Waves are generated by an oscillator which has to be powered.

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1 Traveling wave is a moving disturbance. Can transfer energy and momentum from one place to another. Oscillations occur simultaneously in space and time. Waves are characterized by 1. their velocity 2. their frequency (and hence wavelength) 3. their amplitude Waves are generated by an oscillator which has to be powered. Unique property of wave motion is: Waves can pass through each other or in other words they can be superposed. When they overlap they add algebraically.

2 Basic relation between frequency, f; wavelength' ; and velocity, v is v= f Note dimensions work out : v is m/s, f is 1/s and wavelength is m. Frequency is determined by the source generating the waves.

3 Superposition Principle If two or more traveling waves are moving through a medium, the resultant value of the wave function at any point is the algebraic sum of the values of the wave functions of the individual waves Waves that obey the superposition principle are linear waves For mechanical waves, linear waves have amplitudes much smaller than their wavelengths

4 Two pulses on a stretched rope passing through each other. Here the amplitudes add as they are in the same upward direction.

5 In this example the two wave amplitudes subtract from each other when the pulses are superposed

6 Standing Waves are Characteristic Oscillations of the vibrating system. Example: A violin string under tension and fixed at both ends. The systems parameters are: Length of the string - meters, mass per unit length of the string- kg/m, and Tension in the string - Newtons

7 Superposition of Waves Characteristic Oscillations or standing waves Two pulses in opposite directions Two wave trains in the same direction. Creation of standing waves.

8 As every point on a wave is moving with the same velocity, the location of a point on the wave is a function of time given by x(t) = x(t_0) +v(t) or every point satisfies the relation x-vt = constant So a traveling wave must have a functional form which entangles x and t in a specific way: y(x,t) = f(x-vt) for waves moving to the right.

9 Waves can pass through each other. When they overlap they add algebraically this is called superposition. If, when waves add they produce a pattern that is stationary as contrasted to travelling you obtain characteristic oscillations of the system in which waves are being superposed. For traveling or progressive waves the wave amplitude is y(x,t) = f(x-vt) = A sin kx t

10 For stationary waves the wave amplitude splits into two parts, one which depends on position only and the other on time only, see below. y x,t =A f kx g t Concrete example of two waves traveling in opposite directions with same speed and frequency y 1 x,t =A sin kx t y 2 x,t =A sin kx t y x,t =y 1 x,t y 2 x,t =2 A sin kx cos t Which is moving to the right and which to the left?

11 Phenomena of Beats: Two waves travelling to the right with slightly different frequencies with same speed y 1 x,t =A cos k 1 x 1 t y x,t = k y 2 x,t =A cos k 2 x 2 t y x,t =y 1 x,t y 2 x,t v = k y x,t =2 A cos[ k 1 k 2 2 x t]cos[ k 1 k 2 2 x t]

12 Other examples of superposition of waves: Waves traveling in the same direction Waves traveling in opposite directions Same wavelength and frequency and amplitudes but with a difference of phase (or Path difference) between them.

13 Stationary Vibrations of a String tied at both ends End points are fixed only waves which have zero amplitude at the ends can be sustained Superposition of a right going and left going wave occurs due to reflections at the fixed ends.

14 Fundamental No nodes on string First Harmonic one node on string 2 nd Harmonic two nodes on string

15 Which harmonic is shown? L = 1 m

16 In stationary oscillations all points of the system oscillate with the same frequency one of the characteristic frequencies. Points which do not move are nodes. Points which have maximum motion are anti-nodes. Fixed end points are nodes but are not used in defining harmonics. Any system which is confined and in equilibrium if it is oscillated, it can only oscillate at certain characteristic frequencies ( and therefore wavelengths) which are determined by the boundary conditions. This phenomena is called Quantization. In a characteristic state all elements of the system vibrate with the same frequency. Some locations are always at rest and some locations perform oscillations with maximum amplitude. The former are called nodes and the latter anti-nodes.

17 Waves on a string tied at both ends and under Tension The fundamental frequency corresponds to n = 1 It is the lowest frequency, ƒ 1 The frequencies of the remaining natural modes are integer multiples of the fundamental frequency ƒ n = nƒ 1 Frequencies of normal modes that exhibit this relationship form a harmonic series The normal modes are called harmonics

18 Standing waves in pipes open at one end organ pipes The higher harmonics are ƒ n = nƒ 1 = n (v/2l) where n = 1, 2, 3,

19 Standing waves in closed tubes at one end The frequencies are ƒ n = nƒ1= n (v/4l) where n = 1, 3, 5,

20 Standing Waves in Air Columns, Summary In a pipe open at both ends, the natural frequencies of oscillation form a harmonic series that includes all integral multiples of the fundamental frequency In a pipe closed at one end, the natural frequencies of oscillations form a harmonic series that includes only odd integral multiples of the fundamental frequency

21 Resonance in Air Columns, Example water can be calcuated knowing the length of air column and position of nodes. frequency f = wave speed/ wave length

22 Standing Waves in Membranes Note higher characteristici frequencies are not multiples of the fundamental. No musical tone is developed.

23 Quality of Tone: Determined by the admixture of higher harmonics their relative amplitudes Strong 5 th harmonic

24 Strong second harmonic Only fundamental excited.