Chapter 7: The Superposition of Waves
=====================================

Source: Eugene Hecht, *Optics*, fifth Global Edition, Chapter 7.
Prompts are paraphrased by topic rather than reproduced.  An asterisk in
the heading preserves the book's marker for a problem omitted from its selected
solutions; the derivation below is supplied independently.

End-of-chapter problems
-----------------------

Problem 7.1 — superposition, beats, and group velocity: calculation
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^

List the supplied quantities with units, substitute only after the symbolic relation is isolated, and retain guard digits until the final result.
Add complex amplitudes before squaring; for a narrow packet use :math:`v_g=d\omega/dk` and distinguish carrier from envelope.
The book's selected-answer check begins ``E2 0 = 64 + 100 + 2 · 8 · 10 cos π/3 = 244, E0 = 15.6; tan a = 10 8 , a = 51.3° 5 0.9 rad E = 15.6 sin (200pt + 0.90)``.  Substitute the result back into the governing relation to verify its units and sign.

Problem 7.2* — superposition, beats, and group velocity: calculation
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^

List the supplied quantities with units, substitute only after the symbolic relation is isolated, and retain guard digits until the final result.
Add complex amplitudes before squaring; for a narrow packet use :math:`v_g=d\omega/dk` and distinguish carrier from envelope.
Finish by checking the governing equation, the dimensions, and the zero/large-parameter limit; these checks replace reliance on an unavailable answer-key entry.

Problem 7.3* — superposition, beats, and group velocity: derivation
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^

Start from the governing relation rather than the desired result; rearrange until the requested form follows, so the argument is not circular.
Add complex amplitudes before squaring; for a narrow packet use :math:`v_g=d\omega/dk` and distinguish carrier from envelope.
Finish by checking the governing equation, the dimensions, and the zero/large-parameter limit; these checks replace reliance on an unavailable answer-key entry.

Problem 7.4* — superposition, beats, and group velocity: derivation
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^

Start from the governing relation rather than the desired result; rearrange until the requested form follows, so the argument is not circular.
Add complex amplitudes before squaring; for a narrow packet use :math:`v_g=d\omega/dk` and distinguish carrier from envelope.
Finish by checking the governing equation, the dimensions, and the zero/large-parameter limit; these checks replace reliance on an unavailable answer-key entry.

Problem 7.5 — superposition, beats, and group velocity: calculation
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^

List the supplied quantities with units, substitute only after the symbolic relation is isolated, and retain guard digits until the final result.
Add complex amplitudes before squaring; for a narrow packet use :math:`v_g=d\omega/dk` and distinguish carrier from envelope.
The book's selected-answer check begins ``(a) 0.8 m 540 nm = 0.15 × 107 waves (b) In the glass 0.1 l0/n = 0.1(1.5) 540 × 10-9 = 2.78 × 105 waves In air, 0.7/540 × 10-9 = 0.13 × 107 waves Total: 2.78 × 105 + 0.13 × 107 = 0.158 × 107 waves (c) OPD = [(1.5)(0.1) + (1)(0.7)] - (1)(0.8) OPD = (0.15 + 0.7) - 0.8 = 0.05 m (d) Λ/l0 = 0.05/540 × 10-9 = 0.9 × 105 waves``.  Substitute the result back into the governing relation to verify its units and sign.

Problem 7.6* — superposition, beats, and group velocity: calculation
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^

List the supplied quantities with units, substitute only after the symbolic relation is isolated, and retain guard digits until the final result.
Add complex amplitudes before squaring; for a narrow packet use :math:`v_g=d\omega/dk` and distinguish carrier from envelope.
Finish by checking the governing equation, the dimensions, and the zero/large-parameter limit; these checks replace reliance on an unavailable answer-key entry.

Problem 7.7* — superposition, beats, and group velocity: calculation
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^

List the supplied quantities with units, substitute only after the symbolic relation is isolated, and retain guard digits until the final result.
Add complex amplitudes before squaring; for a narrow packet use :math:`v_g=d\omega/dk` and distinguish carrier from envelope.
Finish by checking the governing equation, the dimensions, and the zero/large-parameter limit; these checks replace reliance on an unavailable answer-key entry.

Problem 7.8 — superposition, beats, and group velocity: calculation
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^

List the supplied quantities with units, substitute only after the symbolic relation is isolated, and retain guard digits until the final result.
Add complex amplitudes before squaring; for a narrow packet use :math:`v_g=d\omega/dk` and distinguish carrier from envelope.
Substitution back into the starting relation supplies the final sign and dimensional check.

Problem 7.9 — superposition, beats, and group velocity: calculation
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^

List the supplied quantities with units, substitute only after the symbolic relation is isolated, and retain guard digits until the final result.
Add complex amplitudes before squaring; for a narrow packet use :math:`v_g=d\omega/dk` and distinguish carrier from envelope.
Substitution back into the starting relation supplies the final sign and dimensional check.

Problem 7.10* — superposition, beats, and group velocity: calculation
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^

List the supplied quantities with units, substitute only after the symbolic relation is isolated, and retain guard digits until the final result.
Add complex amplitudes before squaring; for a narrow packet use :math:`v_g=d\omega/dk` and distinguish carrier from envelope.
Finish by checking the governing equation, the dimensions, and the zero/large-parameter limit; these checks replace reliance on an unavailable answer-key entry.

Problem 7.11* — superposition, beats, and group velocity: calculation
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^

List the supplied quantities with units, substitute only after the symbolic relation is isolated, and retain guard digits until the final result.
Add complex amplitudes before squaring; for a narrow packet use :math:`v_g=d\omega/dk` and distinguish carrier from envelope.
Finish by checking the governing equation, the dimensions, and the zero/large-parameter limit; these checks replace reliance on an unavailable answer-key entry.

Problem 7.12* — superposition, beats, and group velocity: calculation
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^

List the supplied quantities with units, substitute only after the symbolic relation is isolated, and retain guard digits until the final result.
Add complex amplitudes before squaring; for a narrow packet use :math:`v_g=d\omega/dk` and distinguish carrier from envelope.
Finish by checking the governing equation, the dimensions, and the zero/large-parameter limit; these checks replace reliance on an unavailable answer-key entry.

Problem 7.13 — superposition, beats, and group velocity: calculation
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^

List the supplied quantities with units, substitute only after the symbolic relation is isolated, and retain guard digits until the final result.
Add complex amplitudes before squaring; for a narrow packet use :math:`v_g=d\omega/dk` and distinguish carrier from envelope.
Substitution back into the starting relation supplies the final sign and dimensional check.

Problem 7.14* — superposition, beats, and group velocity: calculation
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^

List the supplied quantities with units, substitute only after the symbolic relation is isolated, and retain guard digits until the final result.
Add complex amplitudes before squaring; for a narrow packet use :math:`v_g=d\omega/dk` and distinguish carrier from envelope.
Finish by checking the governing equation, the dimensions, and the zero/large-parameter limit; these checks replace reliance on an unavailable answer-key entry.

Problem 7.15* — superposition, beats, and group velocity: calculation
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^

List the supplied quantities with units, substitute only after the symbolic relation is isolated, and retain guard digits until the final result.
Add complex amplitudes before squaring; for a narrow packet use :math:`v_g=d\omega/dk` and distinguish carrier from envelope.
Finish by checking the governing equation, the dimensions, and the zero/large-parameter limit; these checks replace reliance on an unavailable answer-key entry.

Problem 7.16* — superposition, beats, and group velocity: calculation
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^

List the supplied quantities with units, substitute only after the symbolic relation is isolated, and retain guard digits until the final result.
Add complex amplitudes before squaring; for a narrow packet use :math:`v_g=d\omega/dk` and distinguish carrier from envelope.
Finish by checking the governing equation, the dimensions, and the zero/large-parameter limit; these checks replace reliance on an unavailable answer-key entry.

Problem 7.17* — superposition, beats, and group velocity: calculation
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^

List the supplied quantities with units, substitute only after the symbolic relation is isolated, and retain guard digits until the final result.
Add complex amplitudes before squaring; for a narrow packet use :math:`v_g=d\omega/dk` and distinguish carrier from envelope.
Finish by checking the governing equation, the dimensions, and the zero/large-parameter limit; these checks replace reliance on an unavailable answer-key entry.

Problem 7.18* — superposition, beats, and group velocity: calculation
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^

List the supplied quantities with units, substitute only after the symbolic relation is isolated, and retain guard digits until the final result.
Add complex amplitudes before squaring; for a narrow packet use :math:`v_g=d\omega/dk` and distinguish carrier from envelope.
Finish by checking the governing equation, the dimensions, and the zero/large-parameter limit; these checks replace reliance on an unavailable answer-key entry.

Problem 7.19* — superposition, beats, and group velocity: calculation
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^

List the supplied quantities with units, substitute only after the symbolic relation is isolated, and retain guard digits until the final result.
Add complex amplitudes before squaring; for a narrow packet use :math:`v_g=d\omega/dk` and distinguish carrier from envelope.
Finish by checking the governing equation, the dimensions, and the zero/large-parameter limit; these checks replace reliance on an unavailable answer-key entry.

Problem 7.20* — superposition, beats, and group velocity: calculation
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^

List the supplied quantities with units, substitute only after the symbolic relation is isolated, and retain guard digits until the final result.
Add complex amplitudes before squaring; for a narrow packet use :math:`v_g=d\omega/dk` and distinguish carrier from envelope.
Finish by checking the governing equation, the dimensions, and the zero/large-parameter limit; these checks replace reliance on an unavailable answer-key entry.

Problem 7.21 — superposition, beats, and group velocity: derivation
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^

Start from the governing relation rather than the desired result; rearrange until the requested form follows, so the argument is not circular.
Add complex amplitudes before squaring; for a narrow packet use :math:`v_g=d\omega/dk` and distinguish carrier from envelope.
The book's selected-answer check begins ``E = E0 cosvct + E0a cos vmt cosvct = E0 cosvct + E0a 2 [cos (vc vm)t + cos (vc + vm)t] Audible range nm = 20 Hz to 20 × 103 Hz. Maximum modulation frequency nm(max) = 20 × 103 Hz. nc nm(max) … n … nc + nm(max) ∆n = 2nm(max) = 40 × 103 Hz``.  Substitute the result back into the governing relation to verify its units and sign.

Problem 7.22 — superposition, beats, and group velocity: calculation
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^

List the supplied quantities with units, substitute only after the symbolic relation is isolated, and retain guard digits until the final result.
Add complex amplitudes before squaring; for a narrow packet use :math:`v_g=d\omega/dk` and distinguish carrier from envelope.
The book's selected-answer check begins ``v = v/k = 2ak2 , vg = dv/dk = (2a) (3k2 ) = 6ak2``.  Substitute the result back into the governing relation to verify its units and sign.

Problem 7.23* — Fourier-series decomposition: derivation
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^

Start from the governing relation rather than the desired result; rearrange until the requested form follows, so the argument is not circular.
Compute :math:`a_n` and :math:`b_n` over one period, use parity to eliminate terms, and reconstruct enough harmonics to verify the waveform.
Finish by checking the governing equation, the dimensions, and the zero/large-parameter limit; these checks replace reliance on an unavailable answer-key entry.

Problem 7.24* — Fourier-series decomposition: calculation
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^

List the supplied quantities with units, substitute only after the symbolic relation is isolated, and retain guard digits until the final result.
Compute :math:`a_n` and :math:`b_n` over one period, use parity to eliminate terms, and reconstruct enough harmonics to verify the waveform.
Finish by checking the governing equation, the dimensions, and the zero/large-parameter limit; these checks replace reliance on an unavailable answer-key entry.

Problem 7.25* — Fourier-series decomposition: calculation
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^

List the supplied quantities with units, substitute only after the symbolic relation is isolated, and retain guard digits until the final result.
Compute :math:`a_n` and :math:`b_n` over one period, use parity to eliminate terms, and reconstruct enough harmonics to verify the waveform.
Finish by checking the governing equation, the dimensions, and the zero/large-parameter limit; these checks replace reliance on an unavailable answer-key entry.

Problem 7.26* — Fourier-series decomposition: calculation
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^

List the supplied quantities with units, substitute only after the symbolic relation is isolated, and retain guard digits until the final result.
Compute :math:`a_n` and :math:`b_n` over one period, use parity to eliminate terms, and reconstruct enough harmonics to verify the waveform.
Finish by checking the governing equation, the dimensions, and the zero/large-parameter limit; these checks replace reliance on an unavailable answer-key entry.

Problem 7.27* — Fourier-series decomposition: calculation
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^

List the supplied quantities with units, substitute only after the symbolic relation is isolated, and retain guard digits until the final result.
Compute :math:`a_n` and :math:`b_n` over one period, use parity to eliminate terms, and reconstruct enough harmonics to verify the waveform.
Finish by checking the governing equation, the dimensions, and the zero/large-parameter limit; these checks replace reliance on an unavailable answer-key entry.

Problem 7.28* — Fourier-series decomposition: calculation
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^

List the supplied quantities with units, substitute only after the symbolic relation is isolated, and retain guard digits until the final result.
Compute :math:`a_n` and :math:`b_n` over one period, use parity to eliminate terms, and reconstruct enough harmonics to verify the waveform.
Finish by checking the governing equation, the dimensions, and the zero/large-parameter limit; these checks replace reliance on an unavailable answer-key entry.

Problem 7.29 — Fourier-series decomposition: calculation
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^

List the supplied quantities with units, substitute only after the symbolic relation is isolated, and retain guard digits until the final result.
Compute :math:`a_n` and :math:`b_n` over one period, use parity to eliminate terms, and reconstruct enough harmonics to verify the waveform.
The book's selected-answer check begins ``v = A gl 2p = 1g/k vg = v + k dv dk [7.38] dv dk = - 1 2kA g k = v 2k vg = v/2 Z03_HECH6933_05_GE_SOL.indd 695 08/09/16 9:14 pm 696 Solutions to Selected Problems``.  Substitute the result back into the governing relation to verify its units and sign.

Problem 7.30* — Fourier-series decomposition: calculation
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^

List the supplied quantities with units, substitute only after the symbolic relation is isolated, and retain guard digits until the final result.
Compute :math:`a_n` and :math:`b_n` over one period, use parity to eliminate terms, and reconstruct enough harmonics to verify the waveform.
Finish by checking the governing equation, the dimensions, and the zero/large-parameter limit; these checks replace reliance on an unavailable answer-key entry.

Problem 7.31 — Fourier-series decomposition: calculation
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^

List the supplied quantities with units, substitute only after the symbolic relation is isolated, and retain guard digits until the final result.
Compute :math:`a_n` and :math:`b_n` over one period, use parity to eliminate terms, and reconstruct enough harmonics to verify the waveform.
The book's selected-answer check begins ``vg = v + k dv dk and dv dk = dv dv dv dk = vg dv dv . Since v = c/n, dv dv = dv dn dn dv = c n2 dn dv vg = v vgck n2 dn dv = v 1 + (ck/n2 )(dn/dv) = c n + v(dn/dv)``.  Substitute the result back into the governing relation to verify its units and sign.

Problem 7.32* — Fourier-series decomposition: calculation
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^

List the supplied quantities with units, substitute only after the symbolic relation is isolated, and retain guard digits until the final result.
Compute :math:`a_n` and :math:`b_n` over one period, use parity to eliminate terms, and reconstruct enough harmonics to verify the waveform.
Finish by checking the governing equation, the dimensions, and the zero/large-parameter limit; these checks replace reliance on an unavailable answer-key entry.

Problem 7.33* — Fourier-series decomposition: derivation
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^

Start from the governing relation rather than the desired result; rearrange until the requested form follows, so the argument is not circular.
Compute :math:`a_n` and :math:`b_n` over one period, use parity to eliminate terms, and reconstruct enough harmonics to verify the waveform.
Finish by checking the governing equation, the dimensions, and the zero/large-parameter limit; these checks replace reliance on an unavailable answer-key entry.

Problem 7.34* — Fourier-series decomposition: calculation
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^

List the supplied quantities with units, substitute only after the symbolic relation is isolated, and retain guard digits until the final result.
Compute :math:`a_n` and :math:`b_n` over one period, use parity to eliminate terms, and reconstruct enough harmonics to verify the waveform.
Finish by checking the governing equation, the dimensions, and the zero/large-parameter limit; these checks replace reliance on an unavailable answer-key entry.

Problem 7.35* — Fourier-series decomposition: calculation
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^

List the supplied quantities with units, substitute only after the symbolic relation is isolated, and retain guard digits until the final result.
Compute :math:`a_n` and :math:`b_n` over one period, use parity to eliminate terms, and reconstruct enough harmonics to verify the waveform.
Finish by checking the governing equation, the dimensions, and the zero/large-parameter limit; these checks replace reliance on an unavailable answer-key entry.

Problem 7.36* — Fourier-series decomposition: derivation
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^

Start from the governing relation rather than the desired result; rearrange until the requested form follows, so the argument is not circular.
Compute :math:`a_n` and :math:`b_n` over one period, use parity to eliminate terms, and reconstruct enough harmonics to verify the waveform.
Finish by checking the governing equation, the dimensions, and the zero/large-parameter limit; these checks replace reliance on an unavailable answer-key entry.

Problem 7.37* — Fourier-series decomposition: calculation
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^

List the supplied quantities with units, substitute only after the symbolic relation is isolated, and retain guard digits until the final result.
Compute :math:`a_n` and :math:`b_n` over one period, use parity to eliminate terms, and reconstruct enough harmonics to verify the waveform.
Finish by checking the governing equation, the dimensions, and the zero/large-parameter limit; these checks replace reliance on an unavailable answer-key entry.

Problem 7.38* — Fourier-series decomposition: calculation
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^

List the supplied quantities with units, substitute only after the symbolic relation is isolated, and retain guard digits until the final result.
Compute :math:`a_n` and :math:`b_n` over one period, use parity to eliminate terms, and reconstruct enough harmonics to verify the waveform.
Finish by checking the governing equation, the dimensions, and the zero/large-parameter limit; these checks replace reliance on an unavailable answer-key entry.

Problem 7.39* — Fourier-series decomposition: calculation
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^

List the supplied quantities with units, substitute only after the symbolic relation is isolated, and retain guard digits until the final result.
Compute :math:`a_n` and :math:`b_n` over one period, use parity to eliminate terms, and reconstruct enough harmonics to verify the waveform.
Finish by checking the governing equation, the dimensions, and the zero/large-parameter limit; these checks replace reliance on an unavailable answer-key entry.

Problem 7.40 — Fourier-series decomposition: calculation
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^

List the supplied quantities with units, substitute only after the symbolic relation is isolated, and retain guard digits until the final result.
Compute :math:`a_n` and :math:`b_n` over one period, use parity to eliminate terms, and reconstruct enough harmonics to verify the waveform.
The book's selected-answer check begins ``v 7 7 vi, n2 = 1 - Nqe 2 v2 P0me ^ƒi = 1 - Nqe 2 v2 P0me . Using the binomial expansion, we have (1 x)1/2 ≈ 1 - 1 2 x for x 6 6 1 n = 1 - Nqe 2 /v2 P0me2, dn/dv = Nqe 2 /P0mev3 vg = c n + v(dn/dv) = c 1 - Nqe 2 /v2 P0me2 + Nqe 2 /P0mev2 = c 1 + Nqe 2 /P0mev2 2 and vg 6 c, v = c/n = c 1 - Nqe 2 /P0mev2 2 Binomial expans``.  Substitute the result back into the governing relation to verify its units and sign.

Problem 7.41* — Fourier-series decomposition: calculation
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^

List the supplied quantities with units, substitute only after the symbolic relation is isolated, and retain guard digits until the final result.
Compute :math:`a_n` and :math:`b_n` over one period, use parity to eliminate terms, and reconstruct enough harmonics to verify the waveform.
Finish by checking the governing equation, the dimensions, and the zero/large-parameter limit; these checks replace reliance on an unavailable answer-key entry.

Problem 7.42* — Fourier-series decomposition: calculation
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^

List the supplied quantities with units, substitute only after the symbolic relation is isolated, and retain guard digits until the final result.
Compute :math:`a_n` and :math:`b_n` over one period, use parity to eliminate terms, and reconstruct enough harmonics to verify the waveform.
Finish by checking the governing equation, the dimensions, and the zero/large-parameter limit; these checks replace reliance on an unavailable answer-key entry.

Problem 7.43 — Fourier-series decomposition: calculation
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^

List the supplied quantities with units, substitute only after the symbolic relation is isolated, and retain guard digits until the final result.
Compute :math:`a_n` and :math:`b_n` over one period, use parity to eliminate terms, and reconstruct enough harmonics to verify the waveform.
The book's selected-answer check begins ``3 λ 0 sinakx sinbkx dx = 1 2k c3 λ 0 cos [(a b)kx]k dx - 3 λ 0 cos [(a + b)kx]k dxd = 1 2k sin(a b)kx a b ' λ 0 - 1 2k sin (a + b)kx a + b ' λ 0 = 0 if a Z b Whereas if a = b 3 λ 0 sin2 akx dx = 1 2k3 λ 0 (1 + cos 2akx)k dx = λ 2 The other integrals are similar.``.  Substitute the result back into the governing relation to verify its units and sign.

Problem 7.44 — wave packets and temporal coherence: calculation
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^

List the supplied quantities with units, substitute only after the symbolic relation is isolated, and retain guard digits until the final result.
Fourier transform the spectrum or pulse, use :math:`\Delta t\,\Delta\nu` consistently, and check the monochromatic limit.
The book's selected-answer check begins ``Even function, therefore Bm = 0. A0 = 2 λ 3 λ/a -λ/a dx = 2 λ a λ a + λ a b = 4 a Am = 2 λ 3 λ/a -λ/a (1) cos mkx dx Am = 2 mkl sin mkxd λ/a -λ/a Am = 2 mp sin m2p a π 2p π 2 −π 2 u Si(u)``.  Substitute the result back into the governing relation to verify its units and sign.

Problem 7.45* — wave packets and temporal coherence: calculation
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^

List the supplied quantities with units, substitute only after the symbolic relation is isolated, and retain guard digits until the final result.
Fourier transform the spectrum or pulse, use :math:`\Delta t\,\Delta\nu` consistently, and check the monochromatic limit.
Finish by checking the governing equation, the dimensions, and the zero/large-parameter limit; these checks replace reliance on an unavailable answer-key entry.

Problem 7.46* — wave packets and temporal coherence: calculation
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^

List the supplied quantities with units, substitute only after the symbolic relation is isolated, and retain guard digits until the final result.
Fourier transform the spectrum or pulse, use :math:`\Delta t\,\Delta\nu` consistently, and check the monochromatic limit.
Finish by checking the governing equation, the dimensions, and the zero/large-parameter limit; these checks replace reliance on an unavailable answer-key entry.

Problem 7.47* — wave packets and temporal coherence: calculation
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^

List the supplied quantities with units, substitute only after the symbolic relation is isolated, and retain guard digits until the final result.
Fourier transform the spectrum or pulse, use :math:`\Delta t\,\Delta\nu` consistently, and check the monochromatic limit.
Finish by checking the governing equation, the dimensions, and the zero/large-parameter limit; these checks replace reliance on an unavailable answer-key entry.

Problem 7.48* — wave packets and temporal coherence: calculation
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^

List the supplied quantities with units, substitute only after the symbolic relation is isolated, and retain guard digits until the final result.
Fourier transform the spectrum or pulse, use :math:`\Delta t\,\Delta\nu` consistently, and check the monochromatic limit.
Finish by checking the governing equation, the dimensions, and the zero/large-parameter limit; these checks replace reliance on an unavailable answer-key entry.

Problem 7.49* — wave packets and temporal coherence: derivation
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^

Start from the governing relation rather than the desired result; rearrange until the requested form follows, so the argument is not circular.
Fourier transform the spectrum or pulse, use :math:`\Delta t\,\Delta\nu` consistently, and check the monochromatic limit.
Finish by checking the governing equation, the dimensions, and the zero/large-parameter limit; these checks replace reliance on an unavailable answer-key entry.

Problem 7.50 — wave packets and temporal coherence: calculation
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^

List the supplied quantities with units, substitute only after the symbolic relation is isolated, and retain guard digits until the final result.
Fourier transform the spectrum or pulse, use :math:`\Delta t\,\Delta\nu` consistently, and check the monochromatic limit.
The book's selected-answer check begins ``ƒ′(x) = 1 p3 a 0 E0L sinkL/2 kL/2 cos kx dk = E0L p2 3 b 0 sin(kL/2 + kx) kL/2 dk + E0L p2 3 b 0 sin(kL/2 kx) kL/2 dk Let kL/2 = w, (L/2) dk = dw, kx = wx′. ƒ′(x) = E0 π 3 b 0 sin(w + wx′) w dw + E0 π 3 b 0 sin(w wx′) w dw where b = aL/2. Let w + wx′ = t, dw/w = dt/t. 0 … w … b and 0 … t … (x′ + 1)b. Let w wx′ = -t in``.  Substitute the result back into the governing relation to verify its units and sign.

Problem 7.51* — wave packets and temporal coherence: calculation
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^

List the supplied quantities with units, substitute only after the symbolic relation is isolated, and retain guard digits until the final result.
Fourier transform the spectrum or pulse, use :math:`\Delta t\,\Delta\nu` consistently, and check the monochromatic limit.
Finish by checking the governing equation, the dimensions, and the zero/large-parameter limit; these checks replace reliance on an unavailable answer-key entry.

Problem 7.52* — wave packets and temporal coherence: calculation
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^

List the supplied quantities with units, substitute only after the symbolic relation is isolated, and retain guard digits until the final result.
Fourier transform the spectrum or pulse, use :math:`\Delta t\,\Delta\nu` consistently, and check the monochromatic limit.
Finish by checking the governing equation, the dimensions, and the zero/large-parameter limit; these checks replace reliance on an unavailable answer-key entry.

Problem 7.53* — wave packets and temporal coherence: calculation
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^

List the supplied quantities with units, substitute only after the symbolic relation is isolated, and retain guard digits until the final result.
Fourier transform the spectrum or pulse, use :math:`\Delta t\,\Delta\nu` consistently, and check the monochromatic limit.
Finish by checking the governing equation, the dimensions, and the zero/large-parameter limit; these checks replace reliance on an unavailable answer-key entry.

Problem 7.54 — wave packets and temporal coherence: calculation
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^

List the supplied quantities with units, substitute only after the symbolic relation is isolated, and retain guard digits until the final result.
Fourier transform the spectrum or pulse, use :math:`\Delta t\,\Delta\nu` consistently, and check the monochromatic limit.
The book's selected-answer check begins ``By analogy with Eq. (7.61), A(v) = ∆t 2 E0 sinc(vp v) ∆t 2 From Table 1 (π. 681) sinc(π/2) = 63.7%. Not quite 50% actually, sinc a π 1.65 b = 49.8% ' (vp v) ∆t 2 ' 6 π 2 or π ∆t 6 (vp v) 6 π ∆t Thus appreciable values of A(v) lie in a range ∆v ∼ 2p/∆t and ∆n ∆t ≈ 1. The power spectrum is proportional to A2 (v), and [si``.  Substitute the result back into the governing relation to verify its units and sign.

Problem 7.55 — wave packets and temporal coherence: calculation
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^

List the supplied quantities with units, substitute only after the symbolic relation is isolated, and retain guard digits until the final result.
Fourier transform the spectrum or pulse, use :math:`\Delta t\,\Delta\nu` consistently, and check the monochromatic limit.
The book's selected-answer check begins ``∆lc = c ∆tc, ∆lc ≈ c/∆n. But ∆v/∆k0 = v/k0 = c; thus ∆n/∆l0  = n/l0, ∆lc ≈ cl0 ∆l0n ∆lc ≈ l0 2 /∆l0 Or try using the uncertainty principle: ∆λ ≈ h ∆π where π = h/λ and ∆l0 6 6 l0 Z03_HECH6933_05_GE_SOL.indd 696 08/09/16 9:14 pm Solutions to Selected Problems 697 Let E0 + E′ 0 = E″ 0x and E0 - E′ 0 = E″ 0y; then E $ =``.  Substitute the result back into the governing relation to verify its units and sign.

Problem 7.56* — wave packets and temporal coherence: calculation
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^

List the supplied quantities with units, substitute only after the symbolic relation is isolated, and retain guard digits until the final result.
Fourier transform the spectrum or pulse, use :math:`\Delta t\,\Delta\nu` consistently, and check the monochromatic limit.
Finish by checking the governing equation, the dimensions, and the zero/large-parameter limit; these checks replace reliance on an unavailable answer-key entry.

Problem 7.57 — wave packets and temporal coherence: calculation
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^

List the supplied quantities with units, substitute only after the symbolic relation is isolated, and retain guard digits until the final result.
Fourier transform the spectrum or pulse, use :math:`\Delta t\,\Delta\nu` consistently, and check the monochromatic limit.
The book's selected-answer check begins ``∆lc = c ∆tc = 3 × 108 m/s × 10-8 s = 3 m ∆l0 ≈ l0 2 /∆lc = (500 × 10-9 m)2 /3 m ∆l0 ≈ 8.3 × 10-14 m = 8.3 × 10-5 nm ∆l0/l0 = ∆n/n = 8.3 × 10-5 /500 = 1.6 × 10-7 ≈ 1 part in 107``.  Substitute the result back into the governing relation to verify its units and sign.

Problem 7.58 — wave packets and temporal coherence: calculation
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^

List the supplied quantities with units, substitute only after the symbolic relation is isolated, and retain guard digits until the final result.
Fourier transform the spectrum or pulse, use :math:`\Delta t\,\Delta\nu` consistently, and check the monochromatic limit.
The book's selected-answer check begins ``∆n = 54 × 103 Hz ∆n/n = (54 × 103 )(10600 × 10-9 m) (3 × 108 m/s) = 1.91 × 10-9 ∆lc = c ∆tc ≈ c/∆n ∆lc ≈ (3 × 108 m/s) (54 × 103 Hz) = 5.55 × 103 m``.  Substitute the result back into the governing relation to verify its units and sign.

Problem 7.59* — wave packets and temporal coherence: calculation
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^

List the supplied quantities with units, substitute only after the symbolic relation is isolated, and retain guard digits until the final result.
Fourier transform the spectrum or pulse, use :math:`\Delta t\,\Delta\nu` consistently, and check the monochromatic limit.
Finish by checking the governing equation, the dimensions, and the zero/large-parameter limit; these checks replace reliance on an unavailable answer-key entry.

Problem 7.60 — wave packets and temporal coherence: calculation
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^

List the supplied quantities with units, substitute only after the symbolic relation is isolated, and retain guard digits until the final result.
Fourier transform the spectrum or pulse, use :math:`\Delta t\,\Delta\nu` consistently, and check the monochromatic limit.
The book's selected-answer check begins ``∆lc = c ∆tc = 3 × 108 × 10-10 = 3 × 10-2 m ∆n ≈ 1/∆tc = 1010 Hz ∆l0 ≈ l0 2 /∆lc (see Problem 7.55) = (632.8 nm)2 /3 × 10-2 m = 0.013 nm ∆n = 1015 Hz, ∆lc = c × 10-15 = 300 nm ∆l0 ≈ l0 2 /∆lc = 1334.78 nm``.  Substitute the result back into the governing relation to verify its units and sign.

Problem 7.61* — wave packets and temporal coherence: calculation
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^

List the supplied quantities with units, substitute only after the symbolic relation is isolated, and retain guard digits until the final result.
Fourier transform the spectrum or pulse, use :math:`\Delta t\,\Delta\nu` consistently, and check the monochromatic limit.
Finish by checking the governing equation, the dimensions, and the zero/large-parameter limit; these checks replace reliance on an unavailable answer-key entry.

Problem 7.62* — wave packets and temporal coherence: calculation
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^

List the supplied quantities with units, substitute only after the symbolic relation is isolated, and retain guard digits until the final result.
Fourier transform the spectrum or pulse, use :math:`\Delta t\,\Delta\nu` consistently, and check the monochromatic limit.
Finish by checking the governing equation, the dimensions, and the zero/large-parameter limit; these checks replace reliance on an unavailable answer-key entry.

Problem 7.63* — wave packets and temporal coherence: calculation
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^

List the supplied quantities with units, substitute only after the symbolic relation is isolated, and retain guard digits until the final result.
Fourier transform the spectrum or pulse, use :math:`\Delta t\,\Delta\nu` consistently, and check the monochromatic limit.
Finish by checking the governing equation, the dimensions, and the zero/large-parameter limit; these checks replace reliance on an unavailable answer-key entry.