Questions and Answers of Optics

Show thatψ(x, t) = A cos (kx - ωt)is a solution of the differential wave equation.
Prove thatψ(x, t) = A cos (kx - ωt - π/2)is equivalent toψ(x, t) = A sin (kx - ωt)
Show that if the displacement of the string in Fig. 2.12 is given byy(x, t) = A sin [kx - Ï‰t + Îµ]then the hand generating the wave must be moving vertically in simple harmonic
Write the expression for the wavefunction of a harmonic wave of amplitude 103 V/m, period 2.2 × 10-15 s, and speed 3 × 108 m/s. The wave is propagating in the negative x-direction and has a
Consider the pulse described in terms of its displacement at t = 0 bywhere C is a constant. Draw the wave profile. Write an expression for the wave, having a speed v in the negative x-direction, as a
Determine the magnitude of the wavefunction Ψ(z, t) = A cos [k(z + vt) + π] at the point z = 0, when t = τ/2 and when t = 3π/4.
Use Eq. (2.33) to calculate the speed of the wave whose representation in SI units isÏˆ(y, t) = A cos Ï€(3 Ã— 106y + 9 Ã— 1014t) дх (2.33) tv ot Ф
The displacement of a wave on a string is given bywhere the wave travels at 2.00 m/s and has a period of 1/4 s. Determine the displacement of the string 1.50 m from the origin at a time t = 2.2 s. z,
Begin with the following theorem: If z = Æ’(x, y) and x = g(t), y = h(t), thenDerive Eq. (2.34). хр 20 дz dy dz ду dt дх dt dt || -(a /at)x 土ひ= (ap /əx); (2.34)
Using the results of the previous problem, show that for a harmonic wave with a phase Ï†(x, t) = k(x - vt) we can determine the speed by setting dÏ†/dt = 0. Apply the technique to
A Gaussian wave has the form Ψ(x, t) = Ae-α(bx+ct)2. Use the fact that Ψ(x, t) = f(x ∓ vt) to determine its speed and then verify your answer using Eq. (2.34). (y, t) = A cos T(3 × 10°y + 9 x
Determine which of the following describe traveling waves:(a)(b)(c)(d)Where appropriate, draw the profile and find the speed and direction of motion. (y, t) = ea*y²+b²r²–2abty) | V(z, t) = A
Given the traveling wave Ψ(x, t) = 5.0 exp (-αx2 - bt2 - 2√αb xt), determine its direction of propagation. Calculate a few values of Ψ and make a sketch of the wave at t = 0, taking α =
Imagine a sound wave with a frequency of 1.10 kHz propagating with a speed of 330 m/s. Determine the phase difference in radians between any two points on the wave separated by 10.0 cm.
Consider a lightwave having a phase velocity of 3 × 108 m/s and a frequency of 6 × 1014 Hz. What is the shortest distance along the wave between any two points that have a phase difference of 30°?
Write an expression for the wave shown in Fig. P.2.43. Find its wavelength, velocity, frequency, and period.Fig. P.2.43 t = 0 60 40 20 z (nm) 100 500 300 -20 -40 -60 t = 0.66 x 10-15 s z (nm) 100 300
Working with exponentials directly, show that the magnitude of Ψ = Aeiωt is A. Then re-derive the same result using Euler’s formula. Prove that eiαeiβ = ei(α+β).
Show that the imaginary part of a complex number z˜ is given by (z˜ - z˜*)/2i.
Take the complex quantities zËœ1= (x1+ iy1) and zËœ2= (x2+ iy2) and show that Re (ž1 + ž2) = Re (ž1) + Re (ž2)
Take the complex quantities zËœ1= (x1+ iy1) and zËœ2=(x2+ iy2) and show that Re (ž) X Re (ž2) # Re (ž¡ × ž2)
Beginning with Eq. (2.51), verify thatand thatDraw a sketch showing all the pertinent quantities. (x, y, z, t) = Aek»x+k,y+k_z#wt) (2.51) a² + B² + y? = 1
Show that Eqs. (2.64) and (2.65), which are plane waves of arbitrary form, satisfy the three-dimensional differential wave equation. (x, y, z, t) = f(æx + By + yz – vt) b(x, y, z, t) = g(ax + By +
Make up a table with columns headed by values of θ running from -π/2 to 2π in intervals of π/4. In each column place the corresponding value of sin θ, and beneath those the values of 2 sin θ.
Make up a table with columns headed by values of θ running from -π/2 to 2π in intervals of π/4. In each column place the corresponding value of sin θ, and beneath those the values of sin (θ -
With the last two problems in mind, draw a plot of the three functions(a) sin θ(b) sin (θ - 3/4)(c) sin θ + sin (θ - 3π/4).Compare the amplitude of the combined function (c) in this case with
Show that Ψ(vector k• vector r, t) may represent a plane wave where vector k is normal to the wavefront. [Let vector r1 and vector r2 be position vectors drawn to any two points on the plane and
Show explicitly, that the functiondescribes a wave provided that v = Ï‰/k yT, t) = Aexp[i(k•i + ot + £)]
Write an expression in Cartesian coordinates for a harmonic plane wave of amplitude A and frequency ω propagating in the positive x-direction.
Write an expression in Cartesian coordinates for a harmonic plane wave of amplitude A and frequency ω propagating in the direction of the vector k, which in turn lies on a line drawn from the origin
De Broglie’s hypothesis states that every particle has associated with it a wavelength given by Planck’s constant (h = 6.6 × 10-34 J · s) divided by the particle’s momentum. Compare the
Consider the functionwhere A, Î±, and b are all constants, and they have appropriate SI units. Does this represent a wave? If so, what is its speed and direction of propagation? 2,2 2_2

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