Problems

Problem 5.1 A wave pulse on a string is described by $y(x,t)=\dfrac{3.0\,\mathrm{m}}{1+[(x-(2.0\,\mathrm{m/s})t-5.0\,\mathrm{m})/(1.0\,\mathrm{m})]^2}$, where $x$ is in meters and $t$ is in seconds. (a) In which direction is the wave traveling? (b) What is the speed of the wave? (c) Sketch the wave pulse at $t = 0$ and $t = 1.0$ s. (d) Verify that this function satisfies the one-dimensional wave equation.

Problem 5.2 Determine which of the following functions represent traveling waves, and for those that do, find the wave speed and direction: (a) $y_1 = 5\sin(3x - 2t)$ (b) $y_2 = 4\cos^2(x + 4t)$ (c) $y_3 = 6e^{-(x-8t)^2}$ (d) $y_4 = 3x^2 - 2t^2$

Problem 5.3 A harmonic wave is described by $y(x,t) = (0.25~\text{m})\sin[(4.0~\text{m}^{-1})x + (30~\text{s}^{-1})t + \pi/3]$. (a) What are the amplitude, wavelength, period, and frequency of this wave? (b) What is the phase velocity and in which direction does the wave travel? (c) What is the displacement at $x = 2.0$ m and $t = 0.5$ s? (d) At what time does the wave first reach its maximum positive displacement at $x = 0$?

Problem 5.4 Two harmonic waves travel in opposite directions along a string: $y_1 = (0.15~\text{m})\sin(2x - 10t)$ and $y_2 = (0.10~\text{m})\sin(2x + 10t)$ where $x$ is in meters and $t$ is in seconds. (a) What are the wavelength and frequency of each wave? (b) Find the resultant wave $y = y_1 + y_2$. (c) Do fixed nodes (points where $y=0$ at all times) exist when the two amplitudes differ? Explain using the resultant wave.

Problem 5.5 A wave generator produces harmonic waves with frequency 25 Hz. The waves travel at 340 m/s. (a) What is the wavelength? (b) Write the wave equation if the amplitude is 0.02 m and the wave travels in the positive x-direction. (c) What is the phase difference between two points separated by 3.4 m along the direction of propagation?

Problem 5.6 Given the complex numbers $z_1 = 3 + 4i$ and $z_2 = 2e^{i\pi/4}$: (a) Express both numbers in polar form. (b) Calculate $z_1 + z_2$, $z_1 \cdot z_2$, and $z_1/z_2$. (c) Find $z_1^*$ and verify that $z_1 \cdot z_1^* = |z_1|^2$.

Problem 5.7 A harmonic wave is represented by the complex function $\tilde{y} = (2.0~\text{m})e^{i(5x - 20t + \pi/6)}$. (a) What are the real and imaginary parts of this function? (b) If the real part represents the physical displacement, what are the amplitude, wavelength, frequency, and initial phase? (c) Verify that this complex function satisfies the wave equation.

Problem 5.8 Express the following trigonometric wave functions as complex exponentials: (a) $y = 3\cos(kx - \omega t)$ (b) $y = 5\sin(kx - \omega t + \pi/4)$ (c) $y = 2\cos(kx - \omega t) + 2\sin(kx - \omega t)$

Problem 5.9 Two waves are described by $\tilde{y}_1 = A_1 e^{i(kx - \omega t)}$ and $\tilde{y}_2 = A_2 e^{i(kx - \omega t + \phi)}$. (a) Find the resultant wave $\tilde{y} = \tilde{y}_1 + \tilde{y}_2$. (b) What is the amplitude of the resultant wave in terms of $A_1$, $A_2$, and $\phi$? (c) For what values of $\phi$ do the waves interfere constructively and destructively?

Problem 5.10 Show that if $\tilde{y} = Ae^{i(kx - \omega t)}$ satisfies the wave equation $\frac{\partial^2 \tilde{y}}{\partial x^2} = \frac{1}{v^2}\frac{\partial^2 \tilde{y}}{\partial t^2}$, then $v = \omega/k$.

Problem 5.11 A plane wave propagates in a direction that makes angles of 30° with the x-axis, 60° with the y-axis, and 90° with the z-axis. (a) Write the unit vector $\hat{k}$ in the direction of propagation. (b) If the wavelength is 2.0 m, write the propagation vector $\vec{k}$. (c) Write the equation for a plane wave with amplitude 5.0 units and frequency 150 Hz.

Problem 5.12 A spherical wave emanates from a point source at the origin. At a distance of 10 m from the source, the amplitude is 0.5 units. Assume zero initial phase. (a) What is the amplitude at distances of 5 m and 20 m? (b) If the frequency is 1000 Hz and the wave speed is 340 m/s, write the complete wave equation. (c) At what distance from the source is the amplitude reduced to 10% of its value at 1 m?

Problem 5.13 Compare the distance dependence of intensity for waves of the same frequency in the same lossless medium. Normalize each to intensity $I_1$ at a reference distance of 1 m from its source (or reference plane): (a) a plane wave; (b) a spherical wave at distance $r>0$; (c) a cylindrical wave in its far field at distance $\rho>0$.

Problem 5.14 A Gaussian beam has a beam waist of $w_0 = 1.0$ mm at $z = 0$ and a wavelength $\lambda = 633$ nm. (a) Calculate the Rayleigh range $z_R$. (b) Find the beam radius at distances $z = z_R$, $z = 2z_R$, and $z = 5z_R$. (c) What is the far-field divergence angle?

Problem 5.15 A cylindrical wave has the form $\psi = \frac{A}{\sqrt{\rho}}e^{i(k\rho - \omega t)}$. (a) Show that this approximately satisfies the wave equation for large $\rho$. (b) How does the intensity vary with distance from the line source? (c) Compare this with the intensity variation for plane and spherical waves.

Problem 5.16 An electromagnetic wave in vacuum has an electric field amplitude of $E_0 = 500$ V/m. (a) What is the magnetic field amplitude? (b) Calculate the time-averaged irradiance (power per unit area). (c) What are the maximum instantaneous and time-averaged total electromagnetic energy densities? (d) If the same irradiance is uniform over a circular spot of diameter 2.0 mm, what is the power crossing that spot?

Problem 5.17 A radio wave with frequency 100 MHz propagates in the +z direction. The electric field oscillates in the x-direction with amplitude $E_0 = 0.01$ V/m. (a) Write the complete expressions for $\vec{E}(z,t)$ and $\vec{B}(z,t)$. (b) Calculate the wavelength and wave number. (c) Find the Poynting vector and its time average. (d) What is the radiation pressure if this wave is completely absorbed by a surface?

Problem 5.18 A linearly polarized electromagnetic wave can be written as $\vec{E} = E_0[\cos\theta\hat{x} + \sin\theta\hat{y}]\sin(kz - \omega t)$. (a) What angle does the polarization direction make with the x-axis? (b) Find the corresponding magnetic field. (c) Show that the time-averaged irradiance is independent of $\theta$.

Problem 5.19 Circularly polarized light is described by $\vec{E} = E_0[\hat{x}\cos(kz - \omega t) + \hat{y}\sin(kz - \omega t)]$. (a) Show that $|\vec{E}|$ is constant. (b) Describe the motion of the electric field vector. (c) Find the magnetic field vector. (d) Is this right-hand or left-hand circular polarization?

Problem 5.20 Unpolarized light with irradiance $I_0$ passes through two polarizing filters. The first filter has its transmission axis at 30° to the vertical, and the second at 75° to the vertical. (a) What is the irradiance after the first filter? (b) What is the irradiance after the second filter? (c) What percentage of the original light is transmitted?

Problem 5.21 A police radar gun operates at 24.1 GHz. A car approaches the radar gun at 25 m/s. Account for the Doppler shift on both the outgoing and reflected paths. (a) What is the frequency of the radar waves reflected from the car? (b) What is the beat frequency between the transmitted and received signals? (c) How would the result change if the car were moving away from the radar gun?

Problem 5.22 The hydrogen Balmer-α line has a rest wavelength of 656.3 nm. In the spectrum of a distant quasar, this line is observed at 890.5 nm. Treat a special-relativistic Doppler speed as an equivalent speed for comparison; the observed redshift of a distant quasar is generally cosmological. (a) Calculate the redshift $z = (\lambda' - \lambda)/\lambda$. (b) Find the Doppler-equivalent recession speed using the relativistic formula. (c) Compare with the result using the non-relativistic approximation.

Problem 5.23 In a circular, edge-on double-star orbit, Star A moves at 50 km/s about the common center of mass. Neglect the system’s center-of-mass velocity relative to Earth. (a) What is the maximum Doppler shift of spectral lines from Star A? (b) If a spectral line has a rest wavelength of 500 nm, what are the maximum and minimum observed wavelengths? (c) Sketch how the wavelength varies over one orbital period.

Problem 5.24 For irradiance calculations, model a linearly polarized laser beam by the simplified field $E(x,y,z,t)=E_0\frac{w_0}{w(z)}\exp[-(x^2+y^2)/w^2(z)]\sin(kz-\omega t)$, where $w(z)=w_0\sqrt{1+(z/z_R)^2}$ and $z_R=\pi w_0^2/\lambda$. This model omits the wavefront-curvature and Gouy phases of a full Gaussian beam. (a) Find the axial field amplitude as a function of $z$. Explain why the given expression is not a plane wave on the axis. (b) In the paraxial approximation, integrate the time-averaged irradiance over the transverse plane to calculate the total power carried by the beam in terms of $E_0$ and $w_0$. (c) Find the beam divergence angle in the far field ($z >> z_R$).

Problem 5.25 Consider the interference of two plane waves with the same frequency but different propagation directions: $\vec{k}_1 = k\hat{z}$ and $\vec{k}_2 = k(\sin\theta\hat{x} + \cos\theta\hat{z})$. (a) Write the total wave function $\psi = \psi_1 + \psi_2$. (b) For equal-amplitude, equally polarized waves, show that their interference produces stationary spatial fringes. Explain why two non-opposite propagation directions do not form a one-dimensional standing wave. (c) At fixed $z$, find the spacing between adjacent dark fringes in the $x$-direction. (d) This configuration is used in holographic recording. Explain why.