Waves Concept


Interference of waves:-
-From principle of superposition we know that overlaping waves algbrically add togather to produce a net wave without altering the way of each other or the individual waves.
-If two sinusoidal waves of the same amplitude and wavelength travell in the same direction they interfere to produce a resultant sinusoidal wave travelling in that direction.
-The resultant wave due to interference of two sinusoidal waves is given by the relation
y′(x,t)=[2Amcos(υ/2)]sin(ωt-kx+υ/2)where υ is the phase difference between two waves.
-If υ=0n then there would be no phase difference between the travelling waves and the interference would be fully constructive.
-If υ=π then waves would be out of phase and there interference would be distructive.

Reflection of waves:-
-When a apulse or travelling wave encounters any boundary it gets reflected.
-If the boundary is not completely rigid then then a part of wave gets reflected and rest of it's part gets transmitted or refracted.
-A travelling wave at a rigid boundary is reflected with a phase reversal but the reflection at open boundary takes place without phase change.
-if an incident wave is represented by
yi(x,t)=A sin(ωt-kx)then reflected wave at rigid boundary is
yr(x,t)=A sin(ωt+kx+π)
and for reflections at open boundary reflected wave is given by
Standing waves:-
-The interference of two identical waves moving in opposite directions produces standing waves.
-For a string with fixed ends standing wave is given by
y(x,t)=[2Acos(kx)]sin(ωt)above equation does not represent travelling wave since it does not have characterstic form involving (ωt-kx) or (ωt+kx) in the argument of trignometric function.
-In standing waves amplitude of waves is different at different points i.e., at nodes amplitude is zero and at antinodes amplitude is maximumwhich is equal to sum of amplitudes of constituting waves.
-At intermediate points amplitude of wave varies between these two limits of maxima and minima

Normal modes of stretched string:--Frequency of transverse motion of stretched string of length L fixed at both the ends is given by
where n=1,2,3,4,.......
-The set of frequencies given by above relation are called normal modes of oscillation of the system.
-The mode with n=1 is called the fundamental mode with frequancy
f1=v/2L-Similarly second harmonic is the oscillation mode with n=2 and so on.
-Thus the string has infinite number of possible frequency of viberation which are harmonics of fundamental frequency f1 such that fn=nf1

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