C.2 Wave model：練習問題

Which of the following properties is true for both longitudinal and transverse waves?

A point source of sound waves radiates energy uniformly in all directions. At a distance \(d\) from the source, the intensity of the sound is \(I\). If the power output of the source is increased by a factor of 3 and the distance from the source is increased to \(2d\), what is the new intensity of the sound in terms of \(I\)?

A longitudinal wave travels through a medium. A graph shows the variation with distance \(x\) of the displacement \(s\) of the particles in the medium at a particular time \(t\). At a specific point \(P\), the displacement is zero and the gradient \(\frac{ds}{dx}\) is at its maximum negative value. Which of the following correctly identifies what point \(P\) represents at that instant?

A periodic wave of frequency \( f \) and wavelength \( \lambda \) travels through a medium at speed \( v \). If the frequency of the wave source is increased to \( 3f \) while the medium remains the same, what is the new wavelength of the wave?

A ray of light is incident on the surface of a transparent medium at an angle of \(40^\circ\). The refracted ray makes an angle of \(30^\circ\) with the normal to the surface. What is the refractive index of the medium relative to air?

A wave on a string has a frequency of \(50 \text{ Hz}\). Calculate the time period of the oscillation in seconds.

Unpolarized light of intensity \(I_0\) is incident on a polarizer. The light then passes through an analyzer oriented at an angle of \(30^\circ\) relative to the transmission axis of the first polarizer. Calculate the ratio of the final transmitted intensity to the initial intensity \(I_0\).

A light wave travels from medium 1 to medium 2. The angle of incidence at the boundary is \(30^\circ\) and the angle of refraction is \(45^\circ\). If the speed of the wave in medium 1 is \(v\), determine the speed of the wave in medium 2 in terms of \(v\).

A transverse wave travels along a horizontal string. A diagram shows the variation with distance \(x\) of the displacement \(y\) of a part of the string at time \(t = 0\). The frequency of the wave is \(20 \text{ Hz}\). The horizontal distance between the first crest and the second trough is \(0.45 \text{ m}\).
(a) Determine the wavelength of the wave.
(b) Calculate the speed of the wave along the string.
(c) A point \(P\) on the string is at its maximum positive displacement at \(t = 0\). Calculate the displacement of point \(P\) at \(t = 0.075 \text{ s}\) in terms of its amplitude \(A\).
(d) Explain, with reference to the direction of energy transfer, why this wave is classified as a transverse wave.

In a Young's double-slit experiment, monochromatic light of wavelength \(\lambda\) passes through two slits separated by distance \(d\) to form an interference pattern on a screen at distance \(D\). A very thin transparent film of refractive index \(n\) and thickness \(t\) is placed over one of the slits.
(a) Explain why the central maximum of the interference pattern shifts.
(b) Derive an expression for the additional path length introduced by the film.
(c) If the film causes the central maximum to shift to the position previously occupied by the second-order bright fringe (\(m=2\)), calculate the thickness \(t\) in terms of \(\lambda\) and \(n\).