Let f () be an arbitrary (twice differentiable) function.

Verify that the quantity c = bt that appears in the wave equation for a string does indeed have the units of a speed.

The wave equation (16.4) is the equation of motion for a continuous string, as illustrated in Figure 16.1(a). You can obtain this equation as the limit as n oo of the equations for the n discrete masses of Figure 16.1(b). You need to be careful with the limiting process. As n oo, the spacing b between the masses (see Figure 16.20) and the individual masses m must both go to zero in such a way that the linear mass density m lb approaches p, the density of the continuous string. You can guarantee this by taking m = ktb. Write down Newton's second law for the position ui of the ith mass and show that it goes over to the wave equation as b 0.

Let f () be an arbitrary (twice differentiable) function. Show by direct substitution that f ct) is a solution of the wave equation (16.4).

Show that if we make the change of variables = x ct and rl = x ct, then, as in (16.7), 82u a2 r2 " 4C2 a a" ate ax2 a aq

(a) Show that u = g(x ct) is a solution of the wave equation (16.4) for any twice differentiable function g(0. (b) Argue clearly that this solution represents a disturbance that travels undistorted to the left. b Figure 16.20 Problem 16.2

There is a small flaw in Example 16.1 (page 686). In Equation (16.14) I omitted a constant of integration, so the equation should really have read f (x) g(x) = k. Show that this still gives the same final answer (16.15).

[Computer] Make plots of the two triangular waves of Example 16.1 (page 686) at several closely spaced times and then animate them. Describe the motion. For the purposes of the plot you may as well take the speed c, the height of the triangle at time 0, and the half width of the base all equal to 1. Make your plots for lots of times ranging from t = 4 to 4.

[Computer] Make plots similar to Figure 16.5 of the standing wave (16.18) for several equally spaced times from t = 0 to r, the period. Take 2A = 1 and k = w = 27r. Animate your pictures and describe the motion.

The motion of a finite string, fixed at both ends, was determined by the wave equation (16.19) and the boundary conditions (16.20). We solved these by looking for a solution that was sinusoidal in time. A different, and rather more general, approach to problems of this kind is called separation of variables. In this approach, we seek solutions of (16.19) with the separated form u(x , t) = X (x)T (t), that is, solutions that are a simple product of one function of x and a second of t. [As usual, there's nothing to stop us trying to find a solution of this form. In fact, there is a large class of problems (including this one) where this approach is known to produce solutions, and enough solutions to allow expansion of any solution.] (a) Substitute this form into (16.19) and show that you can rewrite the equation in the form T"(t)I T (t) = c2X"(x)/ X (x). (b) Argue that this last equation requires that both sides of this equation are separately equal to the same constant (call it K). It can be shown that K has to be negative.26 Use this to show that the function T (t) has to be sinusoidal which establishes (16.21) and we're back to the solution of Section 16.3. The method of separation of variables plays an important role in several areas, notably quantum mechanics and electromagnetism.

Using the integral (16.33), show that the Fourier coefficients of the triangular wave of Figure 16.7 are zero for n even and given by (16.34) for n odd.

[Computer] Make plots similar to Figure 16.8 of the wave of Example 16.2 but from t = 0 to r, the period, and for more closely spaced times. Animate your pictures and describe the motion.

Consider a semi-infinite string, fixed at the origin x = 0 and extending far out to the right. Let f () be a function that is localized around the origin, such as the function of Figure 16.4(a). (a) Describe the wave given by the function f (x ct) for a large negative time to. (b) One way to solve for the subsequent motion of this wave on the semi-infinite string is called the method of images and is as follows: Consider the function u = f (x ct) f (x ct). (The second term here is called the "image." Can you explain why?) Obviously this satisfies the wave equation for all x and t. Show that it coincides with the given wave of part (a) at the initial time to and everywhere on the semi-infinite string. Show also that it obeys the boundary condition that u = 0 at x = 0. (c) It is a fact that there is a unique wave that obeys the wave equation and any given initial and boundary conditions. Therefore the wave of part (b) is the solution for all times (on our semi-infinite string). Describe the motion on the semi-infinite string for all times.

In connection with Equation (16.31), I claimed that any function on the interval 0 < x < L can be expanded in a Fourier series containing just sine functions. This is at first sight very surprising since one is used to the claim that the general Fourier series requires sines and cosines. In this problem, you'll prove this surprising claim. Let f (x) be any function defined for 0 < x < L. We can define a function f (x) for all x by setting it equal to the given function in the original interval and requiring that f (x) f (x) and f (x 2L) = f (x). (16.143) for all x. Prove that this defines a function which is (1) periodic with period 2L, (2) odd, and (3) the same as the original f (x) on the original interval. Write down the ordinary Fourier expansion for this new f (x) and show that the coefficients of the cosine terms are all zero. This establishes the possibility of expanding the original function in terms of sines alone.27 Bearing in mind that the period of the new function is 2L, write down the standard formula (5.84) for the expansion coefficients and show that your answer agrees with (16.33). The Fourier sine series is especially convenient for discussing functions that are zero at the end points x = 0 and L.

[Computer] A taut string of length L = 1 is released from rest at t = 0, with initial position 2x [0 x 4] u(x, 0) 2(1 x) < x < 1]. (16.144) Take the wave speed on the string to be c = 1. (a) Sketch this initial shape and find the coefficients B, in its Fourier sine series (16.31). (b) Make plots of the sum of the first several terms for several closely spaced times between t = 0 and r, the period. Animate your plots and describe the motion.

Let f () be any function with first two derivatives f V) and f"(0, and let n be an arbitrary fixed unit vector. (a) Show that V f (n r ct) = nf'(n r ct). (b) Hence show that f (n r ct) satisfies the three-dimensional wave equation (16.38). (c) Argue that f (n r ct) represents a signal that is constant in any plane perpendicular to n (at any fixed time t) and propagates rigidly with speed c in the direction of n.

Let f (r) be any spherically symmetric function; that is, when expressed in spherical polar coordinates, (r, 8, 0), it has the form f (r) = f (r), independent of 8 and 0. (a) Starting from the definition (16.38) of V2, prove that 1 8, V2f = -r 0r2(rf). (b) Prove the same result using the formula inside the back cover for V2 in spherical polar coordinates. (Obviously, this second proof is much simpler, but the hard work is hidden in the derivation of the formula for V2.)

In Section 16.1 we derived the wave equation for transverse waves in a taut string. Here you will examine the possibility of longitudinal waves in the same string. Suppose that an element of string whose equilibrium position is x is displaced a short distance in the x direction to x u(x, t). (a) Consider a short piece of string of length 1 and use the definition (16.55) of Young's modulus YM to show that the tension is F = A YM au/ax, where A is the cross sectional area of the string. [If the string is already in tension in its equilibrium position, this F is the additional tension, that is, F = (actual equilibrium).] (b) Now consider the forces on a short section of string dx and show that u obeys the wave equation with wave speed c = ,/YM/Q where p is the density (mass/volume) of the string.

Figure 16.15 is an end view of a triangular prism, whose three faces are labeled by the vectors dA1, etc. (The magnitude of dA1 is the area of the corresponding surface and the direction is normal to it. There are two more faces parallel to the plane of the paper, but these do not concern us.) The ends of the three faces form a closed triangle. Expain clearly why this implies that dA dA2 dA3 = 0.

Let n1 and n2 be any two unit vectors and P a point in a continuous medium. F(n1 dA) is the surface force on a small area dA at P with unit outward normal n1, so n2 F(ni dA) is the component of that force in the direction of n2. Prove Cauchy's reciprocal theorem that n2 F(n1 dA) = n1 F(n2 dA).

It is found that the stress tensor at any point (x, y, z) in a certain continuous medium has the form (with an unspecified, convenient choice of units) XZ z2 0 = [ z2 y 0 _ (16.145) Find the surface force on a small area dA of the surface x2 + y2 + 2z2 = 4 at the point (1, 1, 1).

At any given point P of a continuous medium, the surface forces are given by the stress tensor, which is a real symmetric matrix L It is a well-known theorem of linear algebra (see the appendix) that any such matrix can be brought into diagonal form by a suitable rotation of the Cartesian coordinate axes. Use this to prove that at any point P there are three orthogonal directions (the principal stress axes at P) with the property that the surface force on any surface normal to one of these directions is exactly normal to the surface.

Show that if the stress tensor E is diagonal (all off-diagonal elements zero) with respect to any choice of orthogonal axes, then it is in fact a multiple of the unit matrix. This gives an alternative and elegant proof that if there are no shearing stresses (in any coordinate system) then the pressure forces are independent of direction. To do this problem, you need to know how the elements of a tensor transform as we rotate our coordinate axes, as described in Section 15.17. Assume that with respect to one set of axes E is diagonal but that not all three diagonal elements are equal. (For example, 611 a33.) It is not hard to come up with a rotation that of Equation (15.36) will do such that in the rotated system a13 0.

An important tool in the development of the strain tensor was the decomposition (16.79) of a matrix M into its antisymmetric and symmetric parts. Prove that this decomposition is unique. [Hint: Show that if M = MA + Ms where MA and Ms are respectively antisymmetric and symmetric, then MA = 4 (M - M) and Ms = 4 (M + M).]

Write out the components of the displacement (16.77), u(r) = 0 x r, for a small rotation 0 and verify that the derivatives matrix is given by Equation (16.78).

At a certain point P (which you can choose to be your origin) in a continuous solid, the strain tensor is E. Assume for simplicity that whatever displacements have occurred left P fixed and the neighborhood of P unrotated. (a) Show that the x axis near P is stretched by a factor of (1 + E11). (b) Hence show that any small volume around P has changed by dV I V = tr E. This shows that any two strains that have the same trace dilate volumes by the same amount. In the decomposition E = el + (16.88), the spherical part el changes volumes by the same amount as E itself, while the deviatoric part E' doesn't change volumes at all.

The table below gives the three elastic moduli for several materials. According to (16.100) Young's modulus for any given material can be calculated if we know the bulk and shear moduli. Using the data for BM and SM, calculate YM for each of the materials and compare with the given values in the third column. (The densities will be needed for Problem 16.32.) Elastic Moduli (in GPa) and Densities (in g/cm3) Material BM SM YM Iron 90 40 100 7.8 Steel 140 80 200 7.8 Sandstone 17 6 16 1.9 Perovskite 270 150 390 4.1 Water 2.2 0 0 1.0

Consider a taut wire or rod lying along the x axis. To define Young's modulus YM we apply a pure tension along the axis; that is, a stress with all > 0 and all other at] = 0. (a) Use Equation (16.95) to write down the corresponding strain tensor E. (b) Argue from the definition (16.55) of Young's modulus to show that YM = 1/E11. (c) Combine these two results to verify the expression (16.100) for YM, showing in particular that YM = 9 BM SM/ (3 BM + SM).

Consider again the wire or rod of Problem 16.27. In general, when one stretches the wire longitudinally it will contract in the transverse directions. The ratio of the transverse fractional contrac- tion to the longitudinal fractional stretch is called Poisson's ratio (after the French mathematician and student of Laplace and Lagrange, 1781-1840) and is denoted by the Greek letter "nu," v. (a) Show that v = E22/E11. (b) Use the method of Problem 16.27 to show that v = (3 BM 2 SM)/ (6 BM + 2 SM). (c) Calculate Poisson's ratio for the five materials listed in Problem 16.26. Comment on its value for materials with BM >> SM.

When we change our coordinate axes, the strain tensor changes in accordance with Equation (15.132), which we can rewrite as ER = RER, where R is the (3 x 3) orthogonal rotation matrix. Use the property (15.129) of orthogonal matrices to show that tr ER = tr E; that is, the trace of any tensor is rotationally invariant. Use this result to show that the decomposition E = el + E' is rotationally invariant, in the sense described below Equation (16.92).

If 8i, denotes the Kronecker delta symbol (16.115) and a is a vector with components di (j -= 1, 2, 3), prove that E j = a,. In the same way, show that E j sjiaj= 8 a result we used in proving the important identity (16.116).

A seismograph records the signals arriving from a distant earthquake. If the S waves arrive 12 minutes after the P waves, how far away was the earthquake? Use the speeds found in Example 16.8 (page 722).

[Computer] Using appropriate software, calculate the speeds of longitudinal and transverse waves in the five materials listed in Problem 16.26. Arrange for the software to give you a nice readable table of values.

Write down the equation of motion (16.124) as applied to a static fluid. Assuming that g is uniform and p is constant (independent of r), prove the well-known result from introductory physics that the pressure difference between two points r1 and r2 is just Ap = ggh, where h is the vertical difference in elevations of r1 and T.,.

Equations (16.129) and (16.130) are two different forms of the equation of continuity. Prove that they are equivalent.

A crucial step in showing that the waves in a fluid are necessarily longitudinal was the integral in (16.142). For an arbitrary function f (0, with derivative f V), prove that f f (n r ct)dt = f (n r ct)/c.

To find the speed of sound in air using the result (16.140) requires a little care. (Even the great Newton got this one wrong!) The trouble is to decide on the correct value of the bulk modulus of air. Because the vibrations are so rapid, there is no time for heat transfer and the air expands and contracts adiabatically, so that , where is the so-called "ratio of specific heats," for air. (a) Show that the bulk modulus is . (b) Use the ideal gas law, pV = nRT to show that the density is , where M is the average molecular mass of air . (c) Put these results together to show that the speed of sound is . Find the speed of sound at 0°C, and compare with the accepted value of 331 m/s.

Show that the intensity I of a sound wave is proportional to the square of the pressure increment p'. To do this consider a small sliver of fluid normal to the direction of propagation with area dA. Write down the rate at which this sliver does work on the fluid just in front of it, then divide by dA to show that the time-average intensity is (I) = (p12)1cgo.

A crucial step in deriving the wave equation for waves in a fluid was the neglect of the first term on the right of Equation (16.134). (a) Justify this by using (16.139) to rewrite the right side of (16.125) as . Argue that the ratio of the first to the second term is of order , where is a typical distance over which varies. (A good choice for A would be the wavelength of the proposed wave — of order a centimeter or at most a few meters.) Using the values for water show that the first term is negligible. (You would also reach the same conclusion for air.) (b) Show with a similar argument that the second term on the right of (16.136) is negligible.