Page 1 Important Formulae 1. Stationary Waves (i) Stationary waves are formed by the superposition of two identical waves travelling in opposite directions. (ii) Formation of stationary waves is really the interference of two waves in which coherent (same frequency) sources are required. (iii) By the word 'Identical waves' we mean that they must have same value of v, ? and k. Amplitudes may be different, but same amplitudes are preferred. (iv) In stationary waves all particles oscillate with same value of ? but amplitudes varying from A 1 + A 2 to A 1  A 2 . Points where amplitude is maximum (or A 1 + A 2 ) are called antinodes (or points of constructive interference) and points where amplitude is minimum (or A 1  A 2 ) are called nodes (or points of destructive interference). (v) If A 1 = A 2 = A then amplitude at antinode is 2A and at node is zero. In this case points at node do not oscillate. (vi) Points at antinodes have maximum energy of oscillation and points at nodes have minimum energy of oscillation (zero when A 1 = A 2 ). (vii) Points lying between two successive nodes are in same phase. They are out of phase with the points lying between two neighbouring successive nodes. (viii) Equation of stationary wave is of type, y = 2A sin kx cos ?t ...(i) or y = Acos kx sin ?t etc. This equation can also be written as, y = A x sin ?t or y = A x cos ?t If x = 0 is a node then, A x = A 0 sin kx If x = 0 is an antinode then, A x = A 0 COS kx Here, A 0 is maximum amplitude at antinode. (ix) Energy of oscillation in a given volume can be obtained either by adding energies due to two individual waves travelling in opposite directions or by integration. Because in standing wave amplitude and therefore energy of oscillation varies point to point. 2. Oscillations of Stretched Wire or Organ Pipes (i) Stretched wire Fundamental tone or first harmonic (n = 1) Page 2 Important Formulae 1. Stationary Waves (i) Stationary waves are formed by the superposition of two identical waves travelling in opposite directions. (ii) Formation of stationary waves is really the interference of two waves in which coherent (same frequency) sources are required. (iii) By the word 'Identical waves' we mean that they must have same value of v, ? and k. Amplitudes may be different, but same amplitudes are preferred. (iv) In stationary waves all particles oscillate with same value of ? but amplitudes varying from A 1 + A 2 to A 1  A 2 . Points where amplitude is maximum (or A 1 + A 2 ) are called antinodes (or points of constructive interference) and points where amplitude is minimum (or A 1  A 2 ) are called nodes (or points of destructive interference). (v) If A 1 = A 2 = A then amplitude at antinode is 2A and at node is zero. In this case points at node do not oscillate. (vi) Points at antinodes have maximum energy of oscillation and points at nodes have minimum energy of oscillation (zero when A 1 = A 2 ). (vii) Points lying between two successive nodes are in same phase. They are out of phase with the points lying between two neighbouring successive nodes. (viii) Equation of stationary wave is of type, y = 2A sin kx cos ?t ...(i) or y = Acos kx sin ?t etc. This equation can also be written as, y = A x sin ?t or y = A x cos ?t If x = 0 is a node then, A x = A 0 sin kx If x = 0 is an antinode then, A x = A 0 COS kx Here, A 0 is maximum amplitude at antinode. (ix) Energy of oscillation in a given volume can be obtained either by adding energies due to two individual waves travelling in opposite directions or by integration. Because in standing wave amplitude and therefore energy of oscillation varies point to point. 2. Oscillations of Stretched Wire or Organ Pipes (i) Stretched wire Fundamental tone or first harmonic (n = 1) (a) First overtone or second harmonic (n = 2) (b) Second overtone or third harmonic (n = 3) (c) v fn 2l ?? ? ?? ?? . Here, n = 1, 2, 3,…….. Even and odd both harmonics are obtained. Here, T v ? ? or T S ?Read More
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