The interference pattern between two identical sources (5, and 52), vibrating in phase, is a symmetric pattern of hyperbolic lines of destructive interference (nodal lines - dark fringes) and areas of constructive interference [antinodal lines - bright fringes). Practice 1. In the diagram given, label the lines of (C)onstructive and (Djestructive interference. 2. Indicate whether the points below are a supercrest, a supertrough or a node. HUG):- The two-point-source interference pattern is useful because it allows direct measurement of the wavelength while the interference pattern remains relatively stationary. By taking a closer look at this interference pattern we can develop some mathematical relationships that will be useful in analysing the interference of light. For any point P1 on the first nodal line, the difference in path length from P1 to SL and from P1 to 52 is half a wavelength (U210. For any point P2 on the second nodal line, the difference in path length is BIZA. Continuing this procedure we can arrive at a general relationship for any point P on the nth nodal line. where P,, 51 -* distance from source 1 to a point P on the n'" nodal line (m) P S P S = (n - 1)}, P,, 52 -* distance from source 2 to a point P on the nth nodal line (m) n 1 2 2 n -* nodal line number (n = 1, 2, 3, ...) A A -+ wavelength (m) 3 _ ' This relationship can be used in the ripple tank to nd the wavelengths of interfering waves. A wavelength can be calculated simply by locating a point on a specic nodal _ l line, measuring the path lengths, and substituting in the above equation. l Practice (be sure to include sketches wherer'when possible) 3. Two point sources generate identical waves that interfere in a ripple tank. The sources are located 5.0 cm apart, and the frequency of the waves is 8.0 Hz. A point on the first nodal line is located 10 cm from one source and 11 cm from the other. ,if ri' (a) What is the wavelength of the waves? {2.0 cm} ,Qh. 0,569.3, ff \"i (b) What is the speed of the waves? {15 cn-ys} r . 4. In a ripple tank experiment, a point on the third nodal line from the centre is 35 cm from one source and 42 cm from the other source. The sources are separated by 11.2 cm and vibrate in phase at 10.5 Hz. Calculate (a) the wavelength of the waves, and {2.8 cm} -" l ' (b) the velocity of the waves. {29 cmrs} r' ,l If the wavelengths are too small andfor the point P is too far away from the two sources, / ,-" the difference in path length will be too small to be measured accurately. Thus, another .: l technique is necessary. When PI, is very far away compared to the separation \"d" of the g\" I.\" two sources, the lines PI, 51 and Pn 52 are very nearly parallel. In this case, if; ll angle for the rim nodal line / fI" nodal Iine number (n = 1, 2, 3, ...) / rE wavelength (m) - a, distance between the sources (m) . 1 l where 3,, a = _ _ _ 5m " (n 2);: 111% n A d