Mth 4A October 04, 0 Viktor Grigoryn 4 Vibrtions nd het flow In this lecture we will derive the wve nd het equtions from physicl principles. These re second order constnt coefficient liner PEs, which model mechnicl vibrtions nd therml flow respectively. 4. Vibrting string Consider thin string of length l, which undergoes reltively smll trnsverse vibrtions (think of string of musicl instrument, sy violin string). Let ρ be the liner density of the string, mesured in units of mss per unit of length. We will ssume tht the string is mde of homogeneous mteril nd its density is constnt long the entire length of the string. The displcement of the string from its equilibrium stte t time t nd position will be denoted by u(t, ). We will position the string on the -is with its left endpoint coinciding with the origin of the u coordinte system. Considering the motion of smll portion of the string sitting top the intervl [, b], which hs mss ρ(b ), nd ccelertion u tt, we cn write Newton s second lw of motion (blnce of forces) s follows ρ(b )u tt = Totl force. () Hving thin string with negligible mss, we cn ignore the effect of grvity on the string, s well s ir resistnce, nd other eternl forces. The only force cting on the string is then the tension force. Assuming tht the string is perfectly fleible, the tension force will hve the direction of the tngent vector long the string. At fied time t the position of the string is given by the prmetric equtions { =, u = u(, t), where ( plys the role ) of prmeter. The tngent vector is then (, u ), nd the unit tngent vector will be +u u, +u. Thus, we cn write the tension force s T(, t) = T (, t) (, u ) = (T, T u ), () where T (t, ) is the mgnitude of the tension. Since we consider only smll vibrtions, it is sfe to ssume tht u is smll, nd the following pproimtion vi the Tylor s epnsion cn be used = + u + O(u 4 ). Substituting this pproimtion into (), we rrive t the following form of the tension force T = (T, T u ). With our previous ssumption tht the motion is trnsverse, i.e. there is no longitudinl displcement (long the -is), we rrive t the following identities for the blnce of forces () in the, respectively u directions 0 = T (b, t) T (, t) ρ(b )u tt = T (b, t)u (b, t) T (, t)u (, t). The first eqution bove merely sttes tht in the direction the tensions from the two edges of the smll portion of the string blnce ech other out (no longitudinl motion). From this we cn lso see tht the tension force is independent of the position on the string. Then the second eqution cn be rewritten s ρu tt = T u (b, t) u (, t). b
Pssing to the limit b, we rrive t the wve eqution ρu tt = T u, or u tt c u = 0, where we used the nottion c = T/ρ. As we will see lter, c will be the speed of wve propgtion, similr to the constnt ppering in the trnsport eqution (we ssume tht ρ nd T re independent of time, which is justified by the smllness of the vibrtions). One cn generlize the wve eqution to incorporte effects of other forces. Some emples follow. ) With ir resistnce: force is proportionl to the speed u t u tt c u + ru t = 0, r > 0. b) With trnsverse elstic force: force is proportionl to the displcement u u tt c u + ku = 0, k > 0. c) With n eternlly pplied force u tt c u = f(, t) (inhomogeneous). 4. Vibrting drumhed Similr to the vibrting string, one cn consider vibrting drumhed (elstic membrne), nd look t the dynmic of the displcement u(, y, t), which now depends on the two sptil vribles (, y) denoting the point on the -dimensionl spce of the equilibrium stte. Tking smll region on the drumhed, the tension force will gin be directed tngentilly to the surfce, in this cse long the norml vector to the boundry of the region. Its verticl component will be T u u, where denotes the n n derivtive of u in the norml direction to the boundry of the region. The verticl component of the cumultive tension force will then be T vert = T u n ds = T u n ds, where denotes the boundry of the region, nd the integrl is tken with respect to the length element long the boundry of. The second lw of motion will then tke the form T u n ds = ρu tt ddy. Using Green s theorem, we cn convert the line integrl on the left hnd side to two dimensionl integrl, nd rrive t the following identity (T u) ddy = ρu tt ddy. Since the region ws tken rbitrrily, the integrnds on both sides must be the sme, resulting in ρu tt = T (u + u yy ), or u tt c (u + u yy ) = 0. This is the wve eqution in two sptil dimensions. Three dimensionl vibrtions cn be treted in much the sme wy, leding to the three dimensionl wve eqution Often one mkes use of the opertor nottion to write the wve eqution in ny dimension s u tt c (u + u yy + u zz ) = 0. = = + y +... u tt c u = 0. The opertor is clled Lplce s opertor or the Lplcin.
4.3 Het flow Let u(, t) denote the temperture t time t t the point in some thin insulted rod. Agin, think of the rod s positioned long the -is in the u coordinte system, where the verticl is will mesure the temperture. The het, or therml energy of smll portion of the rod situted t the intervl [, b] is given by H(, t) = b cρu d, where ρ is the liner density of the rod (mss per unit of length), nd c denotes the specific het cpcity of the mteril of the rod. The instntneous chnge of the het with respect to time will be the time derivtive of the bove epression dh b dt = cρu t d. Since the het cnnot be lost or gined in the bsence of n eternl het source, the chnge in the het must be blnced by the het flu cross the cross-section of the cylindricl piece round the intervl [, b] (we ssume tht the lterl boundry of the rod is perfectly insulted). According to Fourier s lw, the het flu cross the boundry will be proportionl to the derivtive of the temperture in the direction of the outwrd norml to the boundry, in this cse the -derivtive. Het flu = κu, where κ denotes the therml conductivity. Using this epression for the het flu, nd noting tht the chnge in the internl het of the portion of the rod is equl to the combined flu through the two ends of this portion, we hve b cρu t d = κ(u (b, t)) u (, t)). ifferentiting this identity with respect to b, we rrive t the het eqution cρu t = κu, or u t ku = 0, where we denoted k = κ cρ. As in the cse of the wve eqution, one cn consider higher dimensionl het flows (het flow in two dimensionl plte, or three dimensionl solid) to rrive t the generl het eqution u t k u = 0. We lso note tht diffusion phenomen led to n eqution which hs the sme form s the het eqution (cf. Struss for the ctul derivtion, where insted of Fourier s lw of het conduction one uses Fick s lw of diffusion). 4.4 Sttionry wves nd het distribution If one looks t vibrtions or het flows where the physicl stte does not chnge with time, then u t = u tt = 0, nd both the wve nd the het equtions reduce to u = 0. (3) This eqution is clled the Lplce eqution. Notice tht in the one dimensionl cse (3) reduces to u = 0, which hs the generl solution u(, t) = c + c (remember tht u is independent of t). The solutions to the Lplce eqution re clled hrmonic functions, nd we will see lter in the course tht one encounters nontrivil hrmonic function in higher dimensions. 3
4.5 Boundry conditions We sw in previous lectures tht PEs generlly hve infinitely mny solutions. One then imposes some uiliry conditions to single out relevnt solutions. Since the equtions tht we study come from physicl considertions, the uiliry conditions must be physiclly relevnt s well. These conditions come in two different vrieties: initil conditions nd boundry vlue conditions. The initil condition, fmilir from the theory of OEs, specifies the physicl stte t some prticulr time t 0. For emple for the het eqution one would specify the initil temperture, which in generl will be different t different points of the rod, u(, 0) = φ(). (4) For the vibrting string, one needs to specify both the initil position of (ech point of) the string, nd the initil velocity, since the eqution is of second order in time, u(, 0) = φ(), u t (, 0) = ψ(). (5) In the physicl emples tht we considered t the beginning of this lecture, it is cler tht there is domin on which the solutions must live. For emple in the cse of the vibrting string of length l, we only look t the mplitude u(t, ), where 0 l. Similrly, for the het conduction in n insulted rod. In higher dimensions the domin is bounded by curves (d), surfces (3d) or higher dimensionl geometric shpes. The boundry of this domin is where the system intercts with the eternl fctors, nd one needs to impose boundry conditions to ccount for these interctions. There re three importnt kinds of boundry conditions: () the vlue (on the boundry) of u is specified (irichlet condition) (N) the vlue of the norml derivtive u/ n is specified (Neumnn condition) (R) the vlue of u/ n + u is specified (Robin condition, is function of, y, z,... nd t) These conditions re usully written s equtions, for emple the irichlet condition u(, t) = f(, t), or the Robin condition u n + u = h(, t). The condition is clled homogeneous, if the right hnd side vnishes for ll vlues of the vribles. In one dimensionl cse, the domin is n intervl (0 < < l), hence the boundry consists of just two points = 0, nd = l. The boundry conditions then tke the simple form () u(0, t) = g(t) nd u(l, t) = h(t) (N) u (0, t) = g(t) nd u (l, t) = h(t) (R) u(0, t) + (t)u (0, t) = g(t) nd u(l, t) + b(t)u (l, t) = h(t) 4.6 Emples of physicl boundry conditions In the cse of vibrting string, one cn impose the condition tht the endpoints of the string remin fied (the cse for strings of musicl instruments). This gives the irichlet conditions u(0, t) = u(l, t) = 0. If one end is llowed to move freely in the trnsverse direction, then the lck of ny tension force t this endpoint cn be epressed s the Neumnn condition u (l, 0) = 0. A Robin condition will correspond to the cse when one endpoint is llowed to move trnsversely, but the motion is restricted by force proportionl to the mplitude u (think of coiled spring ttched to the endpoint). 4
In the cse of het conduction in rod, the perfect insultion of the rod surfce leds to the Neumnn condition u/ n = 0 (no echnge of het with the mbient spce). If the endpoints of the rod re kept in therml blnce with the mbient temperture g(t), then one hs the irichlet condition u = g(t) on the boundry. If one llows therml echnge between the rod nd the mbient spce obeying Newton s lw of cooling, then he boundry condition tkes the form of Robin condition u (l, t) = (u(l, t) g(t)). 4.7 Conclusion We derived the wve nd het equtions from physicl principles, identifying the unknown function with the mplitude of vibrting string in the first cse nd the temperture in rod in the second cse. Understnding the physicl significnce of these PEs will help us better grsp the qulittive behvior of their solutions, which will be derived by purely mthemticl techniques in the subsequent lectures. The physiclity of the initil nd boundry conditions will lso help us immeditely rule out solutions tht do not conform to the physicl lws behind the pproprite problems. 5