Superposition：練習問題

A beam of water waves in a ripple tank is incident on a narrow gap in a barrier. Which change to the system will result in the waves spreading out over a larger angle after passing through the gap?

A diffraction grating is illuminated with light consisting of two wavelengths, \(\lambda_1\) and \(\lambda_2\). It is observed that the third-order maximum for \(\lambda_1\) occurs at the same angle as the fourth-order maximum for \(\lambda_2\). What is the ratio \(\lambda_1 / \lambda_2\)?

A stationary wave is formed in a tube of length \(L\) that is closed at one end and open at the other. If the fundamental frequency of the air column is \(f\), what is the frequency of the first overtone (the second possible mode of vibration)?

A wire of length \(L\) is stretched between two fixed supports and vibrates in its fundamental mode with a frequency \(f\). If the tension in the wire is increased by a factor of 4 while the length remains constant, what will be the frequency of the second harmonic of the wire?

A string of length \(L\) is fixed at both ends and vibrates in its second harmonic. A point \(P\) on the string is at a distance of \(\frac{L}{6}\) from one of the fixed ends. What is the ratio of the amplitude of vibration at \(P\) to the maximum amplitude of vibration on the string?

Explain, with reference to the principle of superposition, why energy is not transferred by a stationary wave in the direction of the component progressive waves.

In a double-slit experiment, the intensity of the light incident on one slit is reduced. Explain the effect this has on the visibility of the interference fringes, specifically regarding the intensity of the minima.

Define the term coherence as it applies to two sources of waves used to demonstrate a stable interference pattern.

(a) State the conditions necessary for two-source interference fringes to be observed on a screen.
(b) In a Young's double-slit experiment, monochromatic light of wavelength \( 6.5 \times 10^{-7} \text{ m} \) is incident on two slits. The distance between the slits is \( 0.20 \text{ mm} \). An interference pattern is observed on a screen placed \( 1.5 \text{ m} \) from the slits. Calculate the distance between the centres of adjacent bright fringes.

Two microwave transmitters, \( X \) and \( Y \), are placed \( 0.80 \text{ m} \) apart. They emit coherent microwaves of wavelength \( 2.5 \text{ cm} \) in phase. A receiver is moved along a line \( PQ \) which is parallel to the line \( XY \) and at a distance of \( 2.0 \text{ m} \) from it.
(a) Calculate the distance from the central maximum at \( M \) (the point on \( PQ \) directly opposite the center of \( XY \)) to the first-order interference maximum.
(b) A point \( Z \) on the line \( PQ \) is at a distance of \( 2.000 \text{ m} \) from transmitter \( X \) and \( 2.050 \text{ m} \) from transmitter \( Y \). Determine the type of interference occurring at \( Z \) and explain your reasoning by calculating the phase difference in radians.