波的疊加原理與駐波的形成：練習題

In a stationary wave formed on a stretched string, what is the phase difference between the oscillations of any two particles located between two adjacent nodes?

A wire of length \(L\), mass \(M\), and under tension \(T\) is used to produce a stationary wave. Which of the following expressions correctly gives the fundamental frequency of vibration?

A microwave transmitter and a vertical metal reflector are placed 60 cm apart to create a stationary wave pattern. A detector moved along the line between them identifies several points of minimum intensity. If the distance between the first and the fourth minimum is 4.5 cm, what is the frequency of the microwaves? (Speed of light \(c = 3.0 \times 10^8\text{ m s}^{-1}\))

A string of length 1.5 m vibrates in its third harmonic with a frequency of 300 Hz. What is the speed of the progressive waves that superpose to form this stationary wave?

A string is fixed at both ends and its fundamental frequency is measured as \(f\). If the tension in the string is kept constant but the length of the string is reduced to half of its original value, what will be the new fundamental frequency \(f'\)?

Distinguish between a node and an antinode in a stationary wave.

In a stationary wave experiment using microwaves and a reflector, 10 successive nodes are detected over a distance of 14.4 cm. Using the speed of light \(c = 3.0 \times 10^8 \text{ m s}^{-1}\), calculate the frequency of the microwaves.

State the principle of superposition of waves.

a) Describe how a stationary sound wave is formed in an open-ended pipe.

b) A 0.80 m long open pipe produces a fundamental frequency of 210 Hz. Calculate the speed of sound in the air inside the pipe.

Two identical progressive waves, \(y_1 = A \sin(kx - \omega t)\) and \(y_2 = A \sin(kx + \omega t)\), travel in opposite directions along a string.

a) Explain the principle of superposition and how it leads to the formation of a stationary wave from these two progressive waves.

b) By considering the superposition of these two waves at times \(t=0\), \(t=T/8\), and \(t=T/4\) (where \(T\) is the period), describe and sketch the resultant shape of the string, indicating the positions of nodes and antinodes.