYFzq+bfNhoxQiOKIYRzXQoo5NcZhPTHK aEdkB9pToksL8mQjOTYeySNVAFbDUBwfjuOjiECSGIUUHhV5FJLPGbTkah3q9CQuxNkSYnoddZnF idmEeU4+SK6YoVA41qjmSmyeV0aUZkKoH5pq/oJlYA1q+SMZSX5Z3ouIinjojmuWWSOGZvm5aCr0 NRojc3CSyadVlFI3ZEBjEXgHpIZuadinllbq0J0qoohqpAb6NeeQeQb6DK5Vzhoni7cFOYetpGqo KymxGrr+aRx+HpYZpgq5CmahYfGqKpNh/TlasbB2yqmvwJropY5N2qStYQ9Kya17VAYLI6gRlluh qdMeq+C35bR77zrDznPunuiCay+zoebqLErSykbvKDgKjC+/0HZErawHT9EYw6wmVvA9kkYqLyt9 WSzucPAaGfGpE3dTIshIZCwry5/qKXG61wKl8qptuhzIdzJPm6zJ4SZqs7shQ/wwTGMWCTOyJ+fL 09BCP200ziid8ebSzu0MtXs1S11R0aKMybHVDJU878AKbv31yDfvQqi33frLNHhWWqmf12m7SqPS bocdtLENY7rvaQgl7TPcLfNnUc1N2R2R2nr/+3bMxQX+LqTTcffIeN2ZIyx2FXmjvK7iYFBOsrNV 783zWziJnjPXRG5+YcJ3dZy1I7U/RTqUsM/Uc9mQX3x2nZq6LjfxedAuqswSsT7UqI6D2LnnZR7E /IW7D3g95mRzji71Zkdj/PBZ9iu54UBfTtXzRI+3ceF00hxO9kipLzLyf50l1VLyVxg+5rInnyZv WCJ3r9kfg7Y3O6yhDz8LBOD4DFggQ7StV+ZrYPAOBxkIeugwnyvOpYDHjL6NrX+8I2H8GoI08j3O e83SoLlcqDGfnI6C6SENAVsEw8U1g3AeRJ3txvVAE8bvH+2bHMWAk8MZ0Q8+LOlgAmdmmyQiznn7 4+H+E9lFIim+Toi7mh8CHQhCxnDxcUGs2//u9z2tjDEga0yf6yYINxayY4lRM6ETkydHh1GxjV2L 3srsNsYbzpGPV/vdz9SFpz2SxIpprJx36Hi3DH6xdgy8GyGvc8nYncxFi1Okb/54ME4qUYvPymQr TCXK1nkyWzN8WypJQcrRQbJxkbihXl55QirucQytJOPrZGnKGq6SbxUEXdP6CMtglo8tgizKBw/5 Q2d2UZLKRBY0etnIHIESk9QcJniQRx4UzlJz3pzizrxGtVuAjVzjVIoGaZQEaMlhApeJJTAbJRgZ 6KCZWISmRvjpMJe9aQ3lsif/dHke6FAghO18ZDX+UycyPawTUNIrZ0sa6iRx0uyhzbLoIAMYJIzq QqRe5OgIhVcvz+TGoOZEqOWieUUxsbR1JoWcKGd6Ukl+0nM0eCVLARrQDNpviArsZDepSUCOMDKj NUVjOfeTFGyq8qg69Z0xzwcaP3KzqlQtpgVD6FRDWq+pLfSoDyW0VJ6Z9Z94w0OttklDiHKVldTb J6XuCNSIkrVVg8mrVYlZvnUukqRccSEgsyrWye11bPWUoknS+hm/VidjbZGqZAhbUjmFVIVdZSZO KQmwjvrwXZ8lIgk3dEccLjaneUETHAe5WpGUForrcp8vJ9tYKLU0coBVUWz/qMzDItarYNTcb7v+ xlFIbuGMKt3IbOU2WBFWUrZD5c0BjytZiH11QtaNq1qNStdsgTalovXuX6fm0vGF0ZEXqS5ji7dW BHnlpkXk3jTn6lleqiq1hdxqeDvb319qdbjLZGN8BQdVpfK3VfLKLsZ26Z/Kcva8ahXsYJ8LJqzo syyWbW9R5Yth8aVvvGBtLm+/u1PqHrhwaF0wEKOD2evqZL3aPJ5WY3zQ9M6kYdO1rY8pDOLG0kMa dW1Oh4tbwBBHK2fAVS4yzWvfE+txxSV+bxQnWV4p4wm7GOWgdEO7MuYG+LZ1zG2PPJOc1/44yu48 LnJTVcgEuZin7i0llw3bXNTW18OjHaWbcVz+Kbfa68UFRrGWDUbl4ODLp3u2cWIznOhfBbpaBCZz D6Hsxj/PVHS4rGifHU3cQl8U0O/j8ZfJC74B5/OZ/z2T2TptYkvHVNYqxq+rTb1d9hIV06BepoXZ quQw4zrXNS7Vh7PcMlLr8BkoJTF3v6nqrui4Os12tq6he+zuftLBk8rmqavMHzFbmX3BZjOxz62w UkbbzISpNo2V1JQ615q0Na2et9MN3kP3WrVk1cYAKc0cecOVYOq990t77CBxE3rLQjX4wcF8wHUT 7c72vioiCyvwJ3+l3Lnsp6KOSc5Px/qFFKcauLG68Yxjlttr89izUe5OhXs6o0IeCMQvHjX+lT+M 5b6Vz81hmlCJd9zW1kI1zl8l8jE/GtgUPTrCE6Pzsyg7hk8/udNl2rtLs7bVP7c60AUVddlyfE1D HvTLcyPzJkf641c3+kezTW6KwNrlxEr7qHGnrLeudMCl6+voAJ4XLOvLgE0EPG6TLqh3Krq2zMy4 Q5OEd73L1+4ht0tpzH5hvsv4K2eW/Kgdn2/Odx7zD9Q84qS2nLmzFfTJ5BKtSF9PyjfvHPbx/F6O DOkptcb2l53wrltbGd5HttGM2oynhI8cwZeVHUgxfO/xYeSxK/31jE/xmpH8a34NyPm+yWf2wb59 5Ft/3qIuoOhhP/59914p3E//1KF3/ur+p3/E7I4/yOe/4/xqVv74tx6x3k9b99d/Acdv/aN6A6h1 WMdxB4iAQEZL0vdtZ9eAsGRJmcSAE9hbEVRNF4iBGThSEAhS/NeB2GZn5NJ+I7h0Q/cf4oeCZEeB t3eCLQh3d+Fm74ZvMviB0AeCX9d2OEhzNOh+MeiDbsRTNXhEIjiEdgaAJ8GCSeiE18aDTyiF6Md2 U2iFMNd1V6iFElh1W+iF8EaFXyiGDld0EZIlK3iDWGgkZ2iCYIiEr8GGF1VsXagvcbhIc5iFa4iG d+iGAmiGe/hPeDgUJECIhWiIh4iIiaiIi8iIFjCGjwiJkSiJk0iJlWiJ3beEKliEK8X+S4RkHTzn czuYVckgQWtUiuoHdZV2iseHbvsnaBHoimE1D06jBd3Wil/zcLCIi1nTU/tAi/F3SbWIihNhORtm fHHzfYsXjC+ViYoiaHwkjBUFQ9F4jH2zg9SIXKB4eUGDjdS3jFAjit2Yi66HjOMoS8KGdqpofpVm jo+Ejs6UQ9pIcvVATwTTjZ7oJmJ3ifvIj/3oj/8IkAEpkANJkAVpkAeJkAmpkAvJkA3pkA8JkREJ g/4kkY8odRW5hfv0jH2Ujxh5hT1FUPAIktLkkVIIFtRoV+9Ykk54kiGTkre4kv1HZC4pNEYYkyRI k+4ijjeJgiPZIdG4kzyJgZygTi8UaSlGKZQZeYRJqYX4w5RiOE9bWAAAOw== ------=_NextPart_000_0000_01C99CDF.12652B70 Content-Type: image/gif Content-Transfer-Encoding: base64 Content-Location: http://www.physics.purdue.edu/people/faculty/ng/tmp/book_strings/img11.gif R0lGODlhdQAOAIAAAAAAAL+/vyH5BAEAAAEALAAAAAB1AA4AAAKqjA+ny+0Po5x0gluzviny9oGa FI6m0wWpaKwqspRnrGLzjWcu3tm56dq1aEPFTjjrIWu1ZZLICMKMUU8iJHshhL3pr+iFgqViVMq3 SnPBD4773S6HtXS2ncU+R5HOHLlKZXdEEji2h/KFWFimt3gHmPfIBJnoyAf1N4dX1wWyJtn05hbn ONeo2cdJJcpaZJN6cjppWPgJkbbKCteiVLkXBMzls5Vl9HoMUAAAOw== ------=_NextPart_000_0000_01C99CDF.12652B70 Content-Type: image/gif Content-Transfer-Encoding: base64 Content-Location: http://www.physics.purdue.edu/people/faculty/ng/tmp/book_strings/img2.gif R0lGODlhXQAeAIAAAAAAAL+/vyH5BAEAAAEALAAAAABdAB4AAAKujI+py+0PEYhn0ouNzWtDAHJi A3qj+ZQOOpJhq12TyrBwMr+t3Wk2f6v4YECFRVcpFYnDWo71Q3qQwiBJAo1pJb3tNLtSiqlGbJcH xhlNSybutyp3zVxrbX2/quX4qn3fBzjnVxcYcOQypqRn6Mc26KO4GNOWISmm9YKSdnh4SeP5h/W5 ock5WimqmrfaqprqGvshS+tWe2uJq7vL2+v7CxwsPExcbHyMnKy8jFwAADs= ------=_NextPart_000_0000_01C99CDF.12652B70 Content-Type: image/gif Content-Transfer-Encoding: base64 Content-Location: http://www.physics.purdue.edu/people/faculty/ng/tmp/book_strings/img13.gif R0lGODlh9AEVAIAAAAAAAL+/vyH5BAEAAAEALAAAAAD0ARUAAAL+jI+py+0Po5wRSEuz3rz7D4bi SJYU9qAZwLatCcfXWbmHvb7yzvf+DzSpUhxWYhhMFjXIhRHxZGKayqr1is06G9HbphugasdhZg0q rhjS5Lb7DScywGXpMU5mcyFR+mqNFyg4mLfXpHeEMkV4hbjl0Adi5MhYaXn552RxmOOFyTMJOLfp l/ZEebMY5vfZ6voKtVengOrCCusxpUorOus5ulq7pohbbNxKRfx4oQtr+4w6ezrXu6xJStMbfczd nXSozMv8m+o982sqdz38VW7+Dt94bSt+Bhl/j54vq5kavY0voMBcfKiN2zeQXDhrDNE4nLArocSJ H/Qkg2hKxzC1RdhWlTulQuMkStDoQXTnqx4nYCgLVqMIM6YadXfscdnkiWOzOjqm8fSX8sfOlzwj xuqHBpHPEwBlOr1k0SAkk4+UhdSnzQsOclVCcWXnUaq/nlsN3pr6NO27jBq/dtgVMZxcrWKxDGna tEdetXzjANxbM1IkbXA/VgNM4u4Sv30bF/s7Qi7OrDkVgnXbtaFLxo47u/rn+Uvbo9ngIA6NOgjo 1IlLs34Nu97m2JLO0b4dWymCAgA7 ------=_NextPart_000_0000_01C99CDF.12652B70 Content-Type: image/gif Content-Transfer-Encoding: base64 Content-Location: http://www.physics.purdue.edu/people/faculty/ng/tmp/book_strings/img14.gif R0lGODlhPAAbAIAAAAAAAL+/vyH5BAEAAAEALAAAAAA8ABsAAAJ0jI+pywjfDDgz2qtq0OtBDF7c 15HhWToNR1UmGrJqtrnw+Tqeqslt7utEejMJ4/MKJnJHIuWocylbMadRuNyJLCPr9Pa80qg1yRc8 IpfXGzZ45dG648W3HXO+3/P6N70PGCg4SFhoeIiYqLjI2Oj4CBnZUAAAOw== ------=_NextPart_000_0000_01C99CDF.12652B70 Content-Type: image/gif Content-Transfer-Encoding: base64 Content-Location: http://www.physics.purdue.edu/people/faculty/ng/tmp/book_strings/img15.gif R0lGODlh+wDeAIAAAAAAAL+/vyH5BAEAAAEALAAAAAD7AN4AAAL+jI+pCA0Lo5y02ouz3rx72xjP R5bmiabqqozsC8fyTIuJg+f6zvf+DwwKh8Si8Yj8lUI3mGv1VEVTU1T1NL1mqtql8/vqksSf7BjH ALPIHnbHzYFvzGs11C7FU1vhmv8PGDghtyVoeIioh5XI2OioQYgR+UhJxVTZVIe5CYXmwLDzOAnC Wbp0CfmJOFrBasqpauLqpflqS4FqNdvWd+sLsTsYPNf7axww3BqYzHd8ywwCrVzrvBn7J41bXE2Z DYm9zZ14vernjSzOeM5bc76ePmj6nkYNTzMvex9uvy+vX89PSi5b+NwF7OdrnsGDltIpRMgwjr2B ewBGnAOHHLr+i83ucHwjyUamjwcWkrygkdRIGw+YpLSiKN9JlOzolWS58d06kzMj7OKSqaWIghDH 9PRJkdZNdEKTVsxj8egwTyVdfLr68qnWrVLxdcvGU+rRdkUjOh0L1WNPr/64Lpp5Fu3Gt2o/sn3m VibHuHKX6k1rti9NuoAD8hVclXCZlfzucks2NYpjGZOrQT4lGbFmm4v/zt2sOdhlzqAFi1ac43Fp kHGyfiPtrPJqv0ZhG5M9W2Td2LnfuAp7u3dNYlEJCh/+evfxsayAb0Hlevk41sppA2uZeQ1ukqOc I4trlXHTEdEZS6cesrjV1Lqf5CLPVDeu7XCJV5cPzPpn+OT+5dBfm1xhNumQ2Gfy8ceUG/+Jld59 383HH3QhYMXXYecV0qCAlSyIloX4wVQKhxd6h55SF5ZTXCrfzSIiYhaSKAkdqbTookoODrZAJDRu pmBZ0+QXzYmAsAHjjz4ZKaQ52qS4ZJMt7NgbGUVK4F9HSR4ixpRA4vjglepQ6eOWg0F5YhdaWomS h17O4NSZtsVj4JohhmkeUu7JaRyTaNqZI56NXOFmnE+a6eecer6JBpKFLtPniuUpeiQogjq5KKM3 XPMomKRkqmmlgmThjYKcUuqpkm+aeN2hpVI2VKKdpRrqqtNNihw9O8kKzlLScAFfDGSW6Rd7rwbL Jq6m5lT+2J3FGvtPryASSxazxfbn6nDh5SrtFwtdItuvI2Km7KnLZtsLgWkeuN443grnEnXuYedI teTS0hxtQjiaWhLYyatvv/7+C3DAQFQ16rUfrqLmvGP6diAs687mLEbZwaKwUdSOGqmhFRMXKJYb Y7jik8N+bJiu5taqMck2akixyhkf7NkrD58kI8soP+vywR23olPOMPeXVxs956zFu1MJxKrKbQ5a GxYJS/RxlgRHbJ8sQvka9ZI1V30KrThnSySd53o1s2UrB03M1T4r1qnNJa6NrdtQRwu31yOjOm6p OorNJd2VTrIzqf8s2h3fZ/uNZ72qZhj3mr8ZDungcj7+vnjfjSfJIuSCH4t503JzbamQp1V+eOjn HU165HUnfePcKEo3Wuugr5432q7PGqXtN+9uOsS6v+35l7ntqjmcovj+OfB3Cw928W3HW1qsqW/O fF+Bl7489KY5T73yf27PvX52X/S0OEML3SjtTpzfGlDqO4T3+OIeU7bM8Vd5UP0Y1c5zuKCUF4t2 wSs+LpkQxr6WP18dMH3za0J4rgZBnCTqP/qzXPh0FpL1RLBXTanTkCpoPP6xDUxA26BIOthAxHkp UGojEIVCBsPyTa8bQ7ogQ0CYwuTdbjU4lJ/svAcaGX7KENfryrqEGL+YHUde33udDn3HoSjaUC7C wob+iIo4vAWq6HgzhJ0WqYTEKTLQWF/8X8t++LfylVFyv1vVo9Rmvy4WyoVVxIsc31e9NuLRGmLc Yw3v6Eci9jGQKhwhIduCxkNqL5GKzCMCN0aONQ4RkHOUVBh798isQUuC1FrkE8kYLmWh8IqDdNwm kQU0T+pRWunajwS75EhDKkwVnSRYyJgoSEo2ko2Z3OWGSulLRoIsmGf8JDExKctjxlKJymyiMJtZ yBgNpId+/A3VfAjNy+3wVsAUXzKDx5XReZCZ5ExiOcGJQB0VUEzGHGY7GbfKAMHMgvM0pz3R+c3s xcyaGhGYP/8J0IAKdKDUzKZBD4rQhO4Rl195jySdw5ASb/HroanC5qci1i5MZFRS1qDaNelyRG+S yWA44WPDLtmpkGLQpAaCIw3RVc9XqbSlErVXHZ0ZSojGVF34OeJHf7XRLu0EpZThFr4cRlKJGhWG Cm2qU58K1ahKdapUrapV2ZnDq+JujFolIkY7iZWufmliHISlWLFE1hPu9Kysg1ZZ2erVtCLLonDV RcPUmqC6ioJQeuUiVgVTAAA7 ------=_NextPart_000_0000_01C99CDF.12652B70--