in the place of x and y. Let us limit ourselves to a completely symmetrical case, in which the displacement z is independent of (J at a given value of r. Then Eq. (7-40) goes over into the fol lowing form: .. ilz 1 az 1 a2z (cylindrical symmetry) - +- - = - - (7-41) iJr2 r ar v2 iJt2 The traveling waves that represent solutions of this equation are expanding circular wavefronts. One can recognize more or less intuitively that the amplitude of vibration becomes less as r increases, because the disturbance is being spread over the perim eters of circles of increasing radius. The precise solutions are obtained in terms of special functions called Bessel's functions. At sufficiently l a r g e r the second term on the right in Eq. (7-41) becomes almost negligible compared to the first, and to some approximation the equation reverts to that for straight wave fronts of constant amplitude. (More accurately, the amplitude falls off approximately as 1/\\!r.) This is, of course, the impression one has if one is very far from the origin of circular waves and sees only a small portion of the perimeter of the wavefront. Finally, we can set up a wave equation for a three-dimen sional medium, such as a block of elastic solid, or air not confined to a tube. This also we quoted in Chapter 6 : a2'iJ! a2'iJ! a2'iJ! 1 a2'iJ! + +iJx2 iJz2 = v2 7ii2 (7-42) ay2 where is'iJ! some variable such as the local magnitude of the pressure. The combination of differential operators on the left hand side is named the Laplacian (after P. S. de Laplace, a near 2 contemporary of Lagrange) and is given the special symbol v' for short (pronounced \"<lei-squared\"). Thus we write Eq. (7-42) in the alternative form (7-43) As with the two-dimensional medium, if we have a system with rectangular symmetries it is appropriate to look for plane-wave solutions of the wave equation: z, + +'iJ!(x, y, t ) = f(ax {3y Yz - v t ) But, on the other hand, if spherical symmetry suggests itself-as with the waves that would be generated if a small explosion took place deep in the ground-then we introduce the radius r and two angles to define the position of a point. For a system in 245 Waves in two and three d i m e n s i o n s
which the wave amplitude depends on r only, the differential equation reduces to the following: 22 . a '1t 2 a\\Jt 1 a '1t -+--= -- (spherical symmetry) (7-44) a,2 r ar v2 a12 It is easy to verify that this equation is satisfied by simple harmonic waves whose amplitude falls off inversely with r : \\Jt(r, t) = �sin 211'(Pt - kr) (7-45) r Remembering that the energy flow for a one-dimensional wave is proportional to the amplitude squared, one can see in Eq. (7-45) the implication that the time average of [\\Jt(r, t )]2, multiplied by the area 411'r2 of a sphere of radius r, defines a rate of outflow of energy that is independent of the distance from a point source that generates the waves. In the absence of dissipation or absorp tion, this is just what one would expect to find. PROBLEMS 7-1 Satisfy yourself that the following equations can all be used to describe the same progressive wave: y = A sin 211'(x - vt)/A y = A sin 211'(kx - Pt) y = A sin 211'[(x/A) - (t/T)] y = - A sin w(t - x/v) y = A Im{exp U211'(kx - pf)]} 7-2 The equation of a transverse wave traveling along a string is given by y = 0.3 sin 11'(0.5x - 50t), where y and x are in centimeters and t is in seconds. (a) Find the amplitude, wavelength, wave number, frequency, period, and velocity of the wave. (b) Find the maximum transverse speed of any particle in the string. 7-3 What is the equation for a longitudinal wave traveling in the negative x direction with amplitude 0.003 m, frequency 5 s e c - 1 , and speed 3000 m/sec? 7-4 A wave of frequency 20 sec \" ! has a velocity of 80 m/sec. (a) How far apart are two points whose displacements are 30° apart in phase? (b) At a given point, what is the phase difference between two displacements occurring at times separated by 0.01 sec? 7-5 A long uniform string of mass density 0.1 kg/m is stretched with 246 Progressive waves
a force of 50 N. One end of the string (x = 0) is oscillated transversely (sinusoidally) with an amplitude of 0.02 m and a period of 0.1 sec, so that traveling waves in the +x direction are set up. (a) What is the velocity of the waves? (b) What is their wavelength? (c) If at the driving end (x = 0) the displacement (y) at t = O is O.ot m with dy/dt negative, what is the equation of the traveling waves? 7-6 It is observed that a pulse requires 0.1 sec to travel from one end to the other of a long string. The tension in the string is provided by passing the string over a pulley to a weight which has 100 times the mass of the string. (a) What is the length of the string? (b) What is the equation of the third normal mode? 7-7 A very long string of the same tension and mass per unit length as that in Problem 7-6 has a traveling wave set up in it with the fol lowing equation: Y,(x, t ) = 0.02 sin ,r(x - vt) where x and y are in meters, t in seconds, and v is the wave velocity (which you can calculate). Find the transverse displacement and velocity of the string at the point x = 5 m at the time t = 0 . 1 sec. 7-8 Two points on a string are observed as a traveling wave passes them. The points are at x 1 = 0 and x2 = 1 m. The transverse mo tions of the two points are found to be as follows: Y l = 0.2 sin 31rt +Yz = 0.2 sin(ht ,r/8) (a) What is the frequency in hertz? (b) What is the wavelength? (c) With what speed does the wave travel? (d) Which way is the wave traveling? Show how you reach this conclusion. ( Warning! Consider carefully if there are any ambiguities allowed by the limited amount of information given.) 7-9 A symmetrical triangular pulse of maximum height 0.4 m and total length 1 . 0 m is moving in the positive x direction on a string on which the wave speed is 24 m/sec. At t = 0 the pulse is entirely located between x = 0 and x = 1 m. Draw a graph of the transverse +velocity versus time at x = xz = 1 m. 7-10 The end (x = 0) of a stretched string is moved transversely with a constant speed of 0.5 m/sec for 0.1 sec (beginning at t = 0) and is returned to its normal position during the next 0.1 sec, again at con stant speed. The resulting wave pulse moves at a speed of 4 m/sec. 247 Problems
(a) Sketch the appearance of the string at t = 0.4 sec and at t = 0.5 sec. (b) Draw a graph of transverse velocity against x at t = 0.4 sec. 7-11 Suppose that a traveling wave pulse is described by the equation b3 +y(x, t) = b2 (x - vt)2 with b = 5 cm and v = 2.5 cm/sec. Draw the profile of the pulse as it would appear at t = 0 and t = 0.2 sec. By direct subtraction of ordinates of these two curves, obtain an appropriate picture of the transverse velocity as a function of x at t = 0.1 sec. Compare with what you obtain by calculating iJy/iJt at an arbitrary t and then putting t = 0.1 sec. 7-12 The figure shows a pulse on a string of length 100 m with fixed ends. The pulse is traveling to the right without any change of shape, at a speed of 40 m/sec. :f 0.1 m 1. 1f1 � 40m 60m .I � � �� (a) Make a clear sketch showing how the transverse velocity of the string varies with distance along the string at the instant when the pulse is in the position shown. (b) What is the maximum transverse velocity of the string (approximately)? (c) If the total mass of the string is 2 kg, what is the tension T in i t? (d) Write an equation for y(x, t ) that numerically describes sinusoidal waves of wavelength 5 m and amplitude 0.2 m traveling to the left (i.e., in the negative x direction) on a very long string made of the same material and under the same tension as above. 7-13 A pulse traveling along a stretched string is described by the following equation: b3 +y(x, t ) = b2 (2x - ut)2 (a) Sketch the graph of y against x for t = 0. (b) What are the speed of the pulse and its direction of travel? (c) The transverse velocity of a given point of the string is defined by iJy V11 = iJt 248 Progressive waves
Calculate Vy as a function of x for the instant t = 0, and show by means of a sketch what this tells us about the motion of the pulse during a short time !:.t. 7-14 A closed loop of uniform string is rotated rapidly at some con stant angular velocity w. The mass of the string is M a n d the radius is R. A tension T is set up circumferentially in the string as a result of its rotation. (a) By considering the instantaneous centripetal acceleration of a small segment of the string, show that the tension must be equal to 2 Mw R/21r. (b) The string is suddenly deformed at some point, causing a kink to appear in it, as shown in the diagram. Show that this could produce a distortion of the string that remains stationary with respect to the laboratory, regardless of the particular values of M, w, and R. But is this the whole story? (Remember that pulses on a string may travel both ways.) 7-15 Two identical pulses of equal but opposite amplitudes approach each other as they propagate on a string. At t = 0 they are as shown in the figure. Sketch to scale the string, and the velocity profile of the string mass elements, at t = 1 sec, t = 1 . 5 sec, t = 2 sec. 10 cm/sec i- rl0cm12cm s�-r2ocm------�r\" 2cm !--10cm - lOcm/sec 7-16 It is desired to study the rather rapid vertical motion of the moving contact of a magnetically operated switch. To do this, the contact is attached to one end (0) of a horizontal fishline of total mass 5 g (5 X 1 0 - 3 kg) and total length 1 2 . 5 m. The other end of the line passes over a small, effectively frictionless pulley, and a mass of 10 kg is hung from it, as shown in the sketch. The contact is actuated so that the switch (initially open) goes into the closed position, remains closed for a short time, and opens again. Shortly thereafter the string -+x (a)� (b) 249 Problems
is photographed, using a high-speed flash, and it is found to be de formed between 5 and 6 m, as shown (x = 0 is the point O where the string is connected to the contact.) (a) For how long was the switch completely closed? (b) Draw a graph of the displacement of the contact as a function of time, taking t = 0 to be the instant at which the contact first began to move. (c) What was the maximum speed of the contact? Did it occur during closing or during opening of the switch? (d) At what value o f t was the photograph taken? 2.) (Assume g = 1 0 m/sec 7-17 The following two waves in a medium are superposed: Y 1 = A sin(5x - 10t) y2 = A sin(4x - 9t) where x is in meters and t in seconds. (a) Write an equation for the combined disturbance. (b) What is its group velocity? (c) What is the distance between points of zero amplitude in the combined disturbance? 7-18 The motion of ripples of short wavelength ( � l cm) on water is controlled by surface tension. The phase velocity of such ripples is given by v = (27T\"S)112 P pX where S is the surface tension and p the density of water. (a) Show that the group velocity for a disturbance made up of wavelengths close to a given X is equal to 3vp/2. (b) What does this imply about the observed motion of a group of ripples traveling over a water surface? (c) If the group consists of just two waves, of wavelengths 0.99 and 1 . 0 1 cm, what is the distance between crests of the group? II 7-19 The relation between frequency v and wave number k for waves in a certain medium is as shown in the graph. Make a qualitative statement (and explain the basis for it) about the relative magnitudes of the group and phase velocities at any wavelength in the range represented. 7-20 Consider a U'-tube of uniform cross section with two vertical arms. Let the total length of the liquid column be/. Imagine the liquid to be oscillating back and forth, so that at any instant the levels in the side arms are at ±y with respect to the equilibrium level, and all the liquid has the speed dy/dt. (a) Write down an expression for the potential energy plus 250 Progressive waves
Y�I 1111 1111 II .......--.TT\"\"+--........ ......c__ 11-..... I 11 I II j_ ----i--\"'-----l-1-1-L-.;::,----,-,--I_J-----....;:--H .-/ --- I I -......-..!...!!.J......- 1 II I --..-l.J Ts . :L J II L J II L JI \\. .J '----------'1\\. ..J l-- 1 ---l A� kinetic energy of the liquid, and hence show that the period of oscilla tion is 1rv2l/g. (b) Imagine that a succession of such tubes can be used to define a succession of crests and troughs as in a water wave (see the diagram). Taking the result of (a), and the condition X >=::: 2/ implied by this analogy, deduce that the speed of waves on water is something like (gX) 112 /1r. (Assume that only a small fraction of the liquid is in the vertical arms of the U-tube.) (c) Use the exact result, v = (gX/21r)112, to calculate the speed of waves of wavelength 500 m in the ocean. 7-21 Consider a system of N coupled oscillators (N >> 1), each sepa rated from its nearest neighbors by a distance /. (a) Find the wavelength and frequency of the nth mode of oscillation. (b) Find the phase and group velocities for this mode. What are +«they for the cases n N and n = N 1? 7-22 You are given the problem of analyzing the dynamics of a line of cars moving on a one-lane highway. One approach to this problem is to assume that the line of cars behaves like a group of coupled oscillators. How would you set this problem up in a tractable way? Make lots of assumptions. 7-23 One end of a stretched string is moved transversely at constant velocity Uy for a time T, and is moved back to its starting point with velocity -uy during the next interval T. As a result, a triangular pulse is set up on the string and moves along it with speed v. Calculate the kinetic and potential energies associated with the pulse, and show that their sum is equal to the total work done by the transverse force that has to be applied at the end of the string. 7-24 Consider a longitudinal sinusoidal wave � = �o cos 21rk(x - vt) traveling down a rod of mass density p, cross-sectional area S, and Young's modulus Y. Show that if the stress in the rod is due solely to the presence of the wave, the kinetic-energy density is !- 2, pS(iJ�/iJt) and the potential-energy density is !-YS(iJ�/iJx)2• Thus show that the kinetic energy per wavelength and the potential energy per wavelength both equal !-(pSX)uo2, where uo is the maximum particle velocity (iJ�/iJt). 7-25 Verify that the wave equation for spherically symmetric waves [Eq. (7-44)) is satisfied by simple harmonic waves whose amplitudes fall off inversely with r. 251 Problems
We shall see in this chapter how sounds quarrel, fight, and when they are of equal strength destroy one another, and give place to silence. ROBERT BALL, Wonders of Acoustics (1867)
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