Explain the formation of stationary waves in stretched strings and hence deduce the law of transverse waves in stretched strings.

When two progressive waves of the same amplitude and frequency travel in a bounded medium with the same speed but in opposite direction, then their superposition results in a stationary or standing wave. The standing waves can be produced by fixing a string under tension by its two ends, (one end may be left free). Plucking it in the middle will give a standing wave pattern. The two fixed ends will have no amplitude and are called the nodes, while the middle point will have maximum amplitude and is called the antinode.The numbers of these points will depend upon the modes of vibration as follows: Different modes of Vibration in stretched string:- a) First mode of vibration:- In this mode of vibration, the string vibration in one segment. This there are two nodes at fixed ends and an anti node in between them. This is the fundamental frequency of vibration. Second mode of vibration: In this mode of vibration the string vibration in two segments. Thus, there are three nodes and two anti nodes between of string This is the frequency of first overtone and second harmonics.The frequency is double of the fundamental frequency. Third mode of vibration: In this mode there will be four nodes and three anti nodes. This is the frequency of second overtone or third harmonics. The frequency is three times of the fundamental frequency. Law of Transverse Waves in stretched string: When a string of length L is fixed at both ends, the condition of maximum and minimum amplitude of a standing wave is given by x = n + 1 2 λ 2 for maxima and x = nλ 2 for minima In First mode of vibration, Let λ 1 be the wavelength of standing wave Then we have L = λ 1 2 or λ 1 = 2 L The frequency is ν 1 = v λ 1 where v = velocity of the wave. Since v = 1 2 L T m , where T = tension of the string and m = mass per unit length of the string. Therefore, the frequency of the first mode of vibration is ν 1 = 1 2 L T m Therefore, law of Transverse Waves in stretched string using the above formula are: (1) Law of length ν ∝ 1 L i.e the frequency is inversely proportional to its length. (2) Law of tension ν ∝ T i.e the frequency is proportional to its tension. (3) Las of mass ν = 1 m frequency is inversely proportional to the square root of the mass per unit length of the string.