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 bepowered.

Unique property of wave motion is:Waves can pass through each other or in other wordsthey can be superposed. When they overlap they addalgebraically.

Traveling wave is a moving disturbance. Can transferenergy and momentum from one place to another.

 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.

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

Two pulses on a stretchedrope passing througheach other.

Here the amplitudes addas they are in the same upward direction.

In this example the twowave amplitudes subtractfrom each other when the pulses are superposed

Standing Waves are Characteristic Oscillationsof the vibrating system.

Example: A violin string under tension and fixedat both ends. The systems parameters are:

Length of the string - meters, mass per unit length of the string- kg/m, andTension in the string - Newtons

Superposition of WavesCharacteristic Oscillations or standing waves

Two pulses inopposite directions

Two wave trains in the same direction.

Creation of standingwaves.

As every point on a wave is moving with the same velocity, the location of a point on the wave isa 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.

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

For stationary waves the wave amplitude splits into twoparts, 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 oppositedirections with same speed and frequency

y1x ,t=A sin kx− ty2x ,t=A sin kx t

y x ,t=y1x ,ty2x ,t=2A sin kx cos t

Which is moving to the right and which to the left ?

Phenomena of Beats: Two waves travelling to the right with slightly different frequencies with same speed

y1x ,t=A cosk1 x−1 t

y2x ,t=A cosk2 x−2 t

y x ,t=y1x ,ty2x ,t

y x ,t=

y x ,t=2Acos [k1−k2

2x−

1−22

t ]cos [k1k2

2x−

122

t ]

v =

k

k

Other examples of superposition of waves:

Waves traveling in the same direction

Waves traveling in opposite directions

Same wavelength and frequency and amplitudesbut with a difference of phase (or Path difference)between them.

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.

Fundamental

First Harmonic

2nd Harmonic

No nodes on string

one node on string

two nodes on string

L = 1 m

Which harmonic is shown ?

In stationary oscillations all points of the systemoscillate with the same frequency – one of the characteristic frequencies. Points which do notmove are nodes. Points which have maximum motion are anti-nodes. Fixed end points are nodesbut are not used in defining harmonics.

Any system which is confined and in equilibrium if itis oscillated, it can only oscillate at certain characteristicfrequencies ( and therefore wavelengths) which aredetermined by the boundary conditions.

This phenomena is called Quantization. In a characteristicstate all elements of the system vibrate with the same frequency. Some locations are always at rest and somelocations perform oscillations with maximum amplitude.The former are called nodes and the latter anti-nodes.

Waves on a string tied at both ends and under Tension

● The fundamental frequency corresponds to n = 1– It is the lowest frequency, ƒ1

● The frequencies of the remaining natural modes are integer multiples of the fundamental frequency– ƒn = nƒ1

● Frequencies of normal modes that exhibit this relationship form a harmonic series

● The normal modes are called harmonics

Standing waves in pipes open at one end – organ pipes

– The higher harmonics are ƒn = nƒ1 = n (v/2L) where n = 1, 2, 3, …

Standing waves in closed tubes – at one end

– The frequencies are ƒn = nƒ1= n (v/4L) where n = 1, 3, 5, …

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

Resonance in Air Columns, Example

water

can be calcuatedknowing the lengthof air column andposition of nodes.

frequency f = wave speed/ wave length

Standing Waves in Membranes

Note higher characteristicifrequencies are not multiplesof the fundamental. No musicaltone is developed.

Quality of Tone: Determined by the admixture ofhigher harmonics – their relative amplitudes

Strong 5th harmonic

Only fundamentalexcited.

Strong second harmonic