A horizontal string is stretched between two fixed points X and Y. The string is made to vibrate vertically so that a stationary wave is formed.

Question 20

(a) State the conditions required for the formation of a stationary wave. [2]

(b) A horizontal string is stretched between two fixed points X and Y. The string is made to vibrate vertically so that a stationary wave is formed. At one instant, each particle of the string is at its maximum displacement, as shown in Fig. 4.1.

Fig. 4.1

P and Q are two particles of the string. The string vibrates with a frequency of 40 Hz. Distance XY is 2.0 m.

(i) State the number of antinodes in the stationary wave. [1]

(ii) Determine the minimum time taken for the particle P to travel from its lowest point to its highest point. [2]

(iii) State the phase difference, with its unit, between the vibrations of particle P and of

particle Q. [1]

(iv) Determine the speed of a progressive wave along the string. [2]

[Total: 8]

Reference: Past Exam Paper – November 2017 Paper 22 Q4

Solution:

(a)

Stationary waves are formed when (two) waves travelling (at same speed) but in opposite directions overlap.

The waves (should be of the same type and) have the same frequency/wavelength.

(b)

(i)

{Antinodes are points of max displacement (e.g. P)}

5

(ii)

{This is half a period.}

Period T = 1 / 40 (= 2.5 × 10^-2) s

time taken = 2.5 × 10^-2 / 2 = 1.3 × 10^-2 s (1.25 × 10^-2 s)

(iii)

{For a stationary wave, points within a loop have the same phase while points in adjacent loops are out of phase by 180°.}

180°

(iv)

v = fλ  

{From the diagram, 2.5 waves corresponds to 20 cm.}

λ = 2.0 / 2.5 (= 0.80 m)

v = 0.80 × 40 = 32 m s^-1