Oscillations and Mechanical Waves

Derives Proposition 28.4. Equation (28.2) is linear, homogeneous and of second order with constant coefficients, so by Theorem 9.13 its solution space is two-dimensional, and by Corollary 9.9 every solution extends to all of \(\R\). The pair \(\cos\omega_{0}t\) and \(\sin\omega_{0}t\) solves it and has Wronskian (Definition 9.14) equal to \(\omega_{0}\neq0\), so by Proposition 9.16 it is a basis; writing the general combination in amplitude–phase form gives Equation (28.4). Uniqueness for given \(x(0),\dot{x}(0)\) is Theorem 9.8. The period is the least \(T>0\) with \(\omega_{0}T=2\pi\), and \(A\) appears in Equation (28.4) only as a multiplicative constant, which a linear equation cannot convert into a shift of \(T\): hence the isochronism, and hence also its exactness being no better than the linearity (Section 28.7).

Derives Proposition 28.5. Substituting Equation (28.4) and \(k=m\omega_{0}^{2}\),

by the Pythagorean identity (Lemma 7.59), which is also the statement that \(E\) does not depend on \(t\). The mean of \(\cos^{2}\) over a period is \(\tfrac{1}{2}\) and so is the mean of \(\sin^{2}\), whence \(\avg{T}=\avg{V}=E/2\). The energy oscillates between the two forms at \(2\omega_{0}\) — twice the frequency of the motion — because both forms are quadratic in a quantity oscillating at \(\omega_{0}\).

Derives Proposition 28.6. The orbit is the level set of \(\Ham\), an ellipse with semi-axes \(x_{\max}=\sqrt{2E/m\omega_{0}^{2}}\) along \(x\) and \(p_{\max}=\sqrt{2mE}\) along \(p\). The enclosed area is \(\pi x_{\max}p_{\max}=\pi\cdot 2E/\omega_{0}\), and \(\oint p\,\dd x\) is that area. The dimension follows from \(\left[\text{energy}\right]/\left[\text{frequency}\right]\).

Derivation. Derives Phenomenon 28.9. A bob of mass \(m\) on a light rod of length \(\ell\), displaced by an angle \(\theta\) from the downward vertical, has moment of inertia \(m\ell^{2}\) about the pivot, while the weight exerts about the pivot the moment \(-mg\ell\sin\theta\). The rotational law of Newtonian Dynamics then gives

in which the mass has cancelled — which accounts for the observed independence of the bob. For \(\abs{\theta}\ll1\) one has \(\sin\theta=\theta+O(\theta^{3})\) by Theorem 7.38, and Equation (28.8) reduces to the harmonic equation \(\ddot{\theta}+\omega_{0}^{2}\theta=0\) with \(\omega_{0}^{2}=g/\ell\), whose general solution is Equation (28.4) with period \(T=2\pi/\omega_{0}\). That is Equation (28.7), and it contains no \(A\): in a linear equation the amplitude enters only as an overall multiplicative constant, and no linear equation can make its period depend on it (Proposition 28.4). Isochronism is therefore exactly as good as the linearization and no better. Retaining the cubic term makes the period grow with the angular amplitude \(\theta_{0}\) as \(T(\theta_{0})=T\left(1+\theta_{0}^{2}/16+O(\theta_{0}^{4})\right)\), derived in Theorem 28.72 and already about one part in \(10^{3}\) at an amplitude of \(7^\circ\) — the failure taken up in Section 28.7.1.

Derives Proposition 28.11. The moment of the weight about the pivot is \(-mgd\sin\theta\) and the angular momentum about it is \(I\dot{\theta}\), so the rotational law of Newtonian Dynamics gives \(I\ddot{\theta}=-mgd\sin\theta\); linearizing as in Phenomenon 28.9 yields the first of Equation (28.9), and comparison with \(\omega_{0}^{2}=g/\ell\) gives the second. The parallel-axis theorem of Rigid Bodies and Rotating Frames gives \(I=m(b^{2}+d^{2})\), whence \(\ell_{\text{eq}}=(b^{2}+d^{2})/d=d+b^{2}/d\). Suspending instead at \(d'=b^{2}/d\) gives \(\ell_{\text{eq}}'=d'+b^{2}/d'=b^{2}/d+d\), the same number.

Derives Proposition 28.12. Parametrize the curve by arc length \(s\) measured from the lowest point, \(\phi=0\). From Equation (28.10), \(\dd x=a(1+\cos\phi)\dd\phi\) and \(\dd y=a\sin\phi\,\dd\phi\), so

whence \(\dd s=2a\cos(\phi/2)\dd\phi\) and \(s=4a\sin(\phi/2)\), the cosine being non-negative on \(\abs{\phi}\leq\pi\). The height above the lowest point is \(h=y=a(1-\cos\phi)=2a\sin^{2}(\phi/2)\), and eliminating \(\phi\) in favour of \(s\) gives \(h=s^{2}/8a\). The potential energy is therefore \(V=mgh=mgs^{2}/8a\), exactly quadratic in the arc length, and the kinetic energy is \(\tfrac{1}{2}m\dot{s}^{2}\) because \(s\) is arc length. So the motion in \(s\) is Equation (28.2) with \(\omega_{0}^{2}=g/4a\), with no approximation anywhere, and Proposition 28.4 gives a period independent of amplitude.

Derivation. Derives Phenomenon 28.13. Let a system of one degree of freedom have potential energy \(V(x)\), twice continuously differentiable, with an equilibrium at \(x_{0}\), so that \(V'(x_{0})=0\), and let that equilibrium be stable, so that \(V''(x_{0})>0\). Taylor's theorem (Theorem 7.38) gives

the linear term being absent precisely because \(x_{0}\) is an equilibrium. Writing \(u:=x-x_{0}\) and keeping the quadratic term, the force is \(F=-\dv{V}{x}=-V''(x_{0})\,u\) and Newton's second law becomes

a harmonic equation with a real frequency exactly because \(V''(x_{0})>0\). Nothing about the system entered except the two numbers \(m\) and \(V''(x_{0})\), which is the universality asserted; and the approximation is controlled by the neglected cubic term, so it degrades as the amplitude grows (Section 28.7). The two excluded cases are instructive. Where \(V''(x_{0})<0\) the same expansion gives an exponentially growing solution, which is what makes the equilibrium unstable; where \(V''(x_{0})=0\) the leading behaviour is quartic or worse and is not harmonic at all, so the period depends on the amplitude however small the amplitude is made.

Derives Proposition 28.18. Substituting \(x=\ee^{\lambda t}\) into Equation (28.15) gives the characteristic equation \(\lambda^{2}+2\beta\lambda+\omega_{0}^{2}=0\) with roots \(\lambda_{\pm}=-\beta\pm\sqrt{\beta^{2}-\omega_{0}^{2}}\). For \(\beta<\omega_{0}\) the roots are \(-\beta\pm\ii\omega_{d}\) and the real combinations give Equation (28.16). For \(\beta>\omega_{0}\) both roots are real and negative, since \(\sqrt{\beta^{2}-\omega_{0}^{2}}<\beta\), giving Equation (28.18). At \(\beta=\omega_{0}\) the roots coincide and \(\ee^{-\omega_{0}t}\) alone spans only one dimension of the two-dimensional solution space of Theorem 9.13; a second, independent solution is \(t\ee^{-\omega_{0}t}\), as substitution confirms and as the Wronskian \(\ee^{-2\omega_{0}t}\neq0\) certifies (Proposition 9.16). Each exponential has strictly negative real part whenever \(\beta>0\), so every solution decays; and an exponential vanishes nowhere, while Equation (28.17) and Equation (28.18) can each cross zero at most once.

Derives Proposition 28.20. Differentiating \(E=\tfrac{1}{2}m\dot{x}^{2}+\tfrac{1}{2}m\omega_{0}^{2} x^{2}\) along Equation (28.15),

which is the power of the resistive force and is never positive. Over one cycle \(\avg{\dot{x}^{2}}=E/m\) by Proposition 28.5 applied to the slowly decaying envelope, so \(\avg{\dd E/\dd t}=-2\beta E\) and \(E=E_{0}\ee^{-2\beta t}\). The fractional loss per cycle is then \(\Delta E/E=2\beta\cdot 2\pi/\omega_{d}\approx4\pi\beta/\omega_{0}\) for \(Q\gg1\), and \(2\pi E/\Delta E=\omega_{0}/2\beta=Q\). The amplitude envelope falls by \(\ee^{-1}\) in the time \(1/\beta=2Q/\omega_{0}\), which is \(Q/\pi\) periods; the energy, decaying twice as fast, does so in \(Q/2\pi\) periods.

Derives Proposition 28.21. For \(\beta<\omega_{0}\) every solution is bounded by \(\text{const}\times\ee^{-\beta t}\) and no better, so the rate is \(\beta<\omega_{0}\). For \(\beta>\omega_{0}\) the slower of the two exponentials in Equation (28.18) decays at

a strictly decreasing function of \(\beta\) on \((\omega_{0},\infty)\). At \(\beta=\omega_{0}\) the rate is \(\omega_{0}\) itself, larger than either branch, the algebraic factor \(t\) in Equation (28.17) being no obstacle since \(t\ee^{-\omega_{0}t}\to0\) faster than \(\ee^{-\beta' t}\) for any \(\beta'<\omega_{0}\).

Derivation. Derives Phenomenon 28.23. Take the linearly damped, sinusoidally driven oscillator

and seek the steady state \(x=A(\omega)\cos(\omega t-\delta)\). Substituting and collecting the coefficients of \(\cos\omega t\) and \(\sin\omega t\) gives

At \(\omega=\omega_{0}\) the first bracket vanishes and \(A(\omega_{0})=F_{0}/(2m\beta\omega_{0})\), whereas the static response, \(\omega=0\), is \(A(0)=F_{0}/(m\omega_{0}^{2})\). Their ratio is

the quality factor: a resonator with \(Q=10^{4}\) answers a drive at its own frequency ten thousand times more strongly than it answers the same drive applied slowly. The denominator of \(\tan\delta\) vanishes at \(\omega=\omega_{0}\), so \(\delta=\pi/2\) there, which is the quarter-cycle lag. Finally \(A^{2}\) falls to half its peak where \(\left(\omega_{0}^{2}-\omega^{2}\right)^{2}=4\beta^{2}\omega^{2}\), that is \(\abs{\omega_{0}-\omega}\simeq\beta\) for light damping, so the full width of the resonance curve is \(\Delta\omega\simeq2\beta=\omega_{0}/Q\). Gain and selectivity are thus the same number: a sharp resonator is necessarily a narrow filter, and a broadband one is necessarily insensitive.

Derives Proposition 28.24. This is the causal Green's function of Definition 10.42 for the operator on the left, and Equation (28.26) is Duhamel's principle (Theorem 10.52) for it; only \(G\) itself has to be identified. An impulse \(F=P\delta(\tau)\) leaves \(x\) continuous and jumps \(\dot{x}\) by \(P/m\), by integrating the equation across \(\tau=0\); the solution of the homogeneous equation with \(x(0)=0\) and \(\dot{x}(0)=P/m\) is, from Equation (28.16), \(x=(P/m\omega_{d})\ee^{-\beta\tau}\sin\omega_{d}\tau\). Dividing by \(P\) gives \(G\). Vanishing for \(\tau<0\) is the causality requirement, which selects the retarded solution of Definition 10.50. The dimension follows from displacement per unit impulse per unit time.

Derives Proposition 28.27. With \(F=F_{0}\cos\omega t\) and \(\dot{x}=-A\omega\sin(\omega t-\delta)\), the average of \(\cos\omega t\,\sin(\omega t-\delta)\) over a period is \(-\tfrac{1}{2}\sin\delta\), so \(\bar{P}=\avg{F\dot{x}}=\tfrac{1}{2}F_{0}A\omega\sin\delta\). Substituting \(A\) and \(\sin\delta=2\beta\omega/\sqrt{(\omega_{0}^{2}-\omega^{2})^{2} +4\beta^{2}\omega^{2}}\) from Equation (28.24) gives Equation (28.28). Writing \(D^{2}\) for the bracket, \(\bar{P}\) is half its peak where \(\beta\omega^{2}/D^{2}=1/8\beta\), that is \(\left(\omega_{0}^{2}-\omega^{2}\right)^{2}=4\beta^{2}\omega^{2}\), or \(\omega^{2}\pm2\beta\omega-\omega_{0}^{2}=0\); the two positive roots are \(\sqrt{\beta^{2}+\omega_{0}^{2}}\mp\beta\) and their difference is \(2\beta\) with no approximation. For the amplitude, differentiating \(D^{2}\) and setting the result to zero gives \(-4\omega(\omega_{0}^{2}-\omega^{2})+8\beta^{2}\omega=0\), hence \(\omega_{r}^{2}=\omega_{0}^{2}-2\beta^{2}\). Finally, maximizing the velocity amplitude \(\omega A\) means maximizing \(\omega^{2}/D^{2}\), whose derivative in \(u=\omega^{2}\) has numerator \(\left(\omega_{0}^{2}-u\right)\left(\omega_{0}^{2}+u\right)\); this vanishes at \(u=\omega_{0}^{2}\) exactly.

Derivation. Derives Phenomenon 28.28. Modulating the stiffness of Equation (28.15) gives Mathieu's equation with damping,

The damping can be removed exactly. Putting \(x=\ee^{-\beta t}y\) turns Equation (28.29) into a Mathieu equation with no first derivative:

From Equation (28.29) (substituting \(x=\ee^{-\beta t}y\) and cancelling the common exponential). Hence \(x\) grows if and only if \(y\) grows faster than \(\ee^{\beta t}\). Take \(\beta\ll\omega_{0}\) and set \(\gamma=2\omega_{0}+\varepsilon\) with \(\abs{\varepsilon}\ll\omega_{0}\). Look for a response at half the modulation frequency, \(y=a(t)\cos(\gamma t/2)+b(t)\sin(\gamma t/2)\), with \(a\) and \(b\) varying slowly enough that \(\ddot{a}\) and \(\ddot{b}\) may be dropped. Substituting, using \(\cos\gamma t\cos(\gamma t/2)=\tfrac{1}{2} \left[\cos(\gamma t/2)+\cos(3\gamma t/2)\right]\) and \(\cos\gamma t\sin(\gamma t/2)=\tfrac{1}{2} \left[\sin(3\gamma t/2)-\sin(\gamma t/2)\right]\), and discarding the non-resonant third harmonics, the coefficients of \(\cos(\gamma t/2)\) and \(\sin(\gamma t/2)\) give, with \(\omega_{0}^{2}-\gamma^{2}/4\approx-\omega_{0}\varepsilon\) and \(\gamma\approx2\omega_{0}\),

Eliminating \(b\) gives \(\ddot{a}=s^{2}a\) with

real — and the amplitude therefore exponentially growing — exactly in the band \(\abs{\varepsilon}<h\omega_{0}/2\) centred on \(\gamma=2\omega_{0}\). Restoring the factor \(\ee^{-\beta t}\), the motion grows only if \(s>\beta\), which at the centre of the band (\(\varepsilon=0\), \(s=h\omega_{0}/4\)) reads

Every feature asserted follows: the response is at \(\gamma/2\) by construction and could not have been at \(\gamma\), since a term \(\cos\gamma t\) multiplying \(x\) produces sum and difference frequencies and only the half-frequency component feeds back on itself; the growth is exponential, not the linear-in-\(t\) growth of an undamped forced oscillator; and there is a threshold, absent from ordinary resonance, because the modulation multiplies the amplitude instead of adding to it.

Derivation. Derives Phenomenon 28.30. Superpose two harmonic motions of equal amplitude and frequencies \(\omega_{1}\) and \(\omega_{2}\), and use the sum-to-product identity

The second factor is a rapid oscillation at the mean frequency; the first is a slow one multiplying it. The ear cannot follow the sign of the slow factor, only the size of the envelope, and \(\abs{\cos}\) has half the period of \(\cos\), so the loudness is heard to vary at \(\abs{\omega_{1}-\omega_{2}}\) — twice the frequency of the slow factor itself. For the coupled pendulums, \(\omega_{1}\) and \(\omega_{2}\) are the frequencies \(\omega_{\pm}\) of Equation (28.36); starting one pendulum alone excites the two modes with equal amplitude, and taking their sum and difference gives one bob proportional to the cosine of the slow factor and the other to its sine. Since \(\cos^{2}+\sin^{2}=1\), the total energy is constant and passes entirely from one pendulum to the other and back, with period \(2\pi/\abs{\omega_{+}-\omega_{-}}\). The weaker the coupling, the closer the two mode frequencies and the slower the exchange — which is why the effect requires a weak spring to be visible at all.

Derives Proposition 28.31. Adding and subtracting the two equations of Equation (28.35) gives \(\ddot{s}=-\omega_{0}^{2}s\) for \(s=x_{1}+x_{2}\), the coupling cancelling because the spring exerts equal and opposite forces, and \(\ddot{d}=-(\omega_{0}^{2}+2\kappa/m)d\) for \(d=x_{1}-x_{2}\). Each is Equation (28.2), so Equation (28.36) follows from Proposition 28.4. For \(\kappa\ll m\omega_{0}^{2}\), \(\omega_{+}=\omega_{0}\sqrt{1+2\kappa/m\omega_{0}^{2}} \approx\omega_{0}+\kappa/m\omega_{0}\) by Theorem 7.38.

Derives Theorem 28.33. Because \(M\) is symmetric and positive definite, the bilinear form \(\avg{\vect{x},\vect{y}}_{M}:=\vect{x}\transpose M\vect{y}\) is an inner product on \(\R^{n}\) in the sense of Definition 5.17. Set \(A:=M^{-1}K\), which exists because a positive-definite matrix is invertible. For all \(\vect{x},\vect{y}\),

the middle step because a \(1\times1\) matrix equals its own transpose and \(K\transpose=K\), the last because \(\avg{\cdot,\cdot}_{M}\) is symmetric: \(A\) is self-adjoint with respect to it.

Theorem 5.75 is stated for a complex inner-product space, so apply it to the complexification. Extend the form to \(\C^{n}\) sesquilinearly, \(\avg{\vect{x},\vect{y}}_{M} :=\bar{\vect{x}}\transpose M\vect{y}\), which is an inner product there because \(M\) is real, symmetric and positive definite; the computation above, with each transpose replaced by a conjugate transpose, again shows \(A\) self-adjoint for it, using only that \(M\) and \(K\) are real and symmetric. Theorem 5.75 then gives \(n\) real eigenvalues and an orthonormal eigenbasis of \(\C^{n}\). Those eigenvectors may be taken real: each \(\omega_{r}^{2}\) is real and \(A\) is a real matrix, so \(A-\omega_{r}^{2}\identity\) is real and its kernel is spanned by real vectors, and applying Gram–Schmidt (Proposition 5.26) in the \(M\)-inner product within each such real kernel — eigenspaces for distinct eigenvalues being already orthogonal — yields a real \(M\)-orthonormal eigenbasis of \(\R^{n}\). Since \(A\vect{u}=\omega^{2}\vect{u}\) is equivalent to \(\left(K-\omega^{2}M\right)\vect{u}=\vect{0}\), and the eigenvalues of \(A\) are the roots of Equation (28.39) by Theorem 5.67, parts (i) and (ii) follow; the second identity of Equation (28.40) is the first one with \(K\vect{u}^{(s)}=\omega_{s}^{2}M\vect{u}^{(s)}\) substituted. For (iii), \(U\transpose KU=\diag(\omega_{r}^{2})\) is a congruence, so by Sylvester's law of inertia Theorem 5.112 the number of positive, zero and negative \(\omega_{r}^{2}\) equals the corresponding count for \(K\) itself; \(K\) is positive definite exactly when all \(n\) are positive.

Derives Proposition 28.36. Substituting \(\vect{\eta}=U\vect{\xi}\) into Equation (28.37) gives \(T=\tfrac{1}{2}\dot{\vect{\xi}}\transpose U\transpose MU \dot{\vect{\xi}}\) and the analogous expression for \(V\), and Equation (28.40) reduces \(U\transpose MU\) to \(\identity\) and \(U\transpose KU\) to the diagonal matrix of the \(\omega_{r}^{2}\). The Euler–Lagrange equations of the resulting Lagrangian are the \(n\) separate equations shown, each of the form Equation (28.2), so by Proposition 28.5 each carries a conserved energy and the total is their sum. The dimension follows from \(T\) being an energy and \(\dot{\xi}_{r}^{2}\) carrying the whole of it.

Derives Proposition 28.38. Expand \(\vect{u}=\sum_{r}c_{r}\vect{u}^{(r)}\) in the \(M\)-orthonormal basis of Theorem 28.33. Then \(\vect{u}\transpose M\vect{u}=\sum_{r}c_{r}^{2}\) and \(\vect{u}\transpose K\vect{u}=\sum_{r}\omega_{r}^{2}c_{r}^{2}\) by Equation (28.40), so the quotient is a weighted average of the \(\omega_{r}^{2}\) with non-negative weights \(c_{r}^{2}\), hence lies between the least and the greatest of them, and attains the least only when every \(c_{r}\) with \(\omega_{r}^{2}>\omega_{1}^{2}\) vanishes.

Derives Proposition 28.40. For small displacements the string segment from bead \(n\) to bead \(n+1\) makes an angle whose sine is \((y_{n+1}-y_{n})/a\) to first order, and pulls bead \(n\) transversely with force \(T(y_{n+1}-y_{n})/a\); adding the pull of the segment on the other side gives Equation (28.44). Substituting \(y_{n}=u\sin(kna)\ee^{-\ii\omega t}\), using \(\sin(k(n+1)a)+\sin(k(n-1)a)=2\cos(ka)\sin(kna)\), and cancelling the common factor \(u\sin(kna)\ee^{-\ii\omega t}\),

which is the dispersion relation in Equation (28.45). The form \(\sin(kna)\) already satisfies \(y_{0}=0\), and \(y_{N+1}=0\) requires \(\sin(k(N+1)a)=0\), that is \(kL=r\pi\). Values \(r\) and \(r+2(N+1)\) give the same set of bead displacements, and \(r=N+1\) gives none at all, so exactly \(r=1,\dots,N\) are distinct and non-trivial — \(N\) modes for \(N\) degrees of freedom, as Theorem 28.33 requires.

Derives Proposition 28.41. Write \(y_{n}(t)=y(na,t)\) for a twice continuously differentiable \(y\). Taylor's theorem (Theorem 7.38) gives \(y_{n\pm1}=y\pm a\,\pp_{x}y+\tfrac{1}{2}a^{2}\pp_{x}^{2}y +O(a^{3})\), so the bracket in Equation (28.44) is \(a^{2}\pp_{x}^{2}y+O(a^{4})\), the odd orders cancelling. Dividing by \(m=\mu a\),

which is Equation (28.46) in the limit. For the frequencies, fix \(r\); then \(k_{r}a=r\pi a/L\to0\), so \(2\omega_{c}\sin(k_{r}a/2)\to\omega_{c}k_{r}a=k_{r}\sqrt{T/\mu}\), using \(\omega_{c}a=a\sqrt{T/\left(ma\right)}=\sqrt{Ta/m}=\sqrt{T/\mu}\). The limit is uniform only on the low modes: for \(r\) comparable with \(N\) the sine cannot be linearized, which is the statement that a continuum has no cutoff and a lattice does.

Derives Proposition 28.42. Take the element between \(x\) and \(x+\dd x\), of mass \(\mu\,\dd x\). The tension pulls along the string at each end; its transverse component is \(T\sin\theta\approx T\tan\theta=T\,\pp_{x}y\) to first order in the slope, and the net transverse force is the difference of that quantity at the two ends, \(T\,\pp_{x}^{2}y\,\dd x\). Newton's second law gives \(\mu\,\dd x\,\pp_{t}^{2}y=T\,\pp_{x}^{2}y\,\dd x\), which is Equation (28.46). The longitudinal components cancel to the same order, and the tension may be treated as constant along the string because the fractional stretch is \(O((\pp_{x}y)^{2})\). For the variational route, the kinetic energy of the element is \(\tfrac{1}{2}\mu(\pp_{t}y)^{2}\dd x\) and the work done in stretching it from \(\dd x\) to \(\dd x\sqrt{1+(\pp_{x}y)^{2}}\) against constant tension is \(\tfrac{1}{2}T(\pp_{x}y)^{2}\dd x\) to the same order; Theorem 16.36 applied to Equation (28.47) returns Equation (28.46). That Equation (28.47) is the limit of Equation (28.37) is Proposition 28.41 applied term by term.

Derives Proposition 28.44. \(\pp_{t}^{2}-v^{2}\pp_{x}^{2}\) is a linear operator (Definition 10.2), so it takes a linear combination to the same combination of the images, each of which vanishes. The boundary conditions are likewise linear and homogeneous.

Derivation. Derives Phenomenon 28.45. Grant the wave equation of Section 28.4.1, \(\pp_{t}^{2}y=v^{2}\pp_{x}^{2}y\) with \(v=\sqrt{T/\mu}\), on the interval \(0\leq x\leq L\), with the clamped ends imposing \(y(0,t)=y(L,t)=0\). Separating variables (Section 10.2), \(y=X(x)\Theta(t)\), gives \(X''/X=\Theta''/(v^{2}\Theta)\); the two sides depend on different variables, so both equal one constant (Lemma 10.28), which must be negative if \(\Theta\) is to stay bounded — write it \(-k^{2}\). Then \(X=a\sin kx+b\cos kx\); the condition \(X(0)=0\) forces \(b=0\), and \(X(L)=0\) then forces \(\sin kL=0\), that is

The temporal factor oscillates at \(\omega_{n}=v k_{n}\), so \(f_{n}=\omega_{n}/2\pi=n v/(2L)\), which is Equation (28.49); the three empirical rules read off it at once, \(f_{1}\propto1/L\), \(f_{1}\propto\sqrt{T}\) and \(f_{1}\propto1/\sqrt{\mu}\). That the modes are the only solutions, and that a general motion is a superposition of them, is the Sturm–Liouville theory of Ordinary Differential Equations and Sturm–Liouville Theory: the spatial problem is Section 9.7 with \(p=r=1\) and \(q=0\), its eigenfunctions are mutually orthogonal by Theorem 9.57, and they are complete by Theorem 17.34. That the overtones are exact integer multiples of the fundamental is special to the non-dispersive string, in which \(\omega=vk\) is linear in \(k\): a stiff bar, a drumhead or a bell has the same discreteness of modes but an inharmonic spectrum, and does not sound a definite pitch in the same way.

Derives Proposition 28.46. Take the end at \(x=0\) and write the field as \(f(x-vt)+g(x+vt)\), with \(f\) the incoming pulse. A clamped end imposes \(y(0,t)=0\) for all \(t\), hence \(g(s)=-f(-s)\): the reflected pulse is the incident one reversed in space and in sign. A free end can support no transverse force, so \(T\pp_{x}y(0,t)=0\), hence \(g'(s)=f'(-s)\) and \(g(s)=f(-s)\) up to a constant fixed by \(y\to0\) far away: reversed in space and not in sign.

Derives Proposition 28.47. Each term of Equation (28.52) solves Equation (28.46) and the boundary conditions, by Phenomenon 28.45, and the sum does too by Proposition 28.44 and termwise differentiation (Theorem 7.51) where the series permits it. Setting \(t=0\) gives \(y_{0}=\sum a_{n}\sin k_{n}x\) and \(v_{0}=\sum b_{n}\omega_{n}\sin k_{n}x\), which are sine series on \([0,L]\); Equation (28.53) is the Euler–Fourier formula Theorem 17.3 for their coefficients, using the orthogonality \(\int_{0}^{L}\sin k_{n}x\sin k_{m}x\,\dd x =(L/2)\delta_{nm}\) of Theorem 9.57. Completeness of the system (Theorem 17.34) is what makes the expansion possible for every square-integrable datum, and uniqueness for the wave equation with given data (Definition 10.24, Theorem 10.55) makes Equation (28.52) the solution and not merely a solution.

Derives Proposition 28.48. Write \(y=\sum_{n}q_{n}(t)\sin k_{n}x\), and \(T_{\text{ten}}\) for the tension, since here \(T\) is the kinetic energy (Notation 28.1). Then, by the orthogonality used in Proposition 28.47,

the second using \(T_{\text{ten}}k_{n}^{2}=\mu\omega_{n}^{2}\). This is Parseval's identity Theorem 17.34 for the sine system, and it is the continuum form of Equation (28.41). With \(q_{n}=a_{n}\cos\omega_{n}t+b_{n}\sin\omega_{n}t\) each bracket \(\dot{q}_{n}^{2}+\omega_{n}^{2}q_{n}^{2}\) equals \(\omega_{n}^{2}(a_{n}^{2}+b_{n}^{2})\) by Proposition 28.5, giving Equation (28.54).

Derivation. Derives Phenomenon 28.50. Linearize the equations of a compressible fluid about rest, writing \(\rho=\rho_{0}+\rho'\) and \(p=p_{0}+p'\) with a velocity \(u\) that is first order, and dropping all products of primed quantities. Conservation of mass and Newton's second law for a fluid element give, in one dimension,

Differentiating the first with respect to \(t\) and the second with respect to \(x\) and subtracting eliminates \(u\):

a wave equation with speed \(v^{2}=\dd p/\dd\rho\). The derivative must be taken along the thermodynamic path the gas actually follows, and this is where the history turns. Newton took it isothermally: with \(p\propto\rho\) at fixed temperature, \(v_{T}=\sqrt{p_{0}/\rho_{0}}\), which for \(p_{0}=101325\,\mathrm{Pa}\) and \(\rho_{0}=1.204\,\mathrm{kg}/\mathrm{m}^{3}\) gives about \(290\,\mathrm{m}/\mathrm{s}\) — some \(15\,\mathrm{\%}\) below the measured value. But a sound wave compresses the air far faster than heat can diffuse across a wavelength, so the correct path is the adiabatic one, \(p\propto\rho^{\gamma}\), whence

and with the diatomic value \(\gamma=1.4\) this is \(343\,\mathrm{m}/\mathrm{s}\), the observed speed. Because \(p_{0}/\rho_{0}\) equals \(RT/M\) for an ideal gas, the speed depends on temperature and on composition but not on pressure; and because Equation (28.56) has a \(k\)-independent coefficient, it carries no dispersion — both exactly as observed. The thermodynamic input is developed in Classical Thermodynamics, and the fluid equations linearized here in Fluid Dynamics.

Derivation. Derives Phenomenon 28.53. The small transverse displacement \(w(\vect{x},t)\) of a thin elastic plate obeys a linear equation \(\rho_{s}\,\pp_{t}^{2}w+\mathcal{A}w=0\), where \(\mathcal{A}\) is a linear spatial operator, self-adjoint under the plate's boundary conditions, determined by the elastic moduli (Continuum Mechanics and Elasticity); its explicit form is not needed here. Time separation (Proposition 10.29) reduces the problem to the eigenvalue problem \(\mathcal{A}w_{r}=\rho_{s}\omega_{r}^{2}w_{r}\) on a bounded domain, which for a self-adjoint operator has a discrete spectrum with mutually orthogonal eigenfunctions (Theorem 9.57,Section 10.2.2, Theorem 12.44) — so the plate has isolated natural frequencies and, by Phenomenon 28.23 applied to each of them, responds appreciably only near one of them. At \(\omega=\omega_{r}\) the response is essentially the single eigenfunction \(w_{r}\), whose zero set is a fixed family of curves in the plate, unmoving in time because \(w=w_{r}(\vect{x})\cos\omega_{r}t\) factorizes. A grain of sand on the plate is thrown clear wherever the vertical acceleration \(\omega_{r}^{2}\abs{w_{r}}\) exceeds \(g\) and is not thrown clear on the zero set; each bounce lands it at random nearby, so it performs a random walk that is absorbed on the nodal curves, where it accumulates. The pattern is therefore the nodal set of \(w_{r}\), and it changes discontinuously with \(r\) because the eigenfunctions do.

Derives Proposition 28.54. The air in the neck moves as a slug of mass \(\rho_{0}SL'\); the air in the cavity is too slow-moving to contribute inertia but supplies the stiffness. Pushing the slug inward a distance \(x\) reduces the cavity volume by \(Sx\), and by the adiabatic relation used in Equation (28.57) the pressure rises by \(\delta p=-\gamma p_{0}\,\delta V/V=\gamma p_{0}Sx/V\). The restoring force on the slug is \(-S\,\delta p=-\gamma p_{0}S^{2}x/V\), so the effective spring constant is \(\gamma p_{0}S^{2}/V\) and

using Equation (28.57). The approximation is that the wavelength greatly exceeds every dimension of the resonator, so that the cavity pressure is uniform and the neck flow incompressible; \(L'\) exceeds the geometric neck length by an end correction of order the neck radius, for the same reason a pipe does.

Derivation. Derives Phenomenon 28.56. Sound is a disturbance of the air, and Equation (28.56) holds in the frame in which the air is at rest; that frame is therefore singled out, and the two motions are physically different situations. Let the source emit at frequency \(f\) and move toward the listener at \(u_{s}\), the listener move toward the source at \(u_{o}\), and the speed of sound be \(v\). In one period \(1/f\) the source advances \(u_{s}/f\) toward the wavefront it emitted a period earlier, so the wavelength ahead of it is compressed to \(\lambda'=(v-u_{s})/f\). The listener, moving into the wave, meets crests at the closing speed \(v+u_{o}\), so

The two velocities enter differently: \(u_{o}\) in the numerator, where it changes the rate of arrival, and \(u_{s}\) in the denominator, where it changes the spacing. Expanding for \(u\ll v\), both give \(f'\approx f(1+u/v)\), so the asymmetry is second order and appears only at \(O(u^{2}/v^{2})\) — which is why it took a moving train to find it, and why no first-order acoustic experiment could have distinguished the cases.

Derivation. Derives Phenomenon 28.59. Superpose two travelling waves of nearly equal wavenumbers \(k\pm\Delta k/2\) and the corresponding frequencies \(\omega\pm\Delta\omega/2\). The identity Equation (28.34), applied with the phase \(kx-\omega t\) in place of \(\omega t\), gives

whose second factor — the carrier — has crests moving at the phase velocity \(v_{p}=\omega/k\), while the first — the envelope — has its maxima moving at \(\Delta\omega/\Delta k\), which in the limit of a narrow band is

The two coincide exactly when \(\omega=vk\) is linear in \(k\), which is the non-dispersive case. For gravity waves on deep water the dispersion relation is \(\omega=\sqrt{gk}\), derived in Fluid Dynamics, so

the observed factor of two. Spreading follows from carrying the same expansion one order further, which is Proposition 28.60: components of a packet with different \(k\) travel at different \(v_{g}\), so the packet's spatial extent grows at a rate set by \(\dd^{2}\omega/\dd k^{2}\), which vanishes identically in the non-dispersive case. The discrete chain of Proposition 28.40 is the mechanical prototype of a dispersive medium, and the same kinematics governs the matter waves of Matter Waves.

Derives Proposition 28.60. Substituting the expansion and setting \(q=k-k_{0}\),

a carrier times an envelope. With \(\omega''\) dropped the envelope is a function of \(x-v_{g}t\) alone — rigid translation at \(v_{g}\), which is the first claim, and which is Proposition 17.30 (a linear phase in \(q\) is a translation in \(x\)). Retaining \(\omega''\) multiplies the transform by a Gaussian of imaginary width; for \(A\propto\ee^{-\sigma_{0}^{2}q^{2}/2}\) the two Gaussians combine to one of complex variance \(\sigma_{0}^{2}+\ii\omega''t\), whose modulus in \(x\) has width \(\abs{\sigma_{0}^{2}+\ii\omega''t}/\sigma_{0}\), which is Equation (28.63). Since \(\sigma(t)\) is strictly increasing unless \(\omega''=0\), spreading is equivalent to curvature of \(\omega(k)\).

Derives Proposition 28.63. Differentiating \(u\) and using Equation (28.46),

a conservation law in the sense of Definition 10.93; the same statement in integral form over a backward cone is Proposition 10.57. For a wave travelling in the \(+x\) direction, \(\pp_{t}y=-v\,\pp_{x}y\) by Equation (28.48), so \(\tfrac{1}{2}\mu(\pp_{t}y)^{2}=\tfrac{1}{2}\mu v^{2}(\pp_{x}y)^{2} =\tfrac{1}{2}T(\pp_{x}y)^{2}\): the kinetic and potential densities are equal pointwise, not merely on average as in Proposition 28.5. With \(y=A\cos(kx-\omega t)\), \(P=TA^{2}k\omega\sin^{2}(kx-\omega t)\), whose mean is \(\tfrac{1}{2}TkA^{2}\omega=\tfrac{1}{2}\mu v\omega^{2}A^{2}\) using \(Tk=\mu v\omega\). The dimension of \(Z=\sqrt{T\mu}\) is \(\sqrt{\mathrm{N}\times\mathrm{kg}/\mathrm{m}}=\mathrm{kg}/\mathrm{s}\).

Derives Proposition 28.64. Write the field as \(\ee^{\ii k_{1}x}+r\ee^{-\ii k_{1}x}\) for \(x<0\) and \(\tau\ee^{\ii k_{2}x}\) for \(x>0\), common time factor suppressed. Continuity of the string requires \(1+r=\tau\); continuity of the transverse force \(T\,\pp_{x}y\), which must hold because a massless junction cannot sustain a net force, requires \(k_{1}\left(1-r\right)=k_{2}\tau\). Since the tension is common and \(k=\omega/v\) with \(Z=T/v\), one has \(k_{i}=\omega Z_{i}/T\), so the second condition reads \(Z_{1}(1-r)=Z_{2}\tau\). Solving the pair gives Equation (28.66). The incident power is \(\tfrac{1}{2}Z_{1}\omega^{2}\) by Equation (28.65), the reflected \(\tfrac{1}{2}Z_{1}\omega^{2}r^{2}\) and the transmitted \(\tfrac{1}{2}Z_{2}\omega^{2}\tau^{2}\); the ratios are as stated, and \(\left(Z_{1}-Z_{2}\right)^{2}+4Z_{1}Z_{2} =\left(Z_{1}+Z_{2}\right)^{2}\) is the conservation of energy.

Derivation. Derives Phenomenon 28.65. Expand the exact pendulum equation Equation (28.8) one order beyond the linearization, \(\sin\theta=\theta-\theta^{3}/6+O(\theta^{5})\), giving

which is Equation (28.70) with \(\epsilon=-\omega_{0}^{2}/6\). Substituting that value into Equation (28.81) gives

and \(T=2\pi/\omega\) is Equation (28.67) to the order retained. The sign is the physical content: \(\epsilon<0\) because the restoring moment \(\sin\theta\) falls short of \(\theta\), so the pendulum is a softening oscillator and its period lengthens. Numerically, \(\theta_{0}=7^\circ=0.1222\,\mathrm{rad}\) gives \(\theta_{0}^{2}/16=9.3\times 10^{-4}\) and \(\theta_{0}=30^\circ=0.5236\,\mathrm{rad}\) gives \(1.7\times 10^{-2}\), as asserted.

Derives Proposition 28.68. Put \(\omega=\omega_{0}+\varepsilon_{d}\) with \(\abs{\varepsilon_{d}}\ll\omega_{0}\) and seek \(x=A\cos(\omega t-\delta)\) with \(A\) and \(\delta\) slowly varying. Substituting into Equation (28.69), using \(\cos^{3}\psi=\tfrac{3}{4}\cos\psi+\tfrac{1}{4}\cos3\psi\) and discarding the third harmonic — which is non-resonant and enters only at higher order — the coefficients of \(\cos(\omega t-\delta)\) and \(\sin(\omega t-\delta)\) give, after dividing by \(2\omega_{0}\),

Squaring and adding eliminates \(\delta\) and gives Equation (28.71). At fixed \(F_{0}\) the amplitude is greatest where the square bracket is least, that is where the first bracket vanishes, which is the backbone; and Equation (28.71) is a cubic in \(A^{2}\) whose discriminant changes sign as \(F_{0}\) grows, so three real positive roots appear above some \(F_{0}\) and over some interval of \(\omega\). Setting \(\epsilon=0\) recovers Equation (28.24) near resonance, and setting \(\beta=0\) and \(F_{0}=0\) recovers Equation (28.81).

Derives Proposition 28.69. Stability is a statement about the approach to a steady state, so the amplitude and phase must be allowed to move; the harmonic balance of Proposition 28.68 assumed them constant and cannot answer the question. Replace it by the exact change of variables

the second line being a constraint on the pair \((u,v)\), equivalent to \(\dot{u}\cos\omega t+\dot{v}\sin\omega t=0\). Differentiating the second of Equation (28.74) gives \(\ddot{x}=\omega\left(-\dot{u}\sin\omega t +\dot{v}\cos\omega t\right)-\omega^{2}x\), so Equation (28.69) reads

Solving Equation (28.75) together with the constraint — two linear equations for \(\dot{u}\) and \(\dot{v}\), of determinant \(\omega\) — gives \(\dot{u}=-S\sin(\omega t)/\omega\) and \(\dot{v}=S\cos(\omega t)/\omega\), still exactly.

Now average over one drive period holding \(u\) and \(v\) fixed, which is legitimate because Equation (28.69) makes them move on the slow time scales \(\beta^{-1}\) and \(\omega_{0}/\abs{\epsilon}A^{2}\), both long compared with \(2\pi/\omega\) under the hypotheses of Proposition 28.68. With \(A^{2}=u^{2}+v^{2}\) the required averages are \(\avg{x\sin\omega t}=v/2\), \(\avg{x\cos\omega t}=u/2\), \(\avg{\dot{x}\sin\omega t}=-\omega u/2\), \(\avg{\dot{x}\cos\omega t}=\omega v/2\), \(\avg{\cos^{2}\omega t}=1/2\), and, since \(x^{3}=\tfrac{3}{4}A^{2}x+(\text{third harmonic})\) by the same identity used above, \(\avg{x^{3}\sin\omega t}=\tfrac{3}{8}A^{2}v\) and \(\avg{x^{3}\cos\omega t}=\tfrac{3}{8}A^{2}u\). Using \(\left(\omega^{2}-\omega_{0}^{2}\right)/2\omega\approx\varepsilon_{d}\) and \(\omega\approx\omega_{0}\) elsewhere,

From Equations (28.69) and (28.75) (averaging the exact variation-of-parameters equations over one drive period). Setting \(\dot{u}=\dot{v}=0\) and eliminating the phase reproduces \(A^{2}\left(\Lambda^{2}+\beta^{2}\right)=f^{2}\), which is Equation (28.71): the fixed points of Equation (28.76) are exactly the steady states already found, and nothing has been changed except that they may now be perturbed.

Linearize Equation (28.76) about a fixed point. Since \(\pp\Lambda/\pp u=-2\kappa u\) and \(\pp\Lambda/\pp v=-2\kappa v\), the Jacobian of Definition 32.12 is

whose trace is \(-2\beta<0\) whatever the state, and whose determinant is

Differentiating Equation (28.72) gives \(G'(s)=\left(\varepsilon_{d}-\kappa s\right)^{2}+\beta^{2} -2\kappa s\left(\varepsilon_{d}-\kappa s\right) =\Lambda^{2}+\beta^{2}-2\kappa\Lambda s\), so

which is the identity announced. The rest is bookkeeping. A real \(2\times2\) matrix with negative trace has both eigenvalues in the left half plane when its determinant is positive and one of each sign when it is negative, and away from \(\det J=0\) the two eigenvalues are distinct or the matrix is a multiple of the identity, so Theorem 32.13 applies: the state is asymptotically stable exactly when \(G'(s)>0\) and is a saddle when \(G'(s)<0\).

\(G\) has leading coefficient \(\kappa^{2}>0\) and \(G(0)=-f^{2}<0\), so it crosses zero once with positive slope, or three times with slopes positive, negative and positive in order of increasing \(s\). That is the branch structure: outer branches stable, middle branch a saddle, and the transitions occur where a root is double, \(G=G'=0\). Those are the turning points of the response curve and hence the two jump frequencies; parametrized by \(\Lambda\) they are \(s=\left(\Lambda^{2}+\beta^{2}\right)/2\kappa\Lambda\) with \(\left(\Lambda^{2}+\beta^{2}\right)^{2}=2\kappa\Lambda f^{2}\) and \(\varepsilon_{d}=\left(3\Lambda^{2}+\beta^{2}\right)/2\Lambda\), which has solutions only for \(\kappa\Lambda>0\) — the folds lie on the side towards which the backbone leans, high for a hardening spring and low for a softening one.

The two folds merge when \(G''\) vanishes with them. From \(G=\kappa^{2}s^{3}-2\kappa\varepsilon_{d}s^{2} +\left(\varepsilon_{d}^{2}+\beta^{2}\right)s-f^{2}\), \(G''=6\kappa^{2}s-4\kappa\varepsilon_{d}\) vanishes at \(s=2\varepsilon_{d}/3\kappa\), where \(\Lambda=\varepsilon_{d}/3\) and \(G'=\beta^{2}-\varepsilon_{d}^{2}/3\); requiring \(G'=0\) as well gives \(\varepsilon_{d}=\pm\sqrt{3}\beta\), and positivity of \(s\) selects the sign of \(\kappa\), that is of \(\epsilon\). At that point \(s=2\beta/\sqrt{3}\abs{\kappa}\) and \(f^{2}=s\left(\Lambda^{2}+\beta^{2}\right) =8\beta^{3}/3\sqrt{3}\abs{\kappa}\), and restoring \(\kappa=3\epsilon/8\omega_{0}\) and \(f=F_{0}/2m\omega_{0}\) gives \(F_{0}^{2}=256\,m^{2}\omega_{0}^{3}\beta^{3}/9\sqrt{3}\abs{\epsilon}\), which is Equation (28.73). Below that drive \(G\) is monotone and the response is single-valued; above it the two folds separate and the hysteresis loop opens.

Derives Proposition 28.71. At order \(\epsilon\), Equation (28.70) gives \(\ddot{x}_{1}+\omega_{0}^{2}x_{1}=-x_{0}^{3}\). With \(x_{0}=A\cos\omega_{0}t\) and \(\cos^{3}\psi=\tfrac{3}{4}\cos\psi+\tfrac{1}{4}\cos3\psi\), the right side is \(-A^{3}\left(\tfrac{3}{4}\cos\omega_{0}t +\tfrac{1}{4}\cos3\omega_{0}t\right)\). The second piece drives the oscillator off resonance and contributes the bounded terms shown. The first drives it exactly on resonance, and the particular solution of \(\ddot{y}+\omega_{0}^{2}y=C\cos\omega_{0}t\) is \(y=\left(C/2\omega_{0}\right)t\sin\omega_{0}t\), which with \(C=-3A^{3}/4\) gives the first term of Equation (28.80). Since \(\epsilon x_{1}\) must stay small against \(x_{0}\) for the expansion to mean anything, the bound is \(\epsilon A^{2}t/\omega_{0}\ll1\).

Derives Theorem 28.72. Introduce the stretched time \(\tau=\omega t\) with \(\omega\) unknown, and expand both the solution and the frequency,

requiring every \(x_{j}\) to be \(2\pi\)-periodic in \(\tau\) — which is the whole of the method: periodicity in \(\tau\) is imposed, and \(\omega\) is whatever makes it possible. Since \(\dd/\dd t=\omega\,\dd/\dd\tau\), Equation (28.70) becomes \(\omega^{2}x''+\omega_{0}^{2}x+\epsilon x^{3}=0\) with the prime denoting \(\dd/\dd\tau\). At order \(\epsilon^{0}\), \(x_{0}''+x_{0}=0\), so \(x_{0}=A\cos\tau\) after choosing the origin of time. At order \(\epsilon^{1}\),

A term in \(\cos\tau\) on the right drives the left-hand operator on resonance and produces exactly the secular term of Proposition 28.71, destroying the periodicity demanded. It is absent if and only if \(2\omega_{0}\omega_{1}A=\tfrac{3}{4}A^{3}\), that is \(\omega_{1}=3A^{2}/8\omega_{0}\), which is Equation (28.81). With that choice the remaining equation \(x_{1}''+x_{1}=-\left(A^{3}/4\omega_{0}^{2}\right)\cos3\tau\) has the bounded periodic solution \(x_{1}=\left(A^{3}/32\omega_{0}^{2}\right)\cos3\tau\), to which a multiple of \(\cos\tau\) may be added; fixing the amplitude of the fundamental at \(A\) makes it \(\left(A^{3}/32\omega_{0}^{2}\right)(\cos3\tau-\cos\tau)\).