Chapter 2: Wave Motion
======================

Source: Eugene Hecht, *Optics*, fifth Global Edition, Chapter 2.
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 2.1* — wave-equation test: derivation
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^

Start from the governing relation rather than the desired result; rearrange until the requested form follows, so the argument is not circular.
Write the field as :math:`F(x\mp vt)` and verify :math:`\partial_x^2\psi=v^{-2}\partial_t^2\psi`.
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 2.2* — wave-equation test: derivation
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^

Start from the governing relation rather than the desired result; rearrange until the requested form follows, so the argument is not circular.
Write the field as :math:`F(x\mp vt)` and verify :math:`\partial_x^2\psi=v^{-2}\partial_t^2\psi`.
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 2.3* — wave-equation test: calculation
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^

List the supplied quantities with units, substitute only after the symbolic relation is isolated, and retain guard digits until the final result.
Write the field as :math:`F(x\mp vt)` and verify :math:`\partial_x^2\psi=v^{-2}\partial_t^2\psi`.
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 2.4* — wave-equation test: calculation
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^

List the supplied quantities with units, substitute only after the symbolic relation is isolated, and retain guard digits until the final result.
Write the field as :math:`F(x\mp vt)` and verify :math:`\partial_x^2\psi=v^{-2}\partial_t^2\psi`.
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 2.5* — wave-equation test: derivation
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^

Start from the governing relation rather than the desired result; rearrange until the requested form follows, so the argument is not circular.
Write the field as :math:`F(x\mp vt)` and verify :math:`\partial_x^2\psi=v^{-2}\partial_t^2\psi`.
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 2.6 — wavelength, frequency, and speed: calculation
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^

List the supplied quantities with units, substitute only after the symbolic relation is isolated, and retain guard digits until the final result.
Use :math:`v=f\lambda`, :math:`k=2\pi/\lambda`, and :math:`\omega=2\pi f` with one consistent unit system.
The book's selected-answer check begins ``Number of waves, N = d/λ = (100 × 10-6 )/(532 × 10-9 ) ≈188. D = Nl = Nc/n = 188(3 × 108 )/(2.45 × 109 ) ≈23 m``.  Substitute the result back into the governing relation to verify its units and sign.

Problem 2.7* — wavelength, frequency, and speed: calculation
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^

List the supplied quantities with units, substitute only after the symbolic relation is isolated, and retain guard digits until the final result.
Use :math:`v=f\lambda`, :math:`k=2\pi/\lambda`, and :math:`\omega=2\pi f` with one consistent unit system.
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 2.8* — wavelength, frequency, and speed: calculation
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^

List the supplied quantities with units, substitute only after the symbolic relation is isolated, and retain guard digits until the final result.
Use :math:`v=f\lambda`, :math:`k=2\pi/\lambda`, and :math:`\omega=2\pi f` with one consistent unit system.
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 2.9* — wavelength, frequency, and speed: calculation
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^

List the supplied quantities with units, substitute only after the symbolic relation is isolated, and retain guard digits until the final result.
Use :math:`v=f\lambda`, :math:`k=2\pi/\lambda`, and :math:`\omega=2\pi f` with one consistent unit system.
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 2.10* — wavelength, frequency, and speed: calculation
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^

List the supplied quantities with units, substitute only after the symbolic relation is isolated, and retain guard digits until the final result.
Use :math:`v=f\lambda`, :math:`k=2\pi/\lambda`, and :math:`\omega=2\pi f` with one consistent unit system.
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 2.11 — wavelength, frequency, and speed: calculation
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^

List the supplied quantities with units, substitute only after the symbolic relation is isolated, and retain guard digits until the final result.
Use :math:`v=f\lambda`, :math:`k=2\pi/\lambda`, and :math:`\omega=2\pi f` with one consistent unit system.
The book's selected-answer check begins ``In air: λ = v/n = 343/440; λ≈78 cm. In water: λ = v/n = 1500/440; λ≈3.41 m.``.  Substitute the result back into the governing relation to verify its units and sign.

Problem 2.12* — wavelength, frequency, and speed: calculation
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^

List the supplied quantities with units, substitute only after the symbolic relation is isolated, and retain guard digits until the final result.
Use :math:`v=f\lambda`, :math:`k=2\pi/\lambda`, and :math:`\omega=2\pi f` with one consistent unit system.
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 2.13* — wavelength, frequency, and speed: calculation
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^

List the supplied quantities with units, substitute only after the symbolic relation is isolated, and retain guard digits until the final result.
Use :math:`v=f\lambda`, :math:`k=2\pi/\lambda`, and :math:`\omega=2\pi f` with one consistent unit system.
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 2.14* — harmonic-wave graph: construction
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^

Evaluate the boundary and representative interior values, then draw the requested curve or ray construction to scale.
Identify amplitude, period, phase, and propagation sign from :math:`A\cos(kx\mp\omega t+\phi)` before evaluating the requested samples.
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 2.15* — harmonic-wave graph: construction
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^

Evaluate the boundary and representative interior values, then draw the requested curve or ray construction to scale.
Identify amplitude, period, phase, and propagation sign from :math:`A\cos(kx\mp\omega t+\phi)` before evaluating the requested samples.
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 2.16* — harmonic-wave graph: construction
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^

Evaluate the boundary and representative interior values, then draw the requested curve or ray construction to scale.
Identify amplitude, period, phase, and propagation sign from :math:`A\cos(kx\mp\omega t+\phi)` before evaluating the requested samples.
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 2.17* — harmonic-wave graph: calculation
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^

List the supplied quantities with units, substitute only after the symbolic relation is isolated, and retain guard digits until the final result.
Identify amplitude, period, phase, and propagation sign from :math:`A\cos(kx\mp\omega t+\phi)` before evaluating the requested samples.
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 2.18* — harmonic-wave graph: calculation
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^

List the supplied quantities with units, substitute only after the symbolic relation is isolated, and retain guard digits until the final result.
Identify amplitude, period, phase, and propagation sign from :math:`A\cos(kx\mp\omega t+\phi)` before evaluating the requested samples.
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 2.19* — harmonic-wave graph: calculation
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^

List the supplied quantities with units, substitute only after the symbolic relation is isolated, and retain guard digits until the final result.
Identify amplitude, period, phase, and propagation sign from :math:`A\cos(kx\mp\omega t+\phi)` before evaluating the requested samples.
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 2.20* — harmonic-wave graph: calculation
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^

List the supplied quantities with units, substitute only after the symbolic relation is isolated, and retain guard digits until the final result.
Identify amplitude, period, phase, and propagation sign from :math:`A\cos(kx\mp\omega t+\phi)` before evaluating the requested samples.
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 2.21 — harmonic-wave graph: calculation
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^

List the supplied quantities with units, substitute only after the symbolic relation is isolated, and retain guard digits until the final result.
Identify amplitude, period, phase, and propagation sign from :math:`A\cos(kx\mp\omega t+\phi)` before evaluating the requested samples.
The book's selected-answer check begins ``c = A sin 2p (kx + nt), c1 = 5 sin 2p (0.4x + 2t) (a) n = 2 (b) λ = 1/0.4 = 2.5 (c) t = 1/2 = 0.5 (d) A = 5 (e) v = 5 (f) negative x c = A sin (kx vt), c2 = 2 sin (5x - 1.5t) (a) n = 1.5/2p (b) λ = 2p/5 (c) t = 2p/1.5 (d) A = 2 (e) v = 1.5/5 (f) positive x``.  Substitute the result back into the governing relation to verify its units and sign.

Problem 2.22* — harmonic-wave graph: calculation
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^

List the supplied quantities with units, substitute only after the symbolic relation is isolated, and retain guard digits until the final result.
Identify amplitude, period, phase, and propagation sign from :math:`A\cos(kx\mp\omega t+\phi)` before evaluating the requested samples.
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 2.23* — harmonic-wave graph: calculation
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^

List the supplied quantities with units, substitute only after the symbolic relation is isolated, and retain guard digits until the final result.
Identify amplitude, period, phase, and propagation sign from :math:`A\cos(kx\mp\omega t+\phi)` before evaluating the requested samples.
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 2.24* — traveling-wave phase: derivation
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^

Start from the governing relation rather than the desired result; rearrange until the requested form follows, so the argument is not circular.
Hold :math:`kx\mp\omega t+\phi` constant; differentiation gives the phase speed and fixes the direction without relying on a sketch.
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 2.25* — traveling-wave phase: derivation
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^

Start from the governing relation rather than the desired result; rearrange until the requested form follows, so the argument is not circular.
Hold :math:`kx\mp\omega t+\phi` constant; differentiation gives the phase speed and fixes the direction without relying on a sketch.
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 2.26* — traveling-wave phase: calculation
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^

List the supplied quantities with units, substitute only after the symbolic relation is isolated, and retain guard digits until the final result.
Hold :math:`kx\mp\omega t+\phi` constant; differentiation gives the phase speed and fixes the direction without relying on a sketch.
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 2.27 — traveling-wave phase: derivation
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^

Start from the governing relation rather than the desired result; rearrange until the requested form follows, so the argument is not circular.
Hold :math:`kx\mp\omega t+\phi` constant; differentiation gives the phase speed and fixes the direction without relying on a sketch.
The book's selected-answer check begins ``vy = -vA cos (kx vt + e), ay = -v2 y. Simple harmonic motion, since ay ∝ y.``.  Substitute the result back into the governing relation to verify its units and sign.

Problem 2.28 — traveling-wave phase: calculation
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^

List the supplied quantities with units, substitute only after the symbolic relation is isolated, and retain guard digits until the final result.
Hold :math:`kx\mp\omega t+\phi` constant; differentiation gives the phase speed and fixes the direction without relying on a sketch.
The book's selected-answer check begins ``t = 2.2 × 10-15 s; therefore n = 1/t = 4.5 × 1014 Hz; v = nl, 3 × 108 m/s = (4.5 × 1014 Hz)λ; λ = 6.6 × 10-7 m and k = 2p/λ = 9.5 × 106 m-1 . c(x, t) = (103 V/m) cos [9.5 × 106 m-1 × (x + 3 × 108 m/s t)]. It’s cosine because cos 0 = 1.``.  Substitute the result back into the governing relation to verify its units and sign.

Problem 2.29 — traveling-wave phase: physical interpretation
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^

State the controlling conservation or symmetry principle first, then use it to determine signs, directions, and limiting behavior.
Hold :math:`kx\mp\omega t+\phi` constant; differentiation gives the phase speed and fixes the direction without relying on a sketch.
The book's selected-answer check begins ``y(x, t) = C/[2 + (x + vt)2 ].``.  Substitute the result back into the governing relation to verify its units and sign.

Problem 2.30* — traveling-wave phase: calculation
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^

List the supplied quantities with units, substitute only after the symbolic relation is isolated, and retain guard digits until the final result.
Hold :math:`kx\mp\omega t+\phi` constant; differentiation gives the phase speed and fixes the direction without relying on a sketch.
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 2.31 — traveling-wave phase: calculation
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^

List the supplied quantities with units, substitute only after the symbolic relation is isolated, and retain guard digits until the final result.
Hold :math:`kx\mp\omega t+\phi` constant; differentiation gives the phase speed and fixes the direction without relying on a sketch.
The book's selected-answer check begins ``None of the presented functions can be differentiated twice in a nontrivial way (the second derivatives are just zero in all cases). So they cannot be valid wavefunctions.``.  Substitute the result back into the governing relation to verify its units and sign.

Problem 2.32* — traveling-wave phase: calculation
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^

List the supplied quantities with units, substitute only after the symbolic relation is isolated, and retain guard digits until the final result.
Hold :math:`kx\mp\omega t+\phi` constant; differentiation gives the phase speed and fixes the direction without relying on a sketch.
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 2.33* — traveling-wave phase: calculation
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^

List the supplied quantities with units, substitute only after the symbolic relation is isolated, and retain guard digits until the final result.
Hold :math:`kx\mp\omega t+\phi` constant; differentiation gives the phase speed and fixes the direction without relying on a sketch.
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 2.34 — traveling-wave phase: calculation
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^

List the supplied quantities with units, substitute only after the symbolic relation is isolated, and retain guard digits until the final result.
Hold :math:`kx\mp\omega t+\phi` constant; differentiation gives the phase speed and fixes the direction without relying on a sketch.
The book's selected-answer check begins ``dc dt = 0c 0x dx dt + 0c 0y dy dt and let y = t, whereupon dc dt = 0c 0x (±v) + 0c 0t = 0 and the desired result follows immediately.``.  Substitute the result back into the governing relation to verify its units and sign.

Problem 2.35 — traveling-wave phase: calculation
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^

List the supplied quantities with units, substitute only after the symbolic relation is isolated, and retain guard digits until the final result.
Hold :math:`kx\mp\omega t+\phi` constant; differentiation gives the phase speed and fixes the direction without relying on a sketch.
The book's selected-answer check begins ``dw dt = a 0w 0x ba dx dt b + 0w 0t = ka dx dt b kv. This is zero when dx dt = v which is what it should be. Using it in Problem 2.32, we have a 0w 0z b(v) + 0w 0t = 0 and thus p2 × 104 (v) p6 × 1012 = 0 and from here v = 3 × 108 m/s.``.  Substitute the result back into the governing relation to verify its units and sign.

Problem 2.36* — traveling-wave phase: calculation
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^

List the supplied quantities with units, substitute only after the symbolic relation is isolated, and retain guard digits until the final result.
Hold :math:`kx\mp\omega t+\phi` constant; differentiation gives the phase speed and fixes the direction without relying on a sketch.
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 2.37 — traveling-wave phase: calculation
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^

List the supplied quantities with units, substitute only after the symbolic relation is isolated, and retain guard digits until the final result.
Hold :math:`kx\mp\omega t+\phi` constant; differentiation gives the phase speed and fixes the direction without relying on a sketch.
The book's selected-answer check begins ``c(z, 0) = A sin (kz + e); c(-λ/12, 0) = A sin (-π/6 + e) = 0.866 c(λ/6, 0) = A sin (π/3 + e) = 1/2 c(λ/4, 0) = A sin (π/2 + e) = 0 A sin (π/2 + e) = A(sin π/2 cos e + cos π/2 sin e) = A cos e = 0, e = π/2 A sin (π/3 + π/2) = A sin (5p/6) = 1/2 therefore A = 1, hence c(z, 0) = sin (kz + π/2).``.  Substitute the result back into the governing relation to verify its units and sign.

Problem 2.38 — traveling-wave phase: calculation
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^

List the supplied quantities with units, substitute only after the symbolic relation is isolated, and retain guard digits until the final result.
Hold :math:`kx\mp\omega t+\phi` constant; differentiation gives the phase speed and fixes the direction without relying on a sketch.
The book's selected-answer check begins ``Both (a) and (b) are waves, since they are twice differentiable functions of (z vt) and (x + vt), respectively. Thus for (a) c = a2 (z bt/a)2 and the velocity is b/a in the positive z-direction. For (b) c = a2 (x + bt/a + c/a)2 and the velocity is b/a in the negative x-direction.``.  Substitute the result back into the governing relation to verify its units and sign.

Problem 2.39* — traveling-wave phase: calculation
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^

List the supplied quantities with units, substitute only after the symbolic relation is isolated, and retain guard digits until the final result.
Hold :math:`kx\mp\omega t+\phi` constant; differentiation gives the phase speed and fixes the direction without relying on a sketch.
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 2.40 — traveling-wave phase: calculation
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^

List the supplied quantities with units, substitute only after the symbolic relation is isolated, and retain guard digits until the final result.
Hold :math:`kx\mp\omega t+\phi` constant; differentiation gives the phase speed and fixes the direction without relying on a sketch.
The book's selected-answer check begins ``c(x, t) = 5.0 exp [-a(x + 1b/at)2 ], the propagation direction is negative x; v = 1b/a = 0.6 m/s. c(x, 0) = 5.0 exp (-25x2 ); Solutions to Selected Problems 685 t = 0 t = 2 C 2 3 2 1 −1 −2 0 v = 1m s x y``.  Substitute the result back into the governing relation to verify its units and sign.

Problem 2.41* — traveling-wave phase: calculation
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^

List the supplied quantities with units, substitute only after the symbolic relation is isolated, and retain guard digits until the final result.
Hold :math:`kx\mp\omega t+\phi` constant; differentiation gives the phase speed and fixes the direction without relying on a sketch.
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 2.42 — traveling-wave phase: calculation
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^

List the supplied quantities with units, substitute only after the symbolic relation is isolated, and retain guard digits until the final result.
Hold :math:`kx\mp\omega t+\phi` constant; differentiation gives the phase speed and fixes the direction without relying on a sketch.
The book's selected-answer check begins ``180° corresponds to λ 2 or (1/2)3 × 108 /5 × 1014 = 300 nm.``.  Substitute the result back into the governing relation to verify its units and sign.

Problem 2.43 — traveling-wave phase: derivation
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^

Start from the governing relation rather than the desired result; rearrange until the requested form follows, so the argument is not circular.
Hold :math:`kx\mp\omega t+\phi` constant; differentiation gives the phase speed and fixes the direction without relying on a sketch.
The book's selected-answer check begins ``c = A sin 2pa z λ ± t t b c = 60 sin 2pa z 400 × 10-9 t 1.33 × 10-15 b λ = 400 nm v = 400 × 10-9 /1.33 × 10-15 = 3 × 108 m/s n = (1/1.33) × 10+15 Hz, t = 1.33 × 10-15 s t = 0 0.2 0 0.6 0.4 −0.2 −0.4 −0.6 x c Z03_HECH6933_05_GE_SOL.indd 685 08/09/16 9:13 pm 686 Solutions to Selected Problems 3.15 8cos2 (k $~ r $ vt)9 =``.  Substitute the result back into the governing relation to verify its units and sign.

Problem 2.44* — complex wave representation: derivation
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^

Start from the governing relation rather than the desired result; rearrange until the requested form follows, so the argument is not circular.
Use Euler's identity, perform the algebra on the complex amplitude, and take the real part only after the operation is complete.
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 2.45* — complex wave representation: derivation
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^

Start from the governing relation rather than the desired result; rearrange until the requested form follows, so the argument is not circular.
Use Euler's identity, perform the algebra on the complex amplitude, and take the real part only after the operation is complete.
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 2.46* — complex wave representation: calculation
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^

List the supplied quantities with units, substitute only after the symbolic relation is isolated, and retain guard digits until the final result.
Use Euler's identity, perform the algebra on the complex amplitude, and take the real part only after the operation is complete.
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 2.47* — complex wave representation: calculation
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^

List the supplied quantities with units, substitute only after the symbolic relation is isolated, and retain guard digits until the final result.
Use Euler's identity, perform the algebra on the complex amplitude, and take the real part only after the operation is complete.
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 2.48 — complex wave representation: derivation
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^

Start from the governing relation rather than the desired result; rearrange until the requested form follows, so the argument is not circular.
Use Euler's identity, perform the algebra on the complex amplitude, and take the real part only after the operation is complete.
The book's selected-answer check begins ``c = A exp i(kxx + kyy + kzz) kx = ka ky = kb kz = kg k $ = [(ka)2 + (kb)2 + (kg)2 ]1/2 = k[a2 + b2 + g2 ]12``.  Substitute the result back into the governing relation to verify its units and sign.

Problem 2.49* — complex wave representation: derivation
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^

Start from the governing relation rather than the desired result; rearrange until the requested form follows, so the argument is not circular.
Use Euler's identity, perform the algebra on the complex amplitude, and take the real part only after the operation is complete.
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 2.50* — complex wave representation: calculation
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^

List the supplied quantities with units, substitute only after the symbolic relation is isolated, and retain guard digits until the final result.
Use Euler's identity, perform the algebra on the complex amplitude, and take the real part only after the operation is complete.
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 2.51* — plane waves and matter waves: calculation
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^

List the supplied quantities with units, substitute only after the symbolic relation is isolated, and retain guard digits until the final result.
Use :math:`\mathbf k\cdot\mathbf r-\omega t` for a plane wave and :math:`\lambda=h/p` for a de Broglie wave.
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 2.52 — plane waves and matter waves: calculation
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^

List the supplied quantities with units, substitute only after the symbolic relation is isolated, and retain guard digits until the final result.
Use :math:`\mathbf k\cdot\mathbf r-\omega t` for a plane wave and :math:`\lambda=h/p` for a de Broglie wave.
The book's selected-answer check begins ``λ = h/(mv) = 6.62 × 10-34 /(3.31 × 10-3 × 500) = 4 × 10-34 m n = v/λ = 500/(4 × 10-34 ) = 1.25 × 1036 Hz.``.  Substitute the result back into the governing relation to verify its units and sign.

Problem 2.53 — plane waves and matter waves: calculation
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^

List the supplied quantities with units, substitute only after the symbolic relation is isolated, and retain guard digits until the final result.
Use :math:`\mathbf k\cdot\mathbf r-\omega t` for a plane wave and :math:`\lambda=h/p` for a de Broglie wave.
The book's selected-answer check begins ``k $ can be constructed by forming a unit vector in the proper direction and multiplying it by k. The unit vector is [(4 - 0)î + (2 - 0)ĵ + (1 - 0)k̂]/242 + 22 + 12 = (4î + 2ĵ + k̂)/221 and k $ = k(4î + 2ĵ + k̂)/221. r $ = xî + yĵ + zk̂ hence c(x, y, z, t) = A sin [(4k/221)x + (2k/221)y + (k/221) z vt].``.  Substitute the result back into the governing relation to verify its units and sign.

Problem 2.54* — plane waves and matter waves: calculation
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^

List the supplied quantities with units, substitute only after the symbolic relation is isolated, and retain guard digits until the final result.
Use :math:`\mathbf k\cdot\mathbf r-\omega t` for a plane wave and :math:`\lambda=h/p` for a de Broglie wave.
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 2.55 — plane waves and matter waves: derivation
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^

Start from the governing relation rather than the desired result; rearrange until the requested form follows, so the argument is not circular.
Use :math:`\mathbf k\cdot\mathbf r-\omega t` for a plane wave and :math:`\lambda=h/p` for a de Broglie wave.
The book's selected-answer check begins ``c( r $1, t) = c[ r $2 - ( r $2 r $1), t] = c( k $~ r $1, t) = c[k $~ r $2 k $~ ( r $2 r $1), t] = c(k $~ r $2, t) = c( r $2, t) since k $~ (r $2 r $1) = 0.``.  Substitute the result back into the governing relation to verify its units and sign.

Problem 2.56* — plane waves and matter waves: calculation
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^

List the supplied quantities with units, substitute only after the symbolic relation is isolated, and retain guard digits until the final result.
Use :math:`\mathbf k\cdot\mathbf r-\omega t` for a plane wave and :math:`\lambda=h/p` for a de Broglie wave.
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 2.57* — plane waves and matter waves: calculation
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^

List the supplied quantities with units, substitute only after the symbolic relation is isolated, and retain guard digits until the final result.
Use :math:`\mathbf k\cdot\mathbf r-\omega t` for a plane wave and :math:`\lambda=h/p` for a de Broglie wave.
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 2.58* — plane waves and matter waves: construction
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^

Evaluate the boundary and representative interior values, then draw the requested curve or ray construction to scale.
Use :math:`\mathbf k\cdot\mathbf r-\omega t` for a plane wave and :math:`\lambda=h/p` for a de Broglie wave.
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 2.59* — plane waves and matter waves: calculation
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^

List the supplied quantities with units, substitute only after the symbolic relation is isolated, and retain guard digits until the final result.
Use :math:`\mathbf k\cdot\mathbf r-\omega t` for a plane wave and :math:`\lambda=h/p` for a de Broglie wave.
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 2.60* — plane waves and matter waves: calculation
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^

List the supplied quantities with units, substitute only after the symbolic relation is isolated, and retain guard digits until the final result.
Use :math:`\mathbf k\cdot\mathbf r-\omega t` for a plane wave and :math:`\lambda=h/p` for a de Broglie wave.
Finish by checking the governing equation, the dimensions, and the zero/large-parameter limit; these checks replace reliance on an unavailable answer-key entry.